7cmp06se.AN4.qxd 5/18/06 8:09 AM Page 66
Name_________________________________________________________HR#____________ The Distributive Property and Subtraction
4.3
The rectangles in Problem 4.2 illustrate an important property of numbers
and operations called the Distributive Property. This property shows that
multiplication distributes over addition. When you think about a
3
multiplication problem like 512 3 5 as 500 3 5 + 12 3 5, or 12 3 5 4 as
3
12 3 5 + 12 3 4 , you are using the Distributive Property.
Getting Ready for Problem 4.3
You can use the Distributive Property to rewrite an expression as one that
is easier to calculate or gives new information. You can do this in two ways.
1. Suppose an expression is written as the product of two factors, one of
which is a sum. You can use the Distributive Property to multiply one
factor by each number in the second factor. This is called expanding
the expression.
-3 ? (4 + 8) = -3 ? 4 + (-3) ? 8
With a variable: -2 ? (x + 6) = -2x + (-2) ? 6
2. Suppose an expression is written as a sum and the numbers have a
common factor. You can use the Distributive Property to rewrite the
expression as the common factor multiplied by the sum. This is called
factoring the expression.
5 ? 4 + 5 ? 7 = 5 ? (4 + 7)
With a variable:
•
8 ? 2 + 8x = 8 ? (2 + x)
Do you think the Distributive Property can be used to expand or
factor expressions with subtraction? Explain your reasoning.
66
Accentuate the Negative
7cmp06se.AN4.qxd 5/18/06 8:09 AM Page 67
7cmp06se.AN4.qxd 5/18/06 8:09 AM Page 67
mp06se.AN4.qxd
5/18/06 8:09 AM Page 67
Problem 4.3 The Distributive Property and Subtraction
A. Use the Distributive Property to expand each expression.
Problem 4.3 The Distributive Property and Subtraction
1. 5 ? (3 + 2)
2. 5 ? [3 + (-2)]
3. 5 the
? (3Distributive
- 2)
4. 5 ?each
[3 -expression.
(-2)]
A. Use
Property to expand
The
Property
and Subtraction
Problem
4.3
5.8:09
For
parts
(1)–(4),
find the value of
the
7cmp06se.AN4.qxd
5/18/061.
Page
67 Distributive
(3
+
2)
2.
5 ? expression.
[3 + (-2)]
5 ? AM
6. 5
Does
Distributive Property seem
hold
for subtraction?
3.
? (3 the
- 2)
5 to
? [3
- (-2)]
A. Use
Distributive
Property to expand4.each
expression.
theExplain.
For
the+expression.
1. 55.
? (3
+parts
2) (1)–(4), find the value
2. 5of? [3
(-2)]
mp06se.AN4.qxd 5/18/06
8:09
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PageDistributive
67
B.
Use
Property
to
expand
each
expression.
The
Distributive
Property
and Subtraction
Problem
4.3
6.
Does
the
Distributive
Property
seem
to
hold
3. 5 ? (3 - 2)
4. 5 ? [3 - (-2)]for subtraction?
1. Explain.
-5 ? (3 + 2)
2. -5 ? (3 - 2)
7cmp06se.AN4.qxd 5/18/06
8:09
AM Distributive
Page
67 findProperty
5.
parts
(1)
– (4),
the valuetoofexpand
the expression.
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Use
the
each expression.
3. -5
? [3
+ (-2)]
4. -5
? [3 expression.
- (-2)]
The Distributive
Property
and Subtraction
Problem
4.3
B.
Use
the
Distributive
Property
to
expand
each
6. Does
Property seem2.to5hold
subtraction?
1. 5 ?the
(3 Distributive
+ 2)
? [3 for
+ (-2)]
7cmp06se.AN4.qxd 5/18/065.8:09
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ForAM?parts
find the value of
1. -5
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2)
2. the
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Explain.
? (3Distributive
- 2)
4. 5 ?each
[3 -expression.
(-2)]
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Use5 the
Property to expand
6.
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negative
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expand an
3.
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?
[3
+
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4.
-5
?
[3
Use5.theFor
B.
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parts
(1)–(4),
find the value of
Distributive
Property
and Subtraction
Problem
1. 5expression.
? (3
+The
2)
2. the
5 ? expression.
[3 + (-2)]
4.3
5.
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1. -5
2. seem
-5
--2)(-2)]
6.
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to
for subtraction?
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5? (3
? (3+the
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4. 5? (3
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factored
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the
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to
expand
each
expression.
The
Distributive
Property
and Subtraction
4.3
Explain
how
to
distribute
a
negative
number
to expand
an
Explain.
3.
-5
?
[3
+
(-2)]
4.
-5
?
[3
(-2)]
parts (1)–(4), find the value of the expression.
5. For
1.
6
?
2
+
6
?
3
2.
6
?
2
6
?
3
1.
5 ? (3expression.
+ 2)(1)– (4), find the value 2.
5 ? [3
+ (-2)]
5.
parts
the
expression.
B. For
Use
the Distributive
Distributive
Property
toofexpand
expand
each
expression.
A.
Use
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to
each
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The
Distributive
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and
Subtraction
Problem
6.
Does
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hold
for
subtraction?
4.3
3.
-6
?
2
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(-6)
?
3
4.
-6
?
2
(-6)
?3
3.
5
?
(3
2)
4.
5
?
[3
(-2)]
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in
factored
form.
6. Explain
number
toexpand
an
1. 5-5
? how
(3
2.
? (3
2)
Explain.
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? (3
++
2)to2)distribute a negative2.
5-5
? [3
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5. parts
5x
8x6– ?(4),
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-3x
4x
5.
For
(1)
findProperty
the valuetoofexpand
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expression.
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each
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1.
6
?
2
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3
2.
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?
2
6
?
3
B. 3.
3.
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?
[3
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4.
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5
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4.
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Use the Distributive Property to expand each expression.
7. each
to ?factor
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Does
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to5-6
hold
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5Explain
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2)
? expression.
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?expression
2++how
(-6)
4.
2subtraction.
(-6) ? 3
C. 6.
Write
in3find
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5.
For
parts
(1)–(4),
the
value
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the
5.
For
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–
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find
the
value
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the
expression.
1.
-5
?
(3
+
2)
2.
-5
?
(3
2)
Explain.
1.
D. 63.
Three
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buys
55x+
? (3
2) are going hiking. Lisa
4.
5-3x
? [36(-2)]
5.
8x
6.
4x
?
2
6
?
3
2.
6
?
2
?
3
6.
Explain
how
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a negative
number
to(-2)]
expand
an
6.
Does
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subtraction?
3.
-5
? [3
+
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4.each
-5
?hold
[3 -for
2
bottles
of
water
and
3
packs
of
trail
mix
B. Use
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Distributive
Property
to
expand
expression.
3.
-6
5.
For
parts
(1)
–
(4),
find
the
value
of
the
expression.
7. Explain.
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how? to
? 2 + (-6)
3 factor an expression
4. -6 ?with
2 - subtraction.
(-6) ? 3
expression.
1.
-5
5.
For
parts
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each
of+
them.
?
(3
2)
2.
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?
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2) for subtraction?
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Does
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6.expand
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factored
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Lisa
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each expression.
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?Explain.
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4.trail
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6
?
2
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?
3
2.
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?
2
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?
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1. -5
2)
2. -5
? (3 - 2)
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expression.
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parts
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the valueto
ofexpand
the expression.
each
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Use
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each expression.
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3.
-6
2 a+
(-6)
? sentence
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-6 ?? [3
23.
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1. -5
Can
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?
(3
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?
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2 bottles
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and
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found
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- 4x
5.
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expression.
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6
?
2
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6
?
3
2.
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?
2
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?
3
expression.
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customers
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each3.of-5
them.
? [3another
+ (-2)]number sentence 4.
-5 ? [3 - (-2)]
3. Explain
Write
to find
7.
how
to
an expression
with
subtraction.
6.
Explain
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distribute
atonegative
number
to
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an
3.
-6
?
2
+
(-6)
?
3
4.
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?
2
(-6)
?
3
2.
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a
number
sentence
show
how
C. Write
each
expression
in
factored
form.
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express
checkout
5. expression.
For
(1)– (4),the
find
the value
of the expression.
the
total
number
of
items.
D. lane
Three
friends
are
hiking.
Lisa
buys
1.
you
the going
total
of items.
for
customers
with number
15
or fewer
5.
5x
6.
66.
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2.
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? 2-3x
- 6? 34xto expand an
to
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Chan
a roll3inof
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towels
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and window cleaner
2Mr.
bottles
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water and
packs
of trail
mix
for $1.19
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each
expression
factored
form.
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Write
another
number
to ?find
2. -6
Write
a+number
sentence
tosentence
show
how
7.
how
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factor
an
expression
with
subtraction.
expression.
3.
?
2
(-6)
?
3
4.
-6
2
(-6)
?
3 items.
for $2.69.
In his state there is a 4% sales tax on these
each
of them.
1. 6the
? 2total
+ the
6 number
? total
3
of items.
you
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number
of items.2. 6 ? 2 - 6 ? 3
5.
8xeach
6. form.
-3x
- 4x
D.
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going
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Lisa
buys
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Write
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factored
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she
go are
through
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his
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3.
3.
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?2+
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? roll
3 sentence
4. -6for
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? window
3
Write
another
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to
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E.
Mr.
Chan
bought
a
of
kitchen
towels
and
cleaner
2 bottles
water
andan
3 expression
packs
of fewer
trail
mix
to
with
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+of
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7. Explain
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$2.69.
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sales
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5.
- them.
8xIn his state
-3x
each
of
2.
Write
a
number
sentence
to
show
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3.
-6
?
2
+
(-6)
?
3
4.
-6
? 2 - and
(-6)then
? 3 compute the
D. Three
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Lisa
buys
3.
Suppose
you
add
the
prices
of the for
two
items
7.
cleaner
1.
What
ishow
his
total
bill?
E. Mr.
Chan
bought
athrough
roll
of
kitchen
towels
$1.19
and window
Explain
to
factor
anexpress
expression
with
subtraction.
1.
Can
she
gothe
the
checkout
you
found
total
number
of
items.
2
bottles
of
water
and
3
packs
of
trail
mix
for
tax.Your
friend
findsisthe
taxsales
on 6.
each
item
and
then adds the two
5. 5x
8x
-3x
4x items.
for $2.69.
In
his
state
there
a
4%
tax
on
these
lane
for
customers
with
15
or
fewer
items?
2.
Write
a number
sentence
show
how
you found the total bill.
D. Three
friends
are going
hiking.
Lisa
buys
each
ofWrite
them.
3.
another
number
sentence
to
find
together.
Which
method
isto
better?
Explain.
7.
Explain
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to
factor
an
expression
with
subtraction.
1. What
is his
total
22.
water
and
3express
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trail
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Write
athrough
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sentence
toof
show
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Suppose
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add
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two for
items and then compute the
the
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1. Can
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goof
the
checkout
Homework
starts
on
page
D. lane
Three
are
going
hiking.
Lisa
buys
each
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them.friend
2.
Write
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number
sentence
to
show
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you
found
total
bill.
tax.
Your
finds
the
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each
item
andthe
then
adds
the two
you
found
the
total
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items.
for
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with
15
or
fewer
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E. 2
Mr.
Chan
bought
a
roll
of
kitchen
towels
for
$1.19 and window cleaner
bottles
of
water
and
3
packs
of
trail
mix
for
together.
Which
method
is the
better?
1.
she
through
the
express
checkout
3. Write
Suppose
you
add
the
prices
of
two
items
andthese
then items.
compute the
3. Can
Write
another
number
to Explain.
find
2.
a number
sentence
tosentence
show
how
for
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Ingo
his
state
there
is
a 4%
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for
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oreach
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Your
friend
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and then adds the two
the
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you
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total
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items.
Homework
starts
on
page
69.
1. What
isthe
his
total
bill?
7cmp06se.AN4.qxd 5/18/06 8:09 AM Page 67
5. Explain
For
(1)–(4),
find theavalue
of the
expression.
Explain.
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the parts
Distributive
Property
tonegative
expand
each
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6.
how
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find the
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of expand
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4. expression.
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5 the
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-5
?
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5.1.For
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Explain.
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5
?
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2)
4.
5
?
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1.
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2.
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1.
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3
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3.
?
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4.
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Explain.
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find the value
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-5
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-6
?
2
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3
4.
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5.
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expand
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?3.2For
+
6
?
3
6
?
2
6
?
3
6.
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negative
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-5
?
[3
+
(-2)]
4.
-5
?
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?Explain
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3expand
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expression.
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the Distributive
Property
to expandwith
eachsubtraction.
expression.
expression.
7. Explain.
Explain
to factor
expression
expression
inan
factored
form.
3.Write
-5
?each
[3 +how
(-2)]
4. -5
?4x[3 - (-2)]
8x
6.
-3x
5.C.5x
6. -5
Explain
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- to
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B.
Use
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Distributive
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expand
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C. 5.
Write
each
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inhiking.
factored
form.
D.
Three
going
Lisa
buys
1.For
6 ? parts
2 + 6to
? are
3 (4),
2.
6subtraction.
?expression.
2-6?3
(1)–
find
the
value
of
the
7. Explain
how
factor
an expression
with
expression.
3.
-5
(-2)]
4.
-5
2 bottles
of+
3 packs of trail
mix
1.
-5
(3
+
2)3to and
2.
-5
(3
2)3expand
6-6
? ??2?[3
+
6water
?(-6)
2.
? ??for
2?[3
6to(-2)]
?(-6)
3.friends
2are
+
?hiking.
3 in factored
4. 6-6
2? 3 an
6.1.
Explain
how
distribute
a negative
number
D. Three
going
Lisa
buys
C. 5.
Write
each
expression
form.
each
of?parts
them.
For
(1)–
(4),
find
the
value
of
the
expression.
3.
-5
[3
+
(-2)]
4.
-5
?
[3
(-2)]
3.
-6
?
2
(-6)
?
3
4.
-6
?
2
(-6)
?
3
2 bottles
5.expression.
5xof
8x6 ?and
6.for
-3x
4x
water
3 packs of trail mix
1.
6
?
2
+
3
2.
6
?
2
6
?
3
1.
Can
she
go
through
the
express
checkout
6.
Explain
to distribute
avalue
negative
number
to expand an
5.
parts
(1)–(4),
find
the
of
expression.
each
5.
5x
8xhow
6.the
-3x
4x
ofFor
them.
C. Write
each
expression
in factored
form.
7.
Explain
how
to
factor
an
expression
with
subtraction.
expression.
lane
for
customers
with
15
or
fewer
items?
3. -6 ? 2 + (-6) ? 3
4. -6 ? 2 - (-6) ? 3
6.
how
to
a negative
7.6Explain
Explain
how
to distribute
factor
an expression
with
1.
? 2friends
+ 6through
? 3are
2.buys
6 number
? 2 -subtraction.
6 ?to3 expand an
1.D.Can
she
go
the express
checkout
Three
going
hiking.
Lisa
2.
Write
a expression
number sentence
to show
5. expression.
5x each
- 8x
6.how
-3x - 4x
C. Write
in factored
form.
for?friends
customers
or fewer
items?
D.lane
are total
going
hiking.
Lisa
3.Three
-6
2 of
+ water
(-6)
?with
3 315
4.buys
-6
2 - (-6) ? 3
2 bottles
and
packs
of
trail
mix?for
you
found
the
number
of
items.
7.
Explain
how
to
factor
an
expression
1.
6
?
2
+
6
?
3
2. 6 with
? 2 -subtraction.
6?3
C.
Write
each
expression
in
factored
form.
2each
bottles
of water
and 3 packs
of how
trail
mix
for
ofnumber
them.
2. Write
asentence
to show
5.
5x
8x
6.
-3x
4x
3. -6
Write
another
sentence4.
to -6
find? 2 - (-6) ? 3
3.
2+
+6 (-6)
?number
3
D.you
Three
going
hiking.
Lisa2.
buys
each
them.
the
number
of
items.
1.
6found
? of
2?friends
? total
3are
6with
?26?3
1.Explain
Can
she
go
through
the
express
checkout
7.
how
to factor
an expression
subtraction.
the
total
number
of items.
2
bottles
of
water
and
3
packs
of
trail
mix
for
5.
5x
8x
6.
-3x
4x
lane
for
with
15 or
items?
1. -6
Can
go
through
the express
checkout
3. Write
another
number
tofewer
find
3.
? 2she
+ customers
(-6)
? 3 sentence
4.
-6
? 2 - (-6)
?3
D.
friends
are
going
hiking.
Lisa
buys
each
of
them.
E. Three
Mr.
Chan
bought
a
roll
of
kitchen
towels
forsubtraction.
$1.19 and window cleaner
lane
for
customers
with
orshow
fewerhow
items?
7.2.Explain
how
to
factor
an 15
expression
with
total
number
of
items.
the
Write
a
number
sentence
to
5x
- 8x
6.
-3x
25.
bottles
ofInwater
and there
3 packs
trail
mix
for-on4xthese items.
for
$2.69.
histhrough
state
aof4%
sales
tax
1.
Can
she
go
the is
express
checkout
you
found
the
total
number
of
items.
2.
Write
a
number
sentence
to
show
how
D. each
Three
friends
hiking.
Lisa buys
E. Mr.
Chan
aare
roll
of kitchen
towels
forwith
$1.19
and window cleaner
ofbought
them.
7.
togoing
factor
an 15
expression
subtraction.
lane
for
customers
with
oroffewer
items?
1.Explain
What
ishow
his
total
bill?
you
found
the
total
number
items.
2
bottles
of
water
and
3
packs
of
trail
mix
for
3. Write
another
number
sentence
to
Inshe
his
state
there
is a express
4% salescheckout
taxfind
on these
items.
for
$2.69.
go
through
the
D. 1.
Three
are
going
hiking.
buys
2.Can
Write
number
sentence
toLisa
show
how
2.
Write
aaanother
number
you found the total bill.
each
offriends
them.
1.
the
total
number
of
items.
3.
Write
number
sentence
to
find
What
is his
bill?
lane
for
customers
with
15 or
fewer
items?
2 bottles
oftotal
water
and
3 number
packs
ofof
trail
mix
for
you
found
the
total
items.
the
total
number
of
items.
3.
Suppose
you
add
the
prices
of the
two for
items
andand
then
compute
the
1.
Can
she
go
through
the
express
checkout
E.
Mr.
Chan
bought
a
roll
of
kitchen
towels
$1.19
window
each
of
them.
2. Write
a
number
sentence
to
show
how
you
found
the total
bill. cleaner
2.
Write
a
number
sentence
to
show
how
3. lane
Write
another
number
sentence
to
find
Your
friend
the
ontowels
each
item
and
then
adds
the cleaner
two
forIn
customers
with
15
or
fewer
items?
fortax.
$2.69.
his
state
there
istax
aof
4%
sales
tax
these
E. Mr.
Chan
bought
a finds
roll
of
kitchen
foron
$1.19
anditems.
window
found
the
total
number
items.
1. you
Can
she
go
through
the
express
checkout
3. Suppose
you
add
the
prices
of
the
two
items
and
then
compute
the
the
total
number
of
items.
together.
Which
method
istoabetter?
Explain.
Write
number
sentence
show
how
2.for
$2.69.aisIn
histotal
state
there is
4% sales
tax on these items.
1.Write
What
his
bill?
lane
for
customers
with
15
or
fewer
items?
3.
another
number
sentence
to
find
tax.
Your
friend
finds
the
tax
on
each
item
and then adds the two
E. Mr.
you
found
the
total
number
of
items.
Chan
bought
a
roll
of
kitchen
towels
for $1.19 and window cleaner
1.
What
is
his total
bill?
Homework
starts
on
page
69.
the
total
number
of
items.
2.
Write
a
sentence
to
show
how
you found the total bill.
together.
Which
method
is
better?
Explain.
2.
Write
a
number
sentence
to
show
how
forWrite
$2.69.another
In his state
there
is a 4% to
sales
3.
number
sentence
findtax on these items.
2.
Write
a number
sentence
toof
show
you
found
the
total
bill.
found
the add
number
items.
3. you
Suppose
you
the
of
the how
twofor
items
and
compute
the
E. Homework
Mr.
Chan
bought
atotal
roll
ofprices
kitchen
towels
$1.19
andthen
window
cleaner
on
page
69.
total
number
of
items.
1. the
What
isstarts
his total
bill?
tax. Your
friend
finds
the
each
item
then
adds
the two
3. Write
Suppose
you
add
the prices
ofonthe
two
items
and
then
compute
the
for
$2.69.
In his
state
there
is atax
4%
sales
tax
onand
these
items.
3.
another
number
sentence
to
find
2. Write
a
number
sentence
to
show
how
you
found
the
total
bill.
E. Mr.
Chan
bought
a
roll
of
kitchen
towels
for
$1.19
and
window
cleaner
together.
Which
method
is
better?
Explain.
tax.total
Your
friend
finds
the tax on each item and then adds the two
the
number
of items.
1. What
is his
total bill?
for
In his
state
there
isisa better?
4%
sales
taxitems
on these
items.compute the 3. $2.69.
Suppose
you
addmethod
the prices
of the
two
and then
together.
Which
Explain.
Investigation
4 Properties
of Operations
67
E. 2.
Homework
starts
on
page
69.towels
Write
a
number
sentence
to
show
how
you
found
the
total
bill.
Mr.
Chan
bought
a
roll
of
kitchen
for
$1.19
and
cleaner
tax. Your
friend
finds the tax on each item and thenwindow
adds
the
two
1.
What
is
his
total
bill?
forSuppose
$2.69.
Inyou
his
state
there
is a better?
4%
sales
tax
on these
items.
Homework
starts
onprices
page
69.
3.
add method
the
of
the
two
items
and then
compute
the
together.
Which
Explain.
Investigation
4 total
Properties
67
2. Write
a number
sentence is
to show how
you found the
bill. of Operations
1. tax.
Your
friend
What
is his
totalfinds
bill? the tax on each item and then adds the two 3. together.
Suppose
you
addmethod
the
of
items and then compute the
Homework
starts
onprices
page
69.the two
Which
isto
better?
Explain.
2.
Write
a
number
sentence
show
how
youand
found
the
total
bill.
tax.
Your
friend
finds
the
tax
on
each
item
then
adds
the
two
Investigation
4
Properties
67
starts
on prices
page
69.
3. Homework
Suppose you
addmethod
the
of the two
items and then compute the of Operations
together.
Which
is better?
Explain.
Properties
of Operations
67
tax. Your friend finds the tax on each itemInvestigation
and then adds4the
two
Homework starts on page 69.
together. Which method is better? Explain.
Homework starts on page 69.
Investigation 4 Properties of Operations
67
Investigation 4 Properties of Operations
67
Investigation 4 Properties of Operations
67
Investigation 4 Properties of Operations
67