Name:
Date:
Math 9R, M1L7
Miss Kersting
Lesson 7 Goal: SWBAT apply the commutative and associative properties to expressions
Do Now: Complete the maze below.
State the property that makes the expressions below equal.
1.
2.
____________________________
(
)
(
) ____________________________
Let’s review the four properties of arithmetic & provide an example for each.
Abbreviations:
1. The Commutative Property of Addition:
2. The Associative Property of Addition:
3. The Commutative Property of Multiplication:
4. The Associative Property of Multiplication:
Math 9R, M1L7
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We can combine the properties and create new expressions. We will use 2 strategies to
demonstrate this.
1.
2.
Example 1: Show that (3+4)+2 = 2+(4+3)
List format
(3+4)+2 = 2+(3+4) ___________________
2+(3+4) = 2+(4+3) ___________________
Flow Chart
(4+3)+2
(3+4)+2
2+(4+3)
2+(3+4)
Example 2: Show that (x+y)+z = (z+y)+x
List format
(x+y)+z =
___________________
___________________
___________________
Complete Flow Chart
Math 9R, M1L7
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Example 3: Draw a flow diagram and use it to prove that
numbers x, y, and z.
( xy)z = (zy) x for all real
Example 4: Use these abbreviations for the properties of real numbers and complete
the flow diagram.
C for the commutative property of addition
C for the commutative property of multiplication


A for the associative property of addition
A for the associative property of multiplication


Example 5
Let a, b, c, and d be real numbers. Fill in the missing term of the following diagram to
show that a + b + c + d is sure to equal a + b + c + d .
((
) )
( (
))
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Problem Set
1. The following portion of a flow diagram shows that the expression
equivalent to the expression dc + ba.
ab + cd is
Fill in each circle with the appropriate symbol:
C+ for the Commutative Property of Addition
C´ for the Commutative Property of Multiplication
2. Fill in the blanks to prove that
(w + 5)(w + 2) is equivalent to w2 + 7w + 10.
Write either “Commutative Property,” “Associative Property,” or “Distributive Property”
in each blank.
(w + 5)(w + 2)
= (w + 5)w + (w + 5) ´ 2
= w(w + 5) + (w + 5) ´ 2
= w(w + 5) + 2(w + 5)
= w 2 + w ´ 5 + 2(w + 5)
= w2 + 5w + 2(w + 5)
= w2 + 5w + 2w + 10
= w2 + (5w + 2w) + 10
= w2 + 7w + 10
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3. Fill in each circle of the following flow diagram with one of the letters:
C for Commutative Property (for either addition or multiplication)
A for Associative Property (for either addition or multiplication)
D for Distributive Property
4. Write a mathematical proof of the algebraic equivalency of
(ab)
2
and
a2 b2.
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5. Consider the expression:
(a + b + 4) ´ (b + 2).
Draw a picture to represent the expression. Write an equivalent expression.
6. Here x, y, a, and b are real numbers with x and y non-zero. Replace each of the
following expressions with an equivalent expression in which the variable of the
expression appears only once with a positive number for its exponent.
For example,
7
b2
· b -4 is equivalent to
7
b6
a.
æç
ö æ
ö
16x 2 ÷ ¸ ç16x 5 ÷
è
ø è
ø
b.
(2x) (2x)
c.
æç -2 ö÷æç -1 ö÷ -3
9z
3z
è
øè
ø
d.
ææ
ö æ
öö æ
ö
çç25w 4 ÷ ¸ ç5w 3 ÷÷ ¸ ç5w -7 ÷
ø è
øø è
ø
èè
e.
æç
ö ææ
ö æ
öö
25w 4 ÷ ¸ çç5w 3 ÷ ¸ ç5w -7 ÷÷
è
ø èè
ø è
øø
4
3
Math 9R, M1L7
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Name:
Write a mathematical proof of the algebraic equivalence of
Exit Pass
( pq)r and (qr ) p.