Rational Numbers 3-1-A Explore: The Number Line Previously, you have graphed integers and positive fractions on a number line. Today, you will graph negative.
Download ReportTranscript Rational Numbers 3-1-A Explore: The Number Line Previously, you have graphed integers and positive fractions on a number line. Today, you will graph negative.
Slide 1
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 2
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 3
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 4
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 5
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 6
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 7
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 8
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 9
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 10
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 11
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 12
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 13
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 14
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 15
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 16
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 17
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 18
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 19
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 20
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 21
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 22
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 23
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 24
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 25
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 26
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 27
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 28
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 29
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 30
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 31
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 32
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 33
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 34
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 35
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 36
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 37
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 38
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 39
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 40
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 41
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 42
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 43
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 44
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 45
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 46
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 47
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 48
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 2
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 3
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 4
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 5
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 6
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 7
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 8
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 9
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 10
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 11
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 12
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 13
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 14
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 15
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 16
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 17
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 18
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 19
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 20
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 21
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 22
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 23
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 24
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 25
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 26
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 27
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 28
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 29
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 30
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 31
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 32
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 33
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 34
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 35
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 36
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 37
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 38
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 39
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 40
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 41
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 42
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 43
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 44
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 45
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 46
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 47
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Slide 48
Rational Numbers
3-1-A Explore: The Number Line
Previously, you have graphed integers and positive
fractions on a number line.
Today, you will graph negative fractions.
Letβs graph β
1.
π
π
on a number line
Draw a number line. Place a zero
on the right side an a -1 on the left.
Divide the line into fourths.
2.
Starting from the right, label the
line with -1/4, -2/4, and -3/4.
3.
Draw a dot on the number line on
the -3/4 mark.
-1
-ΒΎ
-Β½
-ΒΌ
0
Graph the pair of numbers on a number line.
Then identify which number is less.
β
5
1
πππ β 1
8
8
Remember the steps!
1. Draw a number line. Place a
zero on the right side an a -2 on
the left. Divide the line into the
appropriate parts.
2. Starting from the right, label the
line with the fractions.
3. Draw a dot on the number line
to mark the values.
Self-Assessment: Try pg. 127 # 1-8 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-B Terminating & Repeating Decimals
The table shows the winning speeds for a
10-year period at the Daytona 500.
1.
2.
3.
Year
Winner
Speed
(mph)
What fraction of the speeds are
between 130 and 145 miles per hour?
1999
J. Gordon
148.295
2000
D. Jarrett
155.669
Express this fraction using words and
then as a decimal.
2001
M. Waltrip
161.783
2002
W. Burton
142.971
2003
M. Waltrip
133.870
2004
D. Earnhardt
Jr.
156.345
2005
J. Gordon
135.173
2006
J. Johnson
142.667
2007
K. Harvick
149.335
2008
R. Newman
152.672
What fraction of the speeds are
between 145 and 165 miles per hour?
Express this fraction using words and
decimals.
Mental Math!
Converting Fractions to Decimals
Because our decimal system is based on powers of 10 such as 10, 100, and
1,000, we can use mental math to convert fractions to decimals.
If the denominator of a fraction is a power or multiple of ten, then you can
use place value to write the fraction as a decimal.
Look at the following example:
β’ 7/20
We can easily change this to have a denominator of 100 by
multiplying the numerator and denominator by 5.
β’ This would make the fraction 35/100 or 0.35.
Now you try!
β’
β’ 5ΒΎ
β’ Think: 75/100
β’ 3/25
β’ Think: 12/100
β’ -6 Β½
β’ Think: 50/100
so 5.75
so 0.12
so -6.5
Fractions to Decimals: Division
Think:
The βtopβ
goes
Any fraction can be written as a decimal by dividing its number
numerator
by its denominator!
βin the house.β
3
8
Write as a decimal.
Write β
1
40
83
as a decimal.
40 1
You should get -0.025.
Remember to keep the negative sign!
You should get 0.375!
Now you try!
Convert the following fractions to decimals using long division.
7
1
9
β
2
7
8
8
You should get
-0.875
2.125 7.45
20
Not all fractions are TERMINATING DECIMALS. Remember,
a TERMINATING DECIMAL is a decimal with digits that end.
REPEATING DECIMALS have a pattern in their digit(s)
that repeats forever!
Consider 1/3.
When you divide 1 by 3, you get 0.3333...
Use BAR NOTATION to indicate a
that a number pattern repeats indefinitely.
A bar is written over only the digit(s) that repeat.
0.12121212121212 = 0. 12
11.38585858585 = 11.385
PRACTICE:
Write each as a decimal.
β’ β
β’
2
3
7
9
3
11
1
8
3
β’ β
β’
---------------------------------------------Use the table to find what fraction of
the fish in an aquarium are goldfish.
Write in simplest form.
Determine the fraction of the
aquarium made up by each fish.
Write the answer in simplest form!
a) molly
b) guppy
c) angelfish
Fish
Amount
Guppy
0.25
Angelfish
0.4
Goldfish
0.15
Molly
0.2
Self-Assessment: Try pg. 131 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-1-C Compare & Order Rational Numbers
The batting average of a softball player is found by comparing the
number of hits to the number of times at bat.
Melissa had 50 hits in 175 at bats.
Harmony had 42 hits in 160 at bats.
1. Write the two batting averages
as fractions.
2. Which girl had the better batting
average? Be ready to explain
how you found your answer.
3. Describe two different methods
you could use to compare the
batting averages.
RATIONAL NUMBERS:
numbers that can be expressed as a ratio of two integers expressed as
a fraction (in which the denominator is not zero). Includes common
fractions, terminating and repeating decimals, percents, and all
integers.
Rational Numbers
0.8
20%
2.2
Β½
Integers
-3
Whole
Numbers
2
1
-1
2/3
-1.44
Use <, >,or = to make the statement true:
5
1
β5 _______ β 5
9
9
You wonβt always be comparing rational numbers that have common
denominators. A COMMON DENOMINATOR is a common multiple
of the denominators of two or more fractions.
The LEAST COMMON DENOMINATOR or LCD is the LCM of the
denominators. The LCD is used to compare fractions!
Compare using <, >, or =:
7
8
______
12
18
What is the least
common
denominator?
What does that
make your
numerators?
21 16
>
36 36
so
7
8
>
12 18
In Mr. Reedβs math class, 20% of the students own Sperry shoes. In Mrs.
Croweβs math class, 5 out of 29 students own Sperry. In which math class
does a greater fraction of students own Sperry?
Express each number as a decimal and then compare.
20% = 0.2
5/29 = -.1724
Since 0.2 > 0.1724, 20% > 5/29
Therefore, a greater fraction of students in Mr. Reedβs class own Sperry
shoes.
Now you try!
In a second period class, 37.5% of students like to
bowl. In a fifth period class, 12 out of 29 students like
to bowl. In which class does a greater fraction of the
students like to bowl?
List the numbers 3.44, π, 3.14, 3. 4 in order from least to greatest.
Remember to line
up the decimal
points and
compare using
place value!
3.44
3.1415926β¦
3.14
3.4444444444
So the order would be: 3.14, π, 3.44, 3. 4.
Class average scores on the last four quizzes were:
4
3
0.82
83%
5
4
List the scores from least to greatest.
Self-Assessment: Try pg. 136 # 1-7 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Add & Subtract Positive Fractions
Sean surveyed ten classmates
to find out which type of tennis
shoe they like to wear!
1. What fraction liked cross
trainers?
2. What fraction liked high
tops?
3. What fraction liked either
cross trainers OR high tops?
Shoe
Type
Number
Cross
Trainer
5
Running
3
High Top
2
Fractions that have the same denominator are called LIKE FRACTIONS.
Fractions that do not have the same denominator
are called UNLIKE FRACTIONS.
You can use FRACTION TILES
as a model to help solve
problems that require addition
and subtraction of fractions.
With your βelbow partnerβ , complete Fraction Discovery #1.
In it, you will be asked to do three things:
1. Draw a model to represent the problem and use
that model to find a solution (no numbers allowed)
2. Draw a model to represent the problem and AT THE SAME TIME, write
an expression using numbers. Find a solution using both methods.
3. Write a numerical expression only to solve the problem.
By 7th grade, you should already know fraction addition &
subtraction rules! But your CHALLENGE is to complete
some of the problems without those rulesο
Key Concepts Review
Add and Subtract Like Fractions
To add or subtract like fractions, add or subtract the
numerators and write the result over the denominator.
Numbers
Algebra
5
2
5+2
7
+
=
=
10 10
10
10
π
π
+ =
π
π
π+π
,
π
where π β 0
11 4
11 β 4
7
β
=
=
12 12
12
12
π
π
π
β
π
πβπ
,
π
where π β 0
=
Key Concepts Review
Add and Subtract Unlike Fractions
To add or subtract like fractions with different
denominators
β’ Rename the fractions using the least common
denominator (LCD)
β’ Add or subtract as with like fractions
β’ If necessary, simplify the sum or difference
3 1
15 4
19
+ β
+
=
4 5
20 20
20
Add & Subtract Negative Fractions
Can fractions be negative?
YES!
Although we may not think about it much,
you use negative fractions when you:
β’ Give part of something away
β’ Eat a part of something
β’ Lose part of something
β’ Pour out part of something
β’ Go part of the way backwards
β’ Go part of the way down
With your βelbow partnerβ, complete Fraction Discovery #2. Today,
you will need PINK fractions for NEGATIVE numbers and
YELLOW fractions POSITIVE.
Use what you already know about INTEGER RULES and FRACTION
OPERATIONS to help you!
Key Concepts Review
5 2
+ =
9 9
3
1
β + β
=
5
5
5+2 7
=
9
9
β3 + (β1) β4
4
=
=β
5
5
5
When you have unlike
denominators, first, find a
COMMON DENOMINATOR!
Then, you can just use the
INTEGER RULES to find the sum
or difference in the numerator!
When you have
like denominators,
keep the
denominator and
use your
INTEGER RULES
to find the sum or
difference in the
numerator!
1
2
β β
=
2
5
5
4
9
β β
=
10
10
10
Practice adding and subtracting with
fraction tiles.
a.
1
3
d.
3
β
8
e.
2
β
3
+
2
3
a.
3
3
β
β
b.
=1
1
8
1
β
3
3
β
7
1
7
+
b. β
d.
4
β
8
e.
1
β
3
c.
2
7
=
2
β
5
+
c. β
1
β
2
2
β
5
4
5
1.
5
7
β
8
8
8
9
2. β β
3. β
7
16
5
9
β β
4.
3
+
4
5.
5
4
+
6
6
2
1
8
4
13
4
β = β1
9
9
4
1
β =β
16
4
2
1
β =β
4
2
9
1
=1
6
2
1. β = β
5
β
4
2.
3
16
3.
4.
5.
Answers
Questions
Practice Without Tiles!
π
π
Chelsea saves of her allowance and
π
spends of her allowance at the mall.
π
What fraction of her allowance
remains?
Eliza was riding a bicycle on a bike
π
path. After riding of a mile, she
π
discovers that she still needed to
π
travel of a mile to reach the end of
π
the path. How long is the bike path?
a.
7
+
8
β
5
8
CHALLENGE!
1
5
1
β
b. + β +
8
6
3
7
12
Self-Assessment: Try pg. 148 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-2-D Add & Subtract Mixed Numbers
Baby
Adelaide
Stephen
Micah
Nora
Birth Weight
1. Write an expression to find how much
1
8
8
15
7
16
13
6
16
7
5
8
15
7
7 β5
16
8
more Stephen weighs than Nora.
2. Rename the fractions using the LCD.
15
14
7 β5
16
16
3. Find the difference of the fractional
parts and then the difference of the
whole numbers.
1
2
16
To add or subtract mixed numbers, first add or
subtract the fractions. If necessary, rename
them using the LCD. Then add or subtract the
whole numbers and simplify if necessary.
Add and write in simplest form.
For these problems, you can
add the whole numbers and the
fractions separately.
Subtract. Write in simplest form.
For these problems, you can
subtract the whole numbers
and the fractions separately.
4
2
7 + 10
9
9
5
1
8 β2
6
3
1
5
6 +2
8
8
3
1
7 β4
4
3
5
1
1 +4
9
6
4
1
9 β5
7
2
2
17
3
3
8
4
13
5
18
6
1
2
3
5
12
4
1
14
Many times, it is not possible
Strategy 1
to subtract the whole
1
2
numbers and fractions
2 β1
separately. In this case, two
3
3
different strategies could be
7 5
used:
β
3 3
1. Convert mixed numbers to
improper fractions
2
OR
2. βBorrowβ from the whole IMPROPER FRACTION:
3
number and add 1 to the Has a numerator that
fraction.
is greater than or
equal to the
denominator.
a.
1
8 β
5
3
3
5
Strategy 2
1
2
2 β1
3
3
4
2
1 β1
3
3
2
3
Now you try!
3
3
11
b. 8 β 3
c. 5 β 4
4
a. 4
3
5
b. 4
8
1
4
c.
11
24
d.
12
d. 8
4
5
2
11 β
5
3
2
5
Jenny is making necklaces and bracelets for a
1
class craft sale. She uses 25 feet of string for the
4
necklaces and
feet of string for the
bracelets. What is the total length of string that
Jenny uses?
5
8
The length of Brookeβs garden is 4 feet. Find
7
the width of Brookeβs garden if it is 2 feet
8
shorter than the length.
1
4
Jill wakes up at 6:00 A.M. It takes her 1 hours to shower,
Real World
Problems!
1
15
12
1
get dressed, and comb her hair. It takes her hour to
2
eat breakfast, brush her teeth, and make her bed. At
what time will she be ready for school?
Self-Assessment: Try pg. 154 # 1-9 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
Fraction Discovery #3
ο With a partner, complete Fraction Discover #3
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an answer WITHOUT
using rules you have learned in the past!
3-3-B Multiply Fractions
For the problem, create a sketch or model to solve.
Two-thirds of the students chose pizza
at lunch. One-half of those students
chose pepperoni pizza.
Represent these two situations with equations.
Are the equations the same or different?
1
Darryl had 12 apricots. He ate 6 of them for lunch.
How many apricots did he eat?
Darryl also had some pizzas, each cut into sixths. If he ate 12 pieces of the
pizza, how many (whole) pizzas did he eat?
Key Concepts
To multiply fractions, multiply the numerators and
multiply the denominators.
β3 2 β3 Γ 2 β6
6
Γ =
=
=β
5
7
5Γ7
35
35
If the numerator and denominator of either
fraction have common factors, you can simplify
before you multiply.
2 14
1 2
2
1 2 14 2
Γ
β Γ
β Γ β
17
7 10
10 5
1 5
5
When multiplying with mixed numbers,
you MUST change the mixed numbers to improper
fractions BEFORE you multiply.
A recipe to make one batch of blueberry muffins calls
2
for 4 cups of flour. How many cups of flour are needed to
3
make 3 batches of blueberry muffins?
2
3
One evening, of the students in Rickβs class
3
watch television. Of those, watched a
8
reality show. Of the students that watched
1
the show, of them recorded the show.
4
What fraction of the students in Rickβs class
watched and recorded a reality TV show?
Evaluate each verbal expression:
a) One-half of five-eighths
b) Four-sevenths of two-thirds
c) Nine-tenths of one-fourth
d) One-third of eleven-sixteenths
Fraction Discovery #4
ο With a partner, complete Fraction Discover #4
ο You will use rectangular models to find the
answer to fraction problems.
ο Your challenge is to find an
answer WITHOUT using rules
you have learned in the
past!
3-3-D Divide Fractions
KEY CONCEPT:
To divide a fraction, multiply by its
multiplicative inverse, or reciprocal.
Numbers
7
8
3
Γ·
4
7
8
= Γ
4
3
Algebra
π
π
π
π
π
π
π
π
Γ· = Γ where π, π, π β 0
PAY ATTENTION!
The divisor (or second fraction) is the ONLY fraction that is
flipped during this process.
DO NOT FLIP THE FIRST FRACTION.
Practice Dividing by Fractions
Show your work in your notes. Simplify when necessary.
3
1
Γ· β
4
2
1
β1
2
3 1
Γ·
4 4
3
4 8
β Γ·
5 9
9
β
10
5
2
β Γ· β
6
3
1
1
4
Evaluate each expression
1
1
for π = β , β = :
6
3π Γ· β
πβ
βΓ·π
2
Practice Dividing by Mixed Numbers
Try these in your notes!
2
3
Γ·
1
2
1
3
3
5Γ·
1
1
3
3
3
4
3
β
4
Γ·
1
β
2
1
1
2
1
2
3
Γ·5
7
15
Ms. Holloway
Mrs. Thompson
1
1
has 8 4 cups of
bought 4 2 gallons of
coffee. If she
ice cream to serve
divides the
at her birthday party.
coffee into
1
If a pint is 8 of a
3
cup servings,
4
gallon, how many
how many
pint-sized servings
servings will she
can
be
made?
Self-Assessment: Try pg. 170 # 1-10 on your own. have?
When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-A Multiply & Divide Monomials
1. Examine the exponents of
the powers in the last
column.
What do you observe?
For each increase on the Richter scale, an
earthquakeβs vibrations, or seismic waves, are 10
times greater! So, an earthquake of magnitude 4
has seismic waves that are 10 times greater than that
of a magnitude 3 earthquake.
2. Write a rule for determining
the exponent of the
product when you multiply
powers with the same
base.
Richter
Scale
Times Greater than
Magnitude 3 Earthquake
Written using
Powers
4
10 x 1 = 10
101
5
10 x 10 = 100
101 x 101 = 102
6
10 x 100 = 1,000
101 x 102 = 103
7
10 x 1,000 = 10, 000
101 x 103 = 104
8
10 x 10,000 = 100,000
101 x 104 = 105
REMEMBER:
Exponents are used to show repeated multiplication.
We can use the definition of an exponent to find a rule for
multiplying powers with the SAME BASE.
23 x 24=(2 x 2 x 2)x(2 x 2 x 2 x 2)=27
PRODUCT OF POWERS
Words:
To multiply powers with the same base,
add their exponents
Symbols:
am x an = am+n
Example:
32x 34 = 32+4= 36
Practice Multiplying Powers!
1. 73 x 71
2. 53 x 54
3. (0.5)2 x (0.5)9
4. 8 x 85
MONOMIAL
A number, variable, or
product of a number and
one or more variables.
Monomials can also be
multiplied using the rule for
the product of powers.
Common Mistake:
When multiplying
powers, do not multiply
(evaluate) the bases
that are the same!
Example:
33 x 35 β 98
33 x 35 = 38
1. x5 (x2)
2. (-4n3)(6n2)
3. -3m(-8m4)
4. 52x2y4 (53xy4)
If we get the PRODUCT OF POWERS using ADDITION, we
should get the QUOTIENT OF POWERS usingβ¦β¦
QUOTIENT OF POWERS
Words:
To divide powers with the same base,
subtract their exponents.
Symbols:
am Γ· an = am-n
Example:
34Γ· 32 = 34-2= 32
π¦5
π¦3
76
72
π¦2
74
Try these!
9π 7
9π 2
49
42
π₯8
π₯6
3π 5
47
π₯2
The table compares the
processing speeds of a specific
type of computer in 1999 and in
2008. Find how many times
faster the computer was in 2008
than in 1999.
Year
Processing
Speed
(instructions per
second)
1999
103
2008
109
The number of fish in a school of
fish is 43. If the number of fish in the
school increased by 42 times the
original number of fish, how many
fish are now in the school?
Evaluate the power.
Self-Assessment: Try pg. 179 # 1-10 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-B Negative Exponents
Take a look at the table:
1. Describe the pattern of the
powers in the first column.
Continue the pattern by
writing the next two values in
the table.
2. Describe the pattern of values
in the second column. Then
complete the second column.
3. Using what you observed in
the table, determine how 3-1
should be defined.
Power Value
26
64
25
32
24
23
22
16
8
4
21
20
2-1
β¬
2
β¬
β¬
β¬
β¬
β¬
KEY CONCEPT
Words:
Any nonzero number raised to the negative n
power is the multiplicative inverse of its nth power.
Symbols:
1
πβπ = π where π β 0
π
Example:
1
1
β4
5 = 4=
5
625
PRACTICE!
Write each expression using a positive exponent.
6-2
x-5
5-6
t-4
1
62
1
π₯5
1
56
1
π‘4
When given a
fraction with a
positive exponent
or square, you can
rewrite it using a
negative exponent.
1
1
= 2 = 3β2
9 3
1
1
= 3 = 2β3
8 2
PRACTICE!
Write each expression using a negative exponent other than -1.
1
π5
π β5
1
16
2β4 or 4β2
1
π‘6
π‘ β6
1
100
10β2
1
π9
πβ9
Perform Operations with Exponents
Method 1
Simplify. π₯ 3 π₯ β5
π₯ 3 π₯ β5 = π₯ 3+(β5) = π₯ β2
Method 2
π₯3
Simplify.
π5
π2
Method 1
π5
= π5β2 = π3
2
π
π₯ β5
1
1
=π₯βπ₯βπ₯β
=
π₯βπ₯βπ₯βπ₯βπ₯ π₯βπ₯
= π₯ β2
Method 2
π5 π β π β π β π β π
=
= π3
2
π
πβπ
Perform Operations with Exponents
Simplify.
π₯ β2
π₯ β1
π₯ β1 or
π β4
1
π₯
Γ
π3
π₯5
π β1 or
1
π
Nanometers are often used to
measure wavelengths.
1 nanometer= 0.000000001 meter.
Write the decimal as a power of 10.
10β3
π β5
π β8
π₯ β8
π₯ β3 or
1
π₯3
π3
10β9
A unit of measure called a micron
equals 0.001 millimeter. Write this
number using a negative exponent.
Self-Assessment: Try pg. 183 # 1-13 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?
3-4-C Scientific Notation
More than 425 million pounds of gold have been discovered
in the world. If all this gold were in one place, It would form
a cube seven stories on each side!
Write 425 million in standard form
1.
ο
425,000,000
Complete: 4.25 x β¬ = 425,000,000
2.
ο
100,000,000
When you deal with very large numbers like 425,000,000, it can be difficult to
keep track of the zeros! You can express numbers such as this in
SCIENTIFIC NOTATION
by writing the number as the product of a factor and a power of 10.
Words:
A number is expressed in scientific notation when it is written as the
of a factor and a power of 10.
The factor must be greater than or equal to 1 and less than 10.
Symbols:
a Γ 10π , where 1 β€ π < 10 and n is an integer
Example:
425,000,000 = 4.25 Γ 108
product
WATCH OUT!
Common mistake:
Make sure you are
counting decimal
places rather than
just adding zeroes.
Check it out!
You try!
Express the following numbers in
7.6 x 106
7,600,000
standard form:
(move the decimal point 6 places)
2.16 x 105
2.16 x 100,000
3.201 x 104
2.16 _ _ _
32,010
2.16 0 0 0
(move the decimal point 4 places)
216,000
(move the decimal point 5 places)
1.25 Γ 102 = 125
1.25 Γ 101 = 12.5
1.25 Γ 100 = 1.25
1.25 Γ 10β1 = 0.125
1.25 Γ 10β2 = 0.0125
1.25 Γ 10β3 = 0.00125
Practice: Express each number in standard form:
5.8 x 10-3
0.0058
(move the decimal 3 places left)
4.7 x 10-5
0.000047
9 x 10-4
0.0009
Practice: Express each number in
scientific notation.
1,457,000
1.457 x 106
0.00063
6.3 x 10-4
35,000
3.5 x 104
0.00722
7.22 x 10-3
The Atlantic Ocean has an area of
3.18 x 107 square miles. The Pacific
Ocean has an area of 6.4 x 107
square miles.
Which ocean has a greater area?
Since the exponents are the same
and 3.18 < 6.4, the Pacific
3
Ocean has a greater area. Earth has an average radius of 6.38 x 10
kilometers. Mercury has an average radius
of 2.44 x 103 kilometers. Which planet has
the greater average radius?
Compare using <, >, or =
4.13 x 10-2_____ 5.0 x 10-3
0.00701_____7.1 x 10-3
5.2 x 102_____ 5,000
Self-Assessment: Try pg. 187 # 1-12 on your own. When signaled, compare
your work with your partnerβs. Are there any differences?