Problem
13.1
Topic 13: Properties
for use after
Bits and Pieces III
Investigation 4
When you are simplifying a numerical expression, you may be asked to
justify or explain your steps. In your justifications, it helps to refer to
properties of rational numbers by name.
A. 1. Copy and complete the table.
B. Name the property shown in each equation.
1. 5.81 + (-5.81) = 0
2. -4(
x
+ 2) =-4
x
+ (-8)
3.
+ 0 =
4. ++
y
=++
y
5. 0.68(-5) =-5(0.68)
6. 23
h
- 46 = 23(
h
- 2)
1
Q
3
5
R
Q
1
R
5
5
3
5
16
1
16
1
2
2
Property
Associative
Commutative
Identity
Inverse
Distributive
Algebra
(
a
?
b
)
?
c
?
a
?
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b
?
c
)
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a
?
b
)
?
c
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(
b
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c
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a
?
b
?
b
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a
a
?
b
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b
?
a
a
?
0
?
0
?
a
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1
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a
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1
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a
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(
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a
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0
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1
a
(
b
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c
)
?
ab
?
ac
ab
?
ac
?
a
(
b
?
c
)
Arithmetic
(
?
7
?
3)
?
9
? ?
7
?
(3
?
9)
(5
?
)
?
10
?
5
?
(
?
10)
?
8.01
?
(
?
12)
? ?
12
?
(
?
8.01)
5
?
4
?
4
?
15
?
0
?
0
? ?
1
?
(
?
)
?
(
?
)
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1
? ?
34
?
(
?
34)
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0
?
18
?
1
?
2
?
(6
?
5)
? ?
2
?
6
?
(
?
2)
?
5
?
12
? ?
16
? ?
(12
?
16)
1
2
2
3
3
4
3
4
3
4
2
3
2
3
12
15
1
18
12
15
12
15
1
2
1
a
Sometimes you will need to use more than one property to simplify an
expression. You can do the simplification in steps, using one property
for each step.
A. The steps below show one way to simplify ? (3 ? 8) + 7 - 8.
? (8 ? 3) + 7 - 12
? (3 ? 8) + 7 - 12
? 3 ? 8 + 7 - 12
1 ? 8 + 7 - 12
8 + 7 - 12
15 - 12
3
Copy and complete the steps above, naming a property or operation to
the right of each step.
B. Simplify each expression. Use a property or operation to justify
each step.
1. 6 - 8
x
+ 4(5 + 2
x
)
2. 7.2
a
+ 4(-3.5 +
a
)
Exercises
Name the property illustrated in each equation.
1. 0.85 + (3.5 + 4.15) = (0.85 + 3.5) + 4.15
2. 3
d
- 15 = 3(
d
- 5)
3. 0 + (-1.6) + 2.4 =-1.6 + 2.4
4. 33 = 1 3
5. 15(2
c
- 8) = 30
c
- 120
6. -3.2 + (-8.5
x
) =-8.5
x
+ (-3.2)
7. 123 + (-43) + 0 + (-15) = 123 + (-43) + (-15)
Simplify each expression. Use a property or operation to justify each step.
8. -4 ++++
9. 5
m
+ 6 + 3(
m
+ 2)
10. -2
++ 6 +
2
3
1
Q
1
3
R
2
k
4
5
7
2
6
5
5
2
1
4
1
4
2
1
1
2
Q
1
R
3
1
3
1
3
1
3
Problem
13.2
Topic 11: Properties
Teaching Guide
Mathematical Goals
• Identify properties of rational numbers
• Use properties of rational numbers to simplify expressions and justify
steps in calculations
Vocabulary
•
Associative Property
of Addition
•
Associative Property
of Multiplication
•
Commutative
Property of Addition
•
Commutative
Property of
Multiplication
•
Identity Property of
Addition
•
Identity Property of
Multiplication
•
Inverse Property of
Addition
•
Inverse Property of
Multiplication
•
Distributive Property
At a Glance
Some students may need to use Topic 10 to review number properties as an
introduction to Topic 11.
Remind students that some properties are similar for addition and
multiplication, such as the associative and commutative properties, while
other properties, such as the inverse properties, may look different.
Understanding the similarities between properties will help students to
differentiate between different properties. For example, the associative
properties involve associating different terms around an operation, while
the inverse properties involve values that add (or multiply) to give the
additive (or multiplicative) inverses.
Until students are comfortable with the properties, they should write out
the complete names (Commutative Property of Addition, Commutative
Property of Multiplication) to prevent confusion with other operations. For
example, the commutative property is only true for addition and
multiplication, but not for subtraction or division.
To summarize Problem 11.2A, ask:
•
What is the difference between a property and an operation?
•
Why might you want to use a property in the first step rather than
multiplying 3 and 8?
•
How does writing each property or operation next to each step help you
check the answer once you have simplified the expression?
To summarize Problem 11.2B1, ask:
•
Which property would you use first?
•
The first step in one student’s solution is to use the Distributive Property
on
4(5 + 2
x
)
. The first step in another student’s solution is to use the
Commutative Property of Addition on
5 + 2
x. Is one solution correct?
•
Can you use the commutative property to write
8
x
? 6 + 4(5 + 2
x
)
in a
simpler form?
PACING
1 day
Assignment Guide for Topic 11
Core
1–10
Answers to Topic 11
Problem 11.1
A.
See Figure 1.
B. 1.
Inverse Property of Addition
2.
Distributive Property
3.
Identity Property of Addition
4.
Associative Property of Addition
5.
Commulative Property of Multiplication
6.
Distributive Property
Problem 11.2
A. 1.
? (8 ? 3) 1 7 2 12
? (3 ? 8) 1 7 2 12 (Comm.Prop.)
? 8 1 7 2 12 (Assoc. Prop.)
1 ? 8 1 7 2 12 (Inverse Prop.)
8 1 7 2 12 (Identity Prop.)
15 2 12 (addition)
3 (subtraction)
B. 1.
6 2 8
x
1 4(5 1 2
x
)
6 2 8
x
1 20 1 8
x
(Distributive Prop.)
6 2 8
x
1 8
x
1 20 (Comm.Prop)
6+0+20 (Inverse Prop.)
6+20 (Identity Prop.)
26 (addition)
2.
7.2
a
1 4(23.5 1
a
)
7.2
a
1 (214) 1 4
a
(Distributive Prop.)
7.2
a
1 4
a
1 (–14) (Comm. Prop.)
11.2
a
1 (214) (addition)
Exercises
1.
Associative Property of Addition
2.
Distributive Property
3.
Identity Property of Addition
4.
Inverse Property of Multiplication
5.
Distributive Property
6.
Commutative Property of Addition
7.
Identity Property of Addition
8.
(Comm. Prop.)
(addition)
24 1 6 1 2 (division)
4 (addition)
9.
5
m
1 6 1 3(
m
1 2)
5
m
1 6 1 3
m
1 6 (Distributive Prop.)
5
m
1 3
m
1 6 1 6 (Comm.Prop.)
8
m
1 12 (addition)
10.
(Distributive Prop.)
(Identity Prop. of Mult.)
(Comm. Prop. of Add.)
2
k
1 6 1 0 (Inverse Prop.)
2
k
1 6 (Identity Prop. of Add.)
2k 1 6 1
Q
2
2
3
R
1
2
3
2k 1
Q
2
2
3
R
1 6 1
2
3
22 3
1
2
k 1
Q
2
2
3
R
1 6 1
2
3
22
Q
1
2
k 1
1
3
R
1 6 1
2
3
24 1
12
2
1
10
5
24 1
5
2
1
7
5
1
6
5
1
4
5
24 1
5
2
1
6
5
1
7
2
1
4
5
Q
1
3
? 3
R
1
3
1
3
Figure 1
Property
Associative
Commutative
Identity
Inverse
Distributive
Algebra
(
a
?
b
)
?
c
?
a
?
(
b
?
c
)
(
a
?
b
)
?
c
?
a
?
(
b
?
c
)
a
?
b
?
b
?
a
a
?
b
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b
?
a
a
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0
?
0
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a
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a
1
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a
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a
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1
?
a
a
?
(
?
a
)
?
0
?
a
?
1
a
(
b
?
c
)
?
ab
?
ac
ab
?
ac
?
a
(
b
?
c
)
Arithmetic
(
?
7
?
3)
?
9
? ?
7
?
(3
?
9)
(5
?
)
?
10
?
5
?
(
?
10)
?
8.01
?
(
?
12)
? ?
12
?
(
?
8.01)
5
?
4
?
4
?
15
?
0
?
0
? ?
1
?
(
?
)
?
(
?
)
?
1
? ?
34
?
(
?
34)
?
0
?
18
?
1
?
2
?
(6
?
5)
? ?
2
?
6
?
(
?
2)
?
5
?
12
? ?
16
? ?
(12
?
16)
1
2
2
3
3
4
3
4
3
4
2
3
2
3
12
15
1
18
12
15
12
15
1
2
1
a