1. Ch 1.6 Commutative & Associative Properties
Objective:
To understand the difference between the Commutative and Associative Properties

2. Definitions a + b = b + a 3 + 5 = 5 + 3 4  7 = 7  4 Commute
(travel)
Commutative Property of Addition
a + b = b + a
a “travels” to the other side of b
3 + 5 = 5 + 3
Example:
Commutative Property of Multiplication
a “travels” to the other side of b
4  7 = 7  4
Example:

3. Are the following operations commutative?
1) Subtraction
Counterexamples
a - b = b - a
2 - 0 = 0 - 2
2 = -2
Therefore, subtraction is
not commutative.
2) Division
Therefore, division is
not commutative.
Counterexample - a single example that proves
a statement false.

4. Examples 2) 8 + 7 = 7 + 8 1) 3 + 1 = 1 + 3 4) 4  9 = 9  4 3) 2  7 =
Apply the commutative property
2) = 7 + 8
1) = 1 + 3
4) 4  9 = 9  4
3) 2  7 = 7  2
5) 5 + (2  6) =
(2  6) + 5
6) (3 + 4)  7 = 7 
(3 + 4)

5. Classwork 7 + 4 = 5 + 6 = 3  8 = 9  2 = (1 + 4)  6 = (2 + 6)  3 =
Apply the commutative property 1)
7 + 4 = 2)
5 + 6 = 3)
3  8 = 4)
9  2 = 5)
(1 + 4)  6 = 6)
(2 + 6)  3 =
1 + (5  7) = 7)
8 + (7  4) = 8) 9)
5  (4 + 6) =
10)
(2  2) + 3 =

6. Definitions ( a + b ) + c = a + ( b + c ) (4 + 11) + 6 = 4 + (11 + 6)
Associate
(partner)
Associative Property of Addition
Parenthesis change “partners” – only the parenthesis move
( a + b ) + c = a + ( b + c )
(4 + 11) + 6 = 4 + (11 + 6)
Example:
Associative Property of Multiplication
Parenthesis change “partners” – only the parenthesis move
Example:

7. Are the following operations associative?
1) Subtraction
(a - b) - c = a - (b - c)
(10 - 5) - 2 = 10 - (5 - 2)
Therefore, subtraction is
not associative. =
3 = 7
2) Division
Therefore, division is
not associative.

8. Examples 1) 5 + (5 + 7) = ( ) 5 + 5 + 7 6  4  5 ( ) 2) (6  4)  5 =
Apply the associative property
1) 5 + (5 + 7) =
( )
6  4  5
( )
2) (6  4)  5 =
3) (9 + 2) + 8 =
( )
4) 5  (2  9) =
( )
5  2  9

9. Classwork (3 + 4) + 1 = (9 + 4) + 6 = (3  4)  5 = (9  2)  10 =
Apply the associative property 1)
(3 + 4) + 1 = 2)
(9 + 4) + 6 = 3)
(3  4)  5 = 4)
(9  2)  10 = 5)
4 + (1 + 6) = 6)
3 + (2 + 6) =
2  (5  7) = 7)
8  (7  4) = 8) 9)
4 + (6 + 5) =
10)
(3  2)  2 =

10. Commutative vs. Associative
( Flip-flop )
Associative
( Re-group )
Flip-flop
Re-grouping