Name: __________________________________________________________________ Period: _____
Unit 5: Properties of the Real Number System
PROPERTY
What is it?
COMMUTATIVE
Change Order: changing the order of
multiplication or addition problem does not
change the answer.
ASSOCIATIVE
Multiplication and addition problems can be
regrouped with parentheses. Grouping does
not change the answer.
IDENTITY
(Answer is always the same
as what you started with)
INVERSE
(Answer is always the
Identity)
DISTRIBUTIVE
Additive Identity: The number that adds on
to give you the same number you started with
as an answer. Answer is the same as what you
started with.
Multiplicative Identity: The number that you
multiply by to give you the same number you
started with as an answer. Answer will always
be the same as what you started with.
Additive Inverse: What you will add on so the
answer is the Identity, always has opposite
sign.
Multiplicative Inverse: What you will multiply
by so the answer is the Identity, always the
reciprocal.
To remove parentheses by multiplying the
outside number through the parentheses, or
to put parentheses in by dividing out a GCF.
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Examples/Information
Additive Identity is
ALWAYS:
Multiplicative Identity is
ALWAYS:
_____ 1) The equation x + 4 = 4 + x is an illustration of the
(1) associative property
(2) distributive property
(3) commutative property
(4) additive inverse
_____ 2) Which statement illustrates the associative property?
1 
(1) 2   = 1
2
(2) 2(3 + 4) = 2(3) + 2(4)
(3) 2(1) = 2
(4) 2(3 • 4) = (2 • 3)4
_____ 3) The equation 6(x + y) = 6x + 6y is an example of the
(1) associative property
(2) additive inverse
(3) commutative property
(4) distributive property
_____ 4) The additive inverse of -3 is
(1) 
1
3
1
3
(4) 0
(3)
(2) 3
_____ 5) If n represents an even integer, the next larger even integer is
(1) n + 1
(2) n + 2
(3) 2n
(4) 2n + 2
Matching: Write the letter of the property that justifies each statement below.
1 
___6) 2   = 1
2
A) Additive Identity
___7) 2(1 + 3) = 2(1) + 2(3)
B) Multiplicative Inverse
___8) 2 + 1 = 1 + 2
C) Distributive Property
___9) (1 + 2) + 3 = 1 + (2 + 3)
D) Associative Property
___10) 2 + 0 = 2
E) Commutative Property
2
Properties & Closure Homework
For each example below, write the name of the property that is being illustrated.
Choose your answers from this list:
Commutative Property
Associative Property
Distributive Property
Additive Identity
Additive Inverse
Multiplicative Identity
Multiplicative Inverse
1) (2 + 3) + 7 = 2 + (3 +7)
_____________________
2) (15)(25) = (25)(15)
_____________________
3) 8(x + 4) = 8x + 32
_____________________
4) 14 + 27 = 27 + 14
_____________________
5) 6(3  5) = (6  3)5
_____________________
6) a(b + c) = ab + ac
_____________________
7) 5 + (  5) = 0
_____________________
1
 =1
5
8) 5 
_____________________
 183(1) =  183
_____________________
10) 1926 + 0 = 1926
_____________________
3
9)
4
5
_____ 1) Which is an example of the commutative property of addition?
(1) (2 + 3) + 4 = 2 + (3 + 4)
(3) 2(3 + 4) = 2 · 3 + 2 · 4
(2) 2 + 3 = 3 + 2
(4) 4 + 0 = 4
_____ 2) Which is an illustration of the associative property?
(1) ab + 0 = ab
(3) a + b = b + a
(2) a(b + c) = ab + ac
(4) a + (b + c) = (a + b) + c
3) Name the additive identity element. __________
4) Explain the meaning of “additive identity.” ________________________________
__________________________________________________________________________
5) Name the multiplicative identity element. __________
6) Explain the meaning of “multiplicative identity.” ___________________________
__________________________________________________________________________
_____ 7) Give a word that has the same meaning as “multiplicative inverse.”
(1) identity
(2) reciprocal
(3) opposite
_____ 8) Give a word that has the same meaning as “additive inverse.”
(1) identity
(2) reciprocal
(3) opposite
Matching: Write the letter of the property that justifies each statement below.
_____9) 5 + (3 + 2) = (5 + 3) + 2
A) Additive Identity
_____ 10) - 5 + 0 = - 5
B) Additive Inverse
_____ 11) 10(8) = 8(10)
C) Associative Property
_____ 12) - 4 + 4 = 0
D) Commutative Property
_____ 13) 3(x + 2) = 3x + 6
E) Distributive Property
_____14) - 8(1) = - 8
F) Multiplicative Identity
æ1 ö
_____ 15) 3 çç ÷
÷= 1
çè3 ø÷
G) Multiplicative Inverse
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Commutative
Associative
“Change Order”
“Parenthesis Occupants Change”
a+b=b+a
a•b=b•a
(a + b) + c = a + (b + c)
(a • b) • c = a • (b • c)
Distributive
“multiply through by the
outside number”
4(x + 8) = 4x + 32
Additive Identity
Multiplicative Identity
“zero plus any number =
that number”
“any number times one =
that number”
a+0=a
a•1=a
Additive Inverse
Multiplicative Inverse
“adding the opposite =
zero”
“any number multiplied by
the reciprocal = 1”
1
a 1
a
a + -a = 0
7