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```CC Algebra I
3- Properties of Real Numbers
Name: __________________________
Number_____
Date: ______________________
Do Now:
Suzy draws a picture of squares to represent the product 3 × 4. Ben moves to the end of
the table and says, “From my new seat, your picture looks like the product
4 × 3.” What picture might Suzy have drawn? Why would Ben see it differently from his
viewpoint?
Property (a, b, and c are
real numbers, variables,
or algebraic expressions)
Examples
3 4  43
ab ba
Meaning
Verbal Hints
The order of the “Commute – to get up
numbers change
and move to a new
but the sum is the
location: switch
same
places”
3  (4  5)  (3  4)  5
a  (b  c)  (a  b)  c
The grouping of
numbers change
but the sum is the
same
“regroup – elements
do not physically
move, they simply
group with a new
friend”
a0 a
40  4
**the identity element
When zero is
number, the
result is the
number itself
“the value that
returns the input
unchanged’
number and its
opposite, the
result is zero
“the value that brings
you back to the
identity element
a  (a)  0
4  ( 4)  0
**the result is the
identity element**
Properties of Multiplication
Property (a, b, and c are
real numbers, variables,
or algebraic
expressions)
Examples
Meaning
Verbal Hints
3 4  43
ab  ba
The order of the
numbers change
but the product is
the same
“Commute – to get up
and move to a new
location: switch
places”
3  (4  5)  (3  4)  5
a  (b  c)  (a  b)  c
The grouping of
numbers change
but the product is
the same
“regroup – elements
do not physically
move, they simply
group with a new
friend”
a 1  a
4 1  4
**the identity element
for multiplication is
ONE**
When ONE is
multiplied to any
number, the
result is the
number itself
“the value that
returns the input
unchanged’
When you
multiply a number
and its reciprocal,
the result is ONE
“the value that brings
you back to the
identity element
under multiplication
5(4  x)  20  5 x
 3( y  5)  3 y  15
The number
outside the
parentheses is
multiplied with
each number
inside.
“multiplication
distributes across
subtraction”
ab a b
 
c
c c
ab a b
 
c
c c
Each term in the
numerator is
divided by the
term in the
denominator
“dividing both sides of
an equation by the
same non-zero value
will not change the
truth value of the
equation”
1
a ( ) 1
a
1
4( ) 1
4
**the result is the
identity element**
Match each example to its corresponding property.
_____ 1) (2  b)  9  2  (b  9)
A. Multiplicative Identity
_____ 2) r (1)  r
B. Distributive Property
_____ 3)
5𝑥+2𝑥
3
=
5𝑥
3
+
2𝑥
3
_____ 4)  3  3  0
D. Associative Property of Multiplication
_____ 5) 6  (4  x)  6  ( x  4)
E. Commutative Property of Multiplication
_____ 6) (ab)c  a(bc)
F. Distributive Property
_____ 7)
3
3
x0 x
2
2
1
_____ 8) a   1
a
_____9) 3(4 x)  (4 x)3
I. Multiplicative Inverse
_____10) 7(c  d )  7c  7d
11) When solving the equation 4(3x 2  2)  9  8x 2  7 Emily wrote 4(3x 2  2)  8x 2  16
as her first step. Which property justifies Emily’s first step?
(3) multiplication property of equality
(4) distributive property of multiplication over addition
12) Use these abbreviations for the properties of real numbers and complete the flow diagram.
C for the commutative property of addition
C x for the commutative property of multiplication
A for the associative property of addition
Ax for the associative property of multiplication
13) Draw a flow diagram, and use both the commutative and associative properties to change
the outcome. You must use one of each property for a minimum of two changes.
a) (xy)z
b) (x + y) + z
Each expression has been simplified or solved one step at a time. Next to each step, identify
the property, transformation or simplification used in the step.
14)
4  5( x  7) _________________________
4  (5 x  35) _________________________
(4  35)  5 x _________________________
39  5x _________________________
15)
x  19  23 _________________________
x  19  (19)  23  (19) _________________________
x  0  23  (19) _________________________
x  4 _________________________
16)
Fill in the blanks of this proof. 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) _________________________
w2  w  5  2(w  5) _________________________
w 2  5w  2(w  5) _________________________
w 2  5w  2w  10 _________________________
w 2  (5w  2w)  10 _________________________
w 2  7 w  10
CC Algebra I
Period ____
Name:__________________________
Date:___________________________
Homework -Number Properties
1) The following portion of a flow diagram shows that the expression ab + cd is equivalent to
the expression dc + ba. Fill in each circle with the appropriate symbol: C for the
commutative property of addition, C x for the commutative property of multiplication, A for
the associative property of addition, Ax for the associative property of multiplication
2) Fill in the blanks of this proof. Write either “Commutative Property”, “Associative Property”, or
“Distributive Property” in each blank.
( x  4)( x  3) = ( x  4) x  ( x  4)  3 __________________________
x( x  4)  ( x  4)  3 _________________________
x( x  4)  3( x  4) ____________________________
x 2  4 x  3x  12 _____________________________
x 2  (4 x  3x)  12 ____________________________
x 2  1x  12 _________________________________
Properties of Real Number Word Bank
Multiplicative Identity
Distributive Property
Associative Property of Multiplication
Commutative Property of Multiplication
Distributive Property
Multiplicative Inverse
Properties of Real Number Word Bank
Multiplicative Identity
Distributive Property
Associative Property of Multiplication
Commutative Property of Multiplication
Distributive Property
Multiplicative Inverse