Ch 1.6 Commutative & Associative

Properties

Objective:To understand the difference between the Commutative and Associative Properties

Definitions

Commutative Property of Additiona + b = b + a

Commutative Property of Multiplication

Example: 3 + 5 = 5 + 3

Example:

a b b a • = •4 7 = 7 4

Commute (travel)

a “travels” to the other side of b

a “travels” to the other side of b

2 12

=

Are the following operations commutative?1) Subtraction

2) Division

a - b = b - a

a b b a ÷ = ÷

Counterexamples2 - 0 = 0 - 2

2 = -2Therefore, subtraction isnot commutative.

Counterexample - a single example that proves a statement false.

Therefore, division isnot commutative.

8 4 4 8 ÷ = ÷

Apply the commutative property

1) 3 + 1 =

Examples

1 3+ 2) 8 + 7 = 7 8+

3) 2 7 = 7 2 4) 4 9 = 9 4

5) 5 + (2 6) = (2 6) 5+

6) (3 + 4) 7 = (3 + 4) 7

Apply the commutative property

1)

3)

5)

7)

9)

2)

4)

6)

8)

10)

Classwork

7 + 4 = 5 + 6 =

3 8 =

(1 + 4) 6 =

8 + (7 4) =

5 (4 + 6) =

9 2 =

(2 + 6) 3 =

1 + (5 7) =

(2 2) + 3 =

Definitions

Associative Property of Addition

Associative Property of Multiplication

Example:

Example:

Associate (partner)

Parenthesis change “partners” – only the parenthesis move

( a + b ) + c = a + ( b + c )(4 + 11) + 6 = 4 + (11 + 6)

Parenthesis change “partners” – only the parenthesis move( ) ( )a b c a b c• • = • •( ) ( )2 5 4 2 5 4• • = • •

10 40 =

3 = 7

Are the following operations associative?1) Subtraction

2) Division

(a - b) - c = a - (b - c)

( ) ( )a b c a b c÷ ÷ = ÷ ÷

(10 - 5) - 2 = 10 - (5 - 2)5 - 2 = 10 - 3

Therefore, subtraction isnot associative.

Therefore, division isnot associative.( ) ( )20 2 80 2÷ = ÷

( ) ( )80 4 2 80 4 2÷ ÷ = ÷ ÷

Apply the associative propertyExamples

1) 5 + (5 + 7) =

2) (6 4) 5 =

3) (9 + 2) + 8 =

4) 5 (2 9) =

5 + 5 + 7 ( )

6 4 5 ( )

9 + 2 + 8 ( )

5 2 9 ( )

Apply the associative property

1)

3)

5)

7)

9)

2)

4)

6)

8)

10)

Classwork

(3 + 4) + 1 = (9 + 4) + 6 =

(3 4) 5 =

4 + (1 + 6) =

8 (7 4) =

4 + (6 + 5) =

(9 2) 10 =

3 + (2 + 6) =

2 (5 7) =

(3 2) 2 =

Commutative vs. Associative

Commutative ( Flip-flop ) Associative ( Re-group )

( ) ( )5 3 7 7 5 3+ + = + + ( ) ( )5 3 7 5 3 7+ + = + +

Flip-flop Re-grouping