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To understand the Commutative
Property you first need to know what
the word “commutative” means.
Commutative means that order
doesn’t matter! You can flip-flop
numbers around an operation, and
you still get the same answer.
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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
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The ORDER in which 2
numbers are added or
multiplied does NOT
change the sum or product.
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Commutative Property of Addition
Order
doesn’t
matter
a+b=b+a
Example:
2+8=8+2
Commutative Property of Multiplication
a • b = b • a
Example:
Flip-Flop
7 • 3 = 3 • 7
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Are the following operations commutative?
Subtraction
Counterexamples
8-5=5-8
3 = -3
a-b=b-a
Division
a ÷ b = b ÷ a
Counterexample a single example
that proves
a statement is false
or not possible.
Therefore, subtraction is
not commutative.
8 ÷ 4 = 4 ÷ 8
1
2
Therefore, division is
not commutative.
2 =
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3
The Associative Property
is like a man and woman getting
married.
First, they are each counted
separately. Then after they are
married they are counted as one,
even though they are still separate
people.
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Before a couple are
married they are
considered as two
separate people.
After the marriage they
are considered as one.
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These numbers are all separate.
4+5+9+3=
6•3•8•4=
The numbers with parentheses around
them are counted as one number.
(4 + 5) + 9 + 3 =
6 • 3 • (8 • 4) =
Count as one!
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Therefore, the operations inside the
parentheses must be done first.
The numbers inside the parenthesis
are “associated” together.
(4 + 5) + 9 + 3 =
6 • 3 • (8 • 4) =
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5
To understand the Associative
Property you first need to know what
the word “associative” means.
Associative means Re-grouping is
ok! You can re-group numbers
together and you still get the same
answer.
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The way 3 numbers are
GROUPED does not change
the sum or product.
(4 + 3) + 9 = 16
4 + (3 + 9) = 16
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Associative Property of Addition
Example:
(a+b)+c = a+(b+c)
3 + (5 + 7) = (3 + 5) + 7
3 + (12) = (8) + 7
15 = 15
Associative Property of Multiplication
(a • b) • c = a • (b • c)
2 • (3 • 1) = (2 • 3) • 1 Re-Grouping
Example:
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Are the following operations associative?
Subtraction
(a - b) - c = a - (b - c)
Therefore, subtraction is
(10 - 5) - 2 = 10 - (5 - 2) not associative.
5 - 2 = 10 - 3
3 = 7
Division
(a ÷ b) ÷ c = a ÷ (b ÷ c)
(80 ÷ 4) ÷ 2 = 80 ÷ ( 4 ÷ 2) Therefore, division is
not associative.
(20) ÷ 2 = 80 ÷ (2)
10 = 40
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Name the property shown below.
1) 7 + 4 = 4 + 7 Commutative Property of Add.
2) 6 • (2 • 8) = (6 • 2) • 8 Associative Property of Multi.
3) 5 • 9 = 9 • 5 Commutative Property of Multi.
4) (4 + 2) + 3 = (2 + 4) + 3 Commutative Property of Add.
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Commutative
(2 + 7) + 8 = (7 + 2) + 8
Associative
(2 + 7) + 8 = 2 + (7 + 8)
Flip-flop
Re-group
(2 + 7) + 8 = 8 + (2 + 7)
Flip-flop
( ) does not imply Associative
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So you are probably thinking…
WHY DO I NEED TO KNOW
THESE PROPERTIES???
Well, knowing how these properties
work can really save you time with
your Math, and can help you make the
math problem easier!
JUST WATCH!!!!
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Problem
What property?
16 + (27 + 84)
16 + (84 + 27)
Commutative Property of Add.
(16 + 84) + 27
Associative Property of Add.
(100) + 27
127
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Problem
What property?
25 • (37 • 4)
25 • (4 • 37) Commutative Property of Mult.
(25 • 4) • 37 Associative Property of Mult.
(100) • 37
3700
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Problem
What property?
12 + (29 + 8)
12 + (8 + 29) Commutative Property of Add.
(12 + 8) + 29 Associative Property of Mult.
(20) + 29
49
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When multiplying a number by 1,
the result is the same as the
original number. The number
does NOT lose it’s identity.
63 x 1 = 63
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When adding a 0 to a number,
the result is the same as the
original number. The number
does NOT lose it’s identity.
5+0=5
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Identity Property of Addition
a+0=a
Identity remains the same
Zero is called the additive identity
Identity Property of Multiplication
a •1 = a
Identity remains the same
One is called the multiplicative identity
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Take out your study guides!
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12
#4
You can change the order in which you add or
multiply, and this does not change the sum or
product. Flip-Flop
Order doesn’t matter
For
For Addition
Addition
a+b=b+a
Example:
For
For Multiplication
Multiplication
a •b = b• a
Example:
7•4 = 4•7
28 = 28
5+12 = 12+5
17 = 17
Doesn’t work for Subtraction
8-5=5-8
3 = -3
Doesn’t work for Division
8 ÷ 4 = 4 ÷ 8
2 =
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1
2
#5
You can regroup numbers when adding or
multiplying, and this does not change their
sum or product. Re-Grouping
For
For Addition
Addition
For
For Multiplication
Multiplication
(a + b) + c = a + (b + c)
Example:
(2+5)+3 = 2+(5+3)
(7) + 3 = 2 + (8)
10 = 10
(a ib)ic = ai(bic)
Example:
(5i4)i3 = 5i(4i3)
(20)i3 = 5i(12)
60 = 60
Doesn’t work for Subtraction or Division
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#6
You can add 0 to a number or you can multiply a
number by 1 and this does NOT change the number.
Identity
Identity Property
Property
of
Addition
of Addition
Identity
Identity Property
Property
of
Multiplication
of Multiplication
a •1 = a
a+ 0 =a
Example:
Example:
−8 • 1 = −8
1 • b = 1b
-3 + 0 = -3
0+x =x
The number does NOT lose it’s identity.
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TEACHER NOTE: The next day insert
the following slides during
homework corrections.
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14
So you are probably thinking…
WHY DO I NEED TO KNOW
THESE PROPERTIES???
Well, knowing how these properties
work can really save you time with
your Math, and can help you make the
math problem easier!
JUST WATCH!!!!
http://www.mathslideshows.com
Problem
What property?
5 • 19 • 2
5 • 2 • 19 Commutative Property of Mult.
10 • 19
190
Order doesn’t matter
Flip-Flop
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15
Problem
What property?
7 + ( 43 + 29 )
( 7 + 43 ) + 29 Associative Prop. of Add.
( 50 ) + 29
Re-Grouping
79
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Name the property shown below.
1) (6 + 3) + 1 = (3 + 6) + 1 Comm. Prop. Of Add.
2) 10 • (8 • 3) = (10 • 8) • 3 Assoc. Prop. Of Mult.
3) 5 + 1 = 1 + 5 Comm. Prop. Of Add.
4) (7 • 2) • 4 = 4 • (7 • 2) Comm. Prop. Of Mult.
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