Transcript Just the facts: Order of Operations and Properties of real
Start thinking of math as a language, not a pile of numbers
Just like any other language, math can help us communicate thoughts and ideas with each other An expression is a thought or idea communicated by language In the same way, a
mathematical expression
can be considered a mathematical thought or idea communicated by the language of mathematics.
Mathematics is a language, and the best way to learn a new language is to immerse yourself in it.
A SSE 1
Just like English has nouns, verbs, and adjectives, mathematics has terms, factors, and coefficients. Well, sort of.
TERMS
are the pieces of the expression that are separated by plus or minus signs, except when those signs are within grouping symbols like parentheses, brackets, curly braces, or absolute value bars. Every mathematical expression has
at least
one term. 3
x
2 Has two terms. 3
x and
2 5 A term that has no variables is often called a
constant
because it never changes.
Within each term, there can be two or more
factors.
The numbers and/or variables multiplied together.
3
x
Has two factors: 3 and
x
. There are always
at least
two factors, though one of them may be the number 1, which isn't usually written. Finally, a
coefficient
is a factor (usually numeric) that is multiplying a variable. Using the example, the 3 in the first term is the coefficient of the variable
x
.
The
order
or
degree
of a mathematical expression is the largest sum of the exponents of the variables when the expression is written as a sum of terms.
3
x
2 Order is 1 Since the variable
x
in the first term has an exponent of 1 and there are no other terms with variables.
5
x
2 3
x
2 Order is 2 3
xy
5 2
x y
3 7
x
32
y
4 Order is 5
Now that we have our words, we can start putting them together and make expressions 3
x
2 Translate mathematical expressions into English "the sum of 3 times a number and 2," "2 more than three times a number" It's much easier to write the mathematical expression than to write it in English
Practice 1.1 Variables and Expressions A-SSE.A.1
x
10
d
5 11
d t
3
d
11
t
3
Practice 1.1 Variables and Expressions A-SSE.A.1
20
x
3 5
d
12
w
7. Write a rule in words and as an algebraic expression to model the relationship in each table. The local video store charges a monthly membership fee of $5 and $2.25 per video.
$5 plus $2.25 times the number of videos; 5 2.25
v
Just the facts: Order of Operations and Properties of real numbers
A GEMS/ALEX Submission Submitted by: Elizabeth Thompson, PhD Summer, 2008
Important things to remember
•
Parenthesis – anything grouped… including information above or below a fraction bar.
•
Exponents – anything in the same family as a ‘power’… this includes radicals (square roots).
•
Multiplication- this includes distributive property (discussed in detail later).
•
Some items are grouped!!!
Multiplication and Division are GROUPED from left to right (like reading a book- do whichever comes first.
•
Addition and Subtraction are also grouped from left to right, do whichever comes first in the problem.
So really it looks like this…..
• • • •
P
arenthesis
E
xponents
M
ultiplication and
D
ivision In order from left to right
A
ddition and
S
ubtraction In order from left to right
SAMPLE PROBLEM #1
16 16 4 ( 3 1 Parenthesis 3 ) 4 ( 2 ) 3 22 22 11 11 16 Exponents 4 ( 8 ) 22 11 This one is tricky!
Remember: Multiplication/Division are grouped from left to right…what comes 1 st ? 4 ( 8 ) 22 11 Division did…now do the multiplication (indicated by parenthesis) 32 22 More division 11 32 Subtraction 2 30
SAMPLE PROBLEM
3 ( 2 3 ) 2 65 2 Parenthesis 3 ( 25 ) 65 2 Remember the division symbol here is grouping everything on top, so work everything up there first….multiplication
3 ( 5 ) 2 65 2 Exponents 75 2 65 Subtraction 10 2 Division – because all the work is done above and below the line 5
Order of Operations-BASICS
Think: PEMDAS Please Excuse My Dear Aunt Sally
• • • • • •
P
arenthesis
E
xponents
M
ultiplication
D
ivision
A
ddition
S
ubtraction
Practice 1.2 Order of Operations and Evaluating Expression A-CED.1
Simplify 2 4 4 16 3. 5 3 5 5 5 125 __________ 4. 16 2 6. 3 15 3 3 3 27 353
Practice 1.2 Order of Operations and Evaluating Expression
Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 7. Area of a triangle:
b
6 and
h
F
:
A
1 2
bh S
:
A
1 (6)(14) 2
A
:
A
42
in
2
Practice 1.2 Order of Operations and Evaluating Expression
Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 8. Volume of a pyramid:
B
18 and
h
F
:
V
1 3
Bh S
:
V
1 (18)(8) 3
A
:
V
48
m
3
Practice 1.2 Order of Operations and Evaluating Expression
Using the PARCC High School Assessment Reference Sheet. Evaluate each expression for the given values of the variables. FSA 9. Find the value of x using the quadratic formula with
a
1,
b
2 3
S F
: :
x x
b
2 4
ac
2
a
( 2) 2 2(1)
x
2 2
x
2 3
x
2 1
10. The cost to rent a hall for school functions is $60 per hour. Write an expression for the cost of renting the hall for h hours. Make a table to find how much it will cost to rent the hall for 2, 6, 8, and 10 hours. 60
h hours
$ 2 6 120 360 8 10 480 600
Lesson Extension
• Can you fill in the missing operations?
1. 2 - (3+5) + 4 = -2 2. 4 + 7 * 3 ÷ 3 = 11 3. 5 * 3 + 5 ÷ 2 = 10
Practice 1.3 Real Number and the Number Line Name the radicand of each of the following, then write in simplified form. 64 radicand 3. 1 36 36 radicand 1 36 25 radicand 1 81 81 ____ 4. 100 radicand 100 9
Practice 1.3 Real Number and the Number Line Estimate the square root by finding the two closest perfect squares.
5. 51 49 perfect square 64 perfect square 9.
U
10.
U
yes no
Practice 1.3 Real Number and the Number Line Circle all the statements that are true.
11. 9 rational 12. 5 irrational 13. 1 integer 14. 0 whole 3 19. 17. 9 whole 18. 0 natural 3 100 rational 20. 2 irrational 21. 2.56 rational 22. 49
An
inequality
is a mathematical sentence that compares the values of two expressions using an inequality symbol. The symbols are: ( >, <, ) 3 3.5
7 23. What is the order of 3.51, 2.1, 9, , and 5 from least to greatest?
2 2 2, 5, 9, 7 2 , 3.51
Properties of Real Numbers
(A listing)
• Associative Properties • Commutative Properties • Inverse Properties • Identity Properties • Distributive Property
All of these rules apply to Addition and Multiplication
Associative Properties
Associate = group It doesn’t matter how you group (associate) addition or multiplication…the answer will be the same!
Rules: Associative Property of Addition (a+b)+c = a+(b+c) Samples: Associative Property of Addition (1+2)+3 = 1+(2+3) Associative Property of Multiplication (ab)c = a(bc) Associative Property of Multiplication (2x3)4 = 2(3x4)
Commutative Properties
Commute = travel (move) It doesn’t matter how you swap addition or multiplication around…the answer will be the same!
Rules: Commutative Property of Addition a+b = b+a Samples: Commutative Property of Addition 1+2 = 2+1 Commutative Property of Multiplication ab = ba Commutative Property of Multiplication (2x3) = (3x2)
Stop and think!
• Does the Associative Property hold true for Subtraction and Division?
Is (5-2)-3 = 5-(2-3)? Is (6/3)-2 the same as 6/(3-2)?
• Does the Commutative Property hold true for Subtraction and Division?
Is 5-2 = 2-5? Is 6/3 the same as 3/6?
Properties of real numbers are only for Addition and Multiplication
Inverse Properties
Think: Opposite
What is the opposite (inverse) of addition?
What is the opposite of multiplication?
Subtraction (add the negative) Division (multiply by reciprocal)
Rules:
Inverse Property of Addition
a+(-a) = 0 Samples:
Inverse Property of Addition
3+(-3)=0
Inverse Property of Multiplication
a(1/a) = 1
Inverse Property of Multiplication
2(1/2)=1
Identity Properties
What can you add to a number & get the same number back?
0 (zero) What can you multiply a number by and get the number back?
1 (one)
Rules:
Identity Property of Addition
a+0 = a
Identity Property of Addition
3+0=3 Samples:
Identity Property of Multiplication
a(1) = a
Identity Property of Multiplication
2(1)=2
Distributive Property
If something is sitting just outside a set of parenthesis, you can distribute it through the parenthesis with multiplication and remove the parenthesis.
Rule: a(b+c) = ab+bc Samples:
4(3+2)=4(3)+4(2)=12+8=20 • 2(x+3) = 2x + 6 • -(3+x) = -3 - x
Practice 1.4 Properties of Real Numbers
A. Associative Property of Addition/Multiplication B. Commutative Property of Addition/Multiplication C. Identity Property of Addition/Multiplication D. Zero Property of Multiplication E. Multiplica tion Property of -1 What property is illustrated by each statement?
x x
(
x
E A
x p
)
y
)
p D
: Give an example
y C m B xyz
yxz m
Practice 1.5 Adding and Subtracting Real Numbers
Find each sum.
3 10 2 6 11 4 4 11 21 5 18 13 9 7 30 5 12 17. 12 2
Absolute Value.
Simplify each expression.
13 Opposites: 11 11 4 21. 1 10 State the opposite of result of each statement.
2 2 4 4 3 3 6 7 7 11 11 10 10
Practice 1.6 Multiplying and Dividing Real Numbers
Find each product/quotient.
3 3. 6 40 21 24 7 18 45 60 15 90 80 112 18 13. 10 2 14. 15 5 15. 5 3 8 8 16. 1 6 3 17. 12 10 8 2 8 1 4 10 18. 19. 1 2 10( 2) 1 7 20 3 6 1 3 6 7 20. 7 18 1 5 4 21. 1 5 5 4 12 2 5 12 5 2 30 25 125
Practice 1.7 Distributive Property What is the simplified form of each expression?
1. 5(
x
7) 2. 12(3 1 6
x
5
x
35 36 2
x
1.2 3.3
c
4. (2
y y
) 5. 2
y
2
y x
8
x
20 5) 6. 7. 4(2
x
2 3
x
8
x
2 12
x
4
x
5) 9. 10
x
2 25
x x x
6) 6 3) 3
x
Practice 1.7 Distributive Property 2 3
x
2 4
x
2
What is the simplified form of each expression?
11. 3
y
y
12. 7
mn
4 5
mn
4 13. 3 2
y
12
mn
4 2 2
y x
y
2
y
2 14.
a
3 7
b
2 5
y
3
x
8 16. 5
y
3
y
10
x
3
x
x
3
y
4
y
7
x
Practice 1.8 An Introduction to Equations Tell whether each equation is true, false or open. Explain.
1. 45 14 22 2.
True
52 3. 7 8 15
False
Tell whether the given number is a solution of each equation.
b
4
x
?
3( 7) 8 13 ?
4( 2) 7 15 ?
21 8 13 ?
8 7 15
Not
?
15 15
Yes
?
12 14 2( 1) ?
12 14 2
NO
Practice 1.8 An Introduction to Equations Write an equation for each sentence.
n
n