9A Aim 11 ­ commutative associative property.notebook
Aim # 11: What are the commutative and associative properties?
Do Now: Simplify each of the following (You may use a calculator!):
1. a) 1 + 2 + 3 + 4
b) 2 + 1 + 3 + 4
c) 4 + 1 + 3 + 2
2. a) 1 - 2 - 3 - 4
b) 2 - 1 - 3 - 4
c) 4 - 1 - 3 - 2
3. a) (1)(2)(3)(4)
b) (2)(1)(3)(4)
c) (4)(1)(3)(2)
4. a) 1 ÷ 2 ÷ 3 ÷ 4
b) 2 ÷ 1 ÷ 3 ÷ 4
c) 4 ÷ 1 ÷ 3 ÷ 2
For which of the above operations, (addition, subtraction, multiplication and
division), can we change the order without changing the final result?
Commutative Property of Addition
a+b=b+a
Commutative Property of Multiplication
(a)(b) = (b)(a)
How can the commutative property help us simplify?
example A: 6 + 7 + 14
example B: 2(17)(5)
.
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9A Aim 11 ­ commutative associative property.notebook
1. a) 1 + 2 + 3 + 4
b) 1 + (2 + 3) + 4
c) 1 + 2 + (3 + 4)
2. a) 1 - 2 - 3 - 4
b) 1 - (2 - 3) - 4
c) 1 - 2 - (3 - 4)
3. a) 1 x 2 x 3 x 4
b) 1 x (2 x 3) x 4
c) 1 x 2 x (3 x 4)
4. a) 1 ÷ 2 ÷ 3 ÷ 4
b) 1 ÷ (2 ÷ 3) ÷ 4
c) 1 ÷ 2 ÷ (3 ÷ 4)
What is being changed in each example?
For which operations does this change not affect the final result?
Associative Property of Addition
a + (b + c) = (a + b) + c
Associative Property of Multiplication
a(bc) = (ab)c
Look at the following simplifications. On each line, write which property was used to
produce the expression on the left.
4+5+6+3
4+6+5+3
(4 + 6) + (5 + 3)
10 + 8
18
_____________________
_____________________
2(4 + x) + 16
2(4) + 2(x) + 16 ___________________
8 + 2x + 16
8 + 16 + 2x
___________________
20 + 2x
2x + 20
___________________
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9A Aim 11 ­ commutative associative property.notebook
Write an equivalent expression to ab + cd by using
1) the commutative property of addition
2) the commutative property of multiplication
State if the following are equivalent to xy + 2. If they are equivalent, state
the property that was used.
1) 2 + xy
2) yx + 2
3) 2x + y
Now, show a mathematical proof of the statement that
(x + y) + z = (z + y) + x
(x + y) + z
= z + (x + y) ________________________
= z + (y + x) ________________________
= (z + y) + x ________________________
Fill in the flow chart
D = distributive
Cx = commutative of multiplication
C+ = commutative of addition
Ax = associative of multiplication
A+ = associative of addition
(xy)z
(yx)z
z(yx)
(zy)x
Let's sum it up!!!
The commutative property allows us to change the __________________.
The associative property allows us to change the _________________.
The commutative and associative properties only hold for the operations of
_________________ & __________________
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