Objective - To simplify expressions using
commutative and associative properties.
Commutative - Order doesn’t matter! You can
flip-flop numbers around an operation.
Commutative Property of Addition
a+b=b+a
5+7=7+5
10 + 3 = 3 + 10
Commutative Property of Multiplication
a b  ba
8 4  48
3  12  12  3
Write an equivalent expression.
1) (6 + 4) = (4 + 6)
2) 3  8  8  3
3) 5  12  12  5
4) 9 + 7 = 7 + 9
5) 3  (4  8)  3  (8  4) or (4  8)  3
Addition and multiplication
are commutative.
Difficult to
prove true!
Is subtraction commutative?
Easy to
prove false!
Counterexample - An example that proves a
statement false.
Counterexample: 8  5  5  8
3  3
Therefore, subtraction is not commutative.
Give a counterexample that shows division is
not commutative.
Counterexample: 4  2  2  4
1
2
2
Therefore, division is not commutative.
State whether each situation below is commutative
or not commutative.
1) Waking up in the morning and going to school.
not commutative
2) Brushing your teeth and combing your hair.
commutative
3) Putting on your socks and putting on your
shoes.
not commutative
4) Eating cereal and drinking orange juice.
commutative
Give the property that justifies each step.
Statement
16 + 27 + 84
Reasons
Given
16 + 84 + 27
Comm. Prop. of Add.
100 + 27
Simplify
127
Simplify
Identities
Identity Property of Addition
x+0=x
Zero is sometimes called the Additive Identity.
Identity Property of Multiplication
x 1  x
One is sometimes called the Multiplicative Identity.
Identify the property shown below.
1) 6 + 8 = 8 + 6
Comm. Prop. of Add.
2) (10  4)  (4 10) Comm. Prop. of Mult.
3) (2 + 10) + 3 = (10 + 2) + 3 Comm. Prop. of Add.
4) 5  (7  4)  (7  4)  5 Comm. Prop. of Mult.
5) 7  0  0 Mult. Prop. of Zero
6) 7 + 0 = 7 Identity Prop. of Add.
7) 7  1  7 Identity Prop. of Mult.
Identity Property of Multiplication
x 1 x
2
2  4  6  8 ...
1
3
3 6 9 12
Many Forms of 1
2
4
23
3
5
...
1     ... 
2 3 4 5
23
Consequently, there are an infinite number of
ways to write any fraction.

2 2
4

3 2
6

2 3
6

3 3
9

2 4
8

3 4
12
Models of Equivalent Fractions
2
3

2 2 4

3 2 6

2 3
6

3 3
9
Find 6 equivalent fractions for the fraction below.
3 6
9 12 15 18 21 ...






5 10 15 20 25 30 35
Most
simplified

3 3
9
5 3   15
3 4
12
5 4   20
3 2
6

5 2
10

3 6
18
5 6   30
3 7
21
5 7   35
3 5
15

5 5
25
Write at least 4 equivalent fractions for the given
one below.
1 2
3
4
5
6
7
8 ...
1) 






5 10 15 20 25 30 35 40
3  6  9  12  15  18  21  24 ...
2)
7 14 21 28 35 42 49 56
6 12 18 24 30 36 42 48 ...
3)







11 22 33 44 55 66 77 88
Circle the most simplified form of each value.
Simplifying Fractions (Reducing)
5
10
2

5


2  2  2  3 12
24
10 2  5
24 2 12
GCF  2
48 12 4

60 12 5
48 2  24 2 12 3 4


60 2 30 2 15 3 5
Identify each fraction below.
3 3  1
6 3 2
Reduce!
Identify each fraction below.
6 2  3
8 2 4
Reduce!
Simplify the following fractions.
16 2  8 2  4 2  2
1)
40 2 202 102 5
16 8  2
40 8 5
GCF  8
15 5 3 3 1
2)


75 5 15 3 5
15 15  1
75 15 5
GCF  15
Simplify the following fractions.
12 6 2
1)

30 6 5
1818  1
4)
7218 4
18 9 2
2)

459 5
28 14 2
5)

42 14 3
16 8  2
3)
24 8 3
364 9
6)
80 4 20
Simplify the variable expressions.
14x
2

7

x

x

x
2x
1)


2
5 7 x  x
5
35x
3
2
14x 7x  2x
2
2
5
35x 7x
3
GCF  7 x 2
25a b c  5 5 a a b b b c  5ac
2)
4
2  5 a b b b b
2b
10ab
2 3
3
25a b c 5ab  5ac
3
4
2b
10ab 5ab
2 3
GCF  5 a b3