Transcript 3 +
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Math 110 online work that is due today at the start of class:
Section 1.2 Homework (no worksheet
to turn in on this one)
The
Syllabus Quiz
can still be redone if you haven’t yet gotten 100%
Any questions on the Section 1.2 homework that was due today?
Please CLOSE YOUR LAPTOPS,
and turn off and put away your cell phones,
and get out your note taking materials.
Gateway Quiz Reminders:
• You will be taking an 8-question Gateway Quiz in class at the next
class session, without a calculator .
The Gateway Quiz will consist of 8 questions similar to the ones on fractions and the order of operations that were due at the beginning of the previous class.
• We will spend some time today on how to do these kinds of problems, and you will have another set of eight practice Gateway problems to do at the end of the homework assignment due at the next class session. • You also have a
Practice Gateway Quiz due
at the start of next class session, with a
REQUIRED paper worksheet
that must be turned in (
worksheet will be handed out now
– make sure you get one before you leave class.) You can try it up to 3 times, and only your best score will count in the final grade. Be sure to put the work on the worksheet for the Practice Quiz with the best score!
Section 1.3
Operations on Real Numbers
Working with positive and negative signs
A
number line
is a line on which each point is associated with a number.
– 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5
Negative numbers Positive numbers
To add two numbers, especially if one (or both) of them is negative, it often helps to picture them on a number line.
Example 1 : 3 + (-5)
•Locate the first number on the number line. • Starting from that number, if the second number is positive, move to the right by that many units. If it’s negative, move to the left that many units.
(-5)
3
(Count 5 units to the left from 3)
– 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5
Answer is: -2
Example 2: -2 + (-3)
•Locate the first number on the number line. • Start from that first number. The second number is negative, so move to the left by that many units.
(-3)
2 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5
Answer is: -5
Example from today’s homework:
Answer: -30
Subtracting Real Numbers
Subtracting real numbers
• Substitute the opposite of the number being subtracted • Add.
• a – b = a + (– b)
Example:
Subtract the following numbers.
( – 5) – 6 – ( – 3) =( – 5) + ( – 6) + 3 = – 8
Multiplying or Dividing Real Numbers
Multiplying or dividing two real numbers with same sign
• Result is a positive number
Multiplying or dividing two real numbers with different signs
• Result is a negative number
Multiplying or Dividing Real Numbers
Examples:
Find each of the following products.
4 · (–2) · 3 = –24 ( – 4) · ( – 5) = 20
Working with zero:
• If b is a real number, 0 · b = b · 0 = 0.
• The quotient of any real number and 0 is undefined.
a
0 • The quotient of 0 and any real number = 0.
0
a
0
a
Sample problem from today’s homework:
Answer: 0
If this problem had been
7/0 ,
the answer would be “
N
” (undefined)
Exponents
We may use
exponential notation
products in a more compact form.
to write Exponential notation for the product of five threes is 3 5 • •
Base
is 3
Exponent
is 5 • The notation means 3 • 3 • 3 • 3 • 3, or 243 Example Evaluate 4 3 .
4 3 = 4 • 4 • 4 = 64
Evaluating Exponential Expressions
Example:
Evaluate each of the following expressions.
3 4 = 3 · 3 · 3 · 3 = 81 ( –5) 2 = ( – 5)(–5) = 25 –6 2 = – (6)(6) = –36 (2 · 4) 3 = (2 · 4)(2 · 4)(2 · 4) = 8 · 8 · 8 = 512 3 · 4 2 = 3 · 4 · 4 = 48
The Order of Operations
Order of Operations
Simplify expressions using the order that follows. If grouping symbols such as parentheses are present, simplify expression within those first, starting with the innermost set. If fraction bars are present, simplify the numerator and denominator separately.
1.
Evaluate exponential expressions, roots, or absolute values in order from left to right.
2.
Multiply or divide in order from left to right.
3.
Add or subtract in order from left to right.
Order of Operations Memory Device:
“Please excuse my dear Aunt Sally”
1.
P lease
( P arentheses) 2.
E
xcuse ( E xponents) 3.
M y D ear
( M ultiply and D ivide) 4.
A unt S
ally ( A dd and S ubtract) … or just remember
PEMDAS
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Using the Order of Operations
Example:
Evaluate: 6 9 3 3 2 6 9 3 3 2 6 9 3 6 ( 9 ) ( 3 ) 9 9 9 1 Write 3 2 as 9. Divide 9 by 3.
Add 3 to 6.
Divide 9 by 9.
More examples Simplify the following expressions.
6 2 2 2 5 6 2 2 32 6 4 32 2 32 3 6 ( 8 5 ) 4 2 2 3 6 ( 3 ) 16 2 3 18 16 2 21 18 3 7 3 6 7 6 34
Example:
Simplify the following.
8 9 3 ( 2 ) 2 ( 3 ) 8 6 9 6 14 3 14 3
Sample problem from today’s homework: 324 7
Evaluating Algebraic Expressions
To
evaluate an algebraic expression
, substitute the numerical value for each variable into the expression and simplify the result using the order of operations.
Example:
Evaluate each expression for the given value.
(a)
5
x
– 2 for
x
= 8 5( 8 ) – 2 = 40 – 2 = 38
(b)
3
a
2 + 2
a
+ 4 for
a
= – 4 3( – 4 ) 2 + 2( – 4 )
+
4 = 3(16) + ( – 8)
+
4 = 44
NOTE
:
Many of today’s homework problems as well as
Gateway problems
7 & 8
all require using the
order of operations
.
Order of operations:
1) First, calculate expressions within grouping symbols (parentheses, brackets, braces,absolute values, fraction bars). If there are nested sets of grouping symbols, start with the innermost ones first and work your way out.
2) Exponential expressions – left to right 3) Multiplication and division – left to right 4) Addition and subtraction – left to right 24
Sample Gateway Problem # 7:
Strategy:
Calculate out the entire top expression and then the entire bottom expression, using the order of operations on each part. Then simplify the resulting fraction, if necessary.
TOP EXPRESSION:
2 4 – 4(7 + 2) Step 1: Parentheses : 2 4 – 4 (7 + 2) = 2 4 – 4 (9) Step 2: Exponents: 2 4 – 4(9) = 2•2•2•2 – 4(9) = 16
(because 2•2•2•2 = 4• 2 •2 = 8 •2 = 16 )
– 4(9) Step 3: Multiply/Divide: 16 – 4(9) = 16 – 4•9 = 16 – 36 Step 4: Add/Subtract: 16 – 36 = -20
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Now calculate the bottom expression: 2(6+2) + 4
Step 1: Parentheses : 2 (6+2) + 4 = 2 (8) + 4 Step 2: Exponents:
There aren’t any in this part.
Step 3: Multiply/Divide: 2(8) + 4 = 2•8 + 4 = 16 + 4 Step 4: Add/Subtract: 16 + 4 = 20
Now put the top over the bottom and simplify the resulting fraction: TOP = 2
4 – 4(7 + 2) = -20 = -1 = -1
BOTTOM 2(6+2) + 4
20 1
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Full Solution to Sample Problem #7:
Here is the work we expect to see on your worksheet:
2 4 – 4(7 + 2) = 2 4 – 4(9) = 16 – 4(9) = 16 – 36 = -20 = -1 = -1 2(6+2) + 4 2(8) + 4 16 + 4 20 20 1 27
Sample Gateway Problem # 8:
Strategy:
Deal with the expressions inside the grouping symbols (parentheses, brackets) first, starting with the innermost set (-3 + 6).
STEP 1: (inside the parentheses) 3[17 + 5 (-3 + 6) - 10] = 3[17 + 5 (3) - 10] STEP 2: (inside the brackets;
multiply first, then add and subtract)
3 [17 + 5(3) -10] = 3 [17 + 5•3 -10] = 3 [17 + 15 - 10] = 3 [ 17 + 15 - 10] = 3 [ 32 - 10] = 3 [ 22 ]
STEP 3: Do the final multiplication: 3
[ 22 ] = 3 • 22 = 66
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Full Solution to Sample Problem #8:
Here is the work we expect to see on your worksheet:
3[17 + 5(-3 + 6) - 10] = 3[17 + 5(3) - 10] = 3[17 + 15 - 10] = 3[32 - 10] = 3[22] = 66 29
Quick review of fractions:
• • • • • • Terminology: numerator, denominator Simplifying fractions Multiplying Dividing Adding: needs common denominator Subtracting: needs common denominator
ALWAYS simplify answer!!
(factor numerator and denominator into their prime factors, then cancel common factors, if any)
If you want more tips on
Gateway Problems 1-6
(and today’s last 8
homework problems
) click on the
PowerPoint Slides
item in today’s
Assignment
folder for
Section 1.3
, then click on the link for
Gateway Problems.
(click here to show link)
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Additional fraction problems to do if we have time: 32
Reminders:
• Today’s
homework assignment on section 1.3
is due at the
start
of our next class session.
• The
Practice Gateway Quiz
is also due at the start of the next class session, and has a
required paper worksheet
grade.
that must be turned in at that time. You can take this quiz up to three times, and only your best score will count in your final •
Links
to these online assignments are in the Section 1.3 Assignments folder .
• At the next class session you will be taking the
in-class Gateway Quiz
(no calculators allowed!)
You may now open your laptops to work on the homework for section 1.3 and/or the practice Gateway Quiz.
If you have finished the homework and practice quiz already, or if you get them both finished before the end of the class period, show your on-screen 100% score to the teacher or TA and you may then leave class early.
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REMEMBER
: Come to the
open lab
you need help. There are more than 40 open lab hours each week.
if
Math TLC Open Lab Hours:
Monday through Thursday 8:00 a.m. – 6:30 p.m
.
This lab is closed on Fridays, on weekends and during breaks.