Properties of Operations with Real Numbers
Property
Commutative
Property
Associative
Property
Key Idea
changing the order of numbers
when adding or multiplying
changing the numbers grouped
in parenthesis without changing
the order when adding or
multiplying
Poem
Commute the numbers
from place to place. A
change in order is what
you’ll face.
Associate the numbers,
that’s all you have to do.
Putting them in groups is
entirely up to you.
Picture
Examples
Commutative Property of Addition
8 + 3 + 10 = 10 + 8 + 3
Commutative Property of Multiplication
3•x•4=3•4•x
Associative Property of Addition
(x + 2) + 9 = x + (2 + 9)
Associative Property of Multiplication
(9 ● 8 ) ● 2 = 9 ● (8 ● 2)
Additive Identity Property
4+0=4
Identity
Property
keeping the number the same;
maintaining the number’s
identity
I look in the mirror and what
do I see? The number’s
identity staring back at me.
(4x + 2 ) + 0 = 4x + 2
Multiplicative Identity Property
x●1=x
(6x + 3) ● 1 = 6x + 3
Additive Inverse Property
7 + -7 = 0
x + -x = 0
Inverse
Property
undoing or canceling a number
Upside down or inside out,
that’s what inverse is all
about.
Multiplicative Inverse Property
4•
•
Distributive
Property
rewriting an expression using
multiplication; distribute (pass
out/share) the number outside
the parentheses to all numbers
inside
Distribute through the
group, each number gets a
turn. Multiply then
add/subtract to show what
you have learned.
Multiplicative
Property of
Zero
multiplying by zero; any number
times zero equals zero
Multiply by zero and see
what you get. The answer
is zero – on that you can
bet.
=1
=1
2(3 + 4) = 2(3) + 2(4)
8(x - 5) = 8(x) - 8(5)
3●0=0
m●0=0
(8x + 1) ● 0 = 0
Identifying Properties
Identify the property shown by each equation
1. 12 · 32 + 12 · 8 = 12 · (32 + 8)
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2. -19 + 19 = 0
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3. 1 · (4 + 6) = (4 + 6) · 1
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4. 97 + (3 + 35) = (97 + 3) + 35
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5. 0 + (7 · 3) = (7 · 3)
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6. 97 + 3 + 35 = 97 + 35 + 3
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7. -8 · 1 = 1 · (-8)
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8. 0 + (10 · 3) = (10 · 3) + 0
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9. 20 · (5 · 87) = (20 · 5) · 87
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10. (7 · 4) + 6 = 6 + (7 · 4)
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11. (13 + 9) + 0 = 13 + 9
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12. 48 · 5 + 48 · 15 = 48(5 + 15)
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13. (4 + 14) · 1 = (4 + 14)
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14. (12+ 3) + 5 = (3 + 12) + 5
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15. 20 · (5 · 87) = 20 · (87 · 5)
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