CHAPTER 6
B
You will need
The Distributive
Property in Reverse
• a calculator
c GOAL
Factor and simplify expressions using the distributive property in reverse.
Learn about the Math
Arithmetic expressions can often be evaluated or simplified in
multiple ways. The distributive property makes it possible to
write the product of one number and the sum or difference
of 2 numbers as the sum or difference of two products.
Using the distributive property in reverse allows the sum or
difference of two products to be written as a product of one
number and the sum or difference of two numbers. Because
the number and the sum or difference are factors of the
resulting product, using the distributive property in this way
is called factoring.
Tom and Maggie are helping to plan the school carnival. They
each bought four prizes for the carnival games. Tom spent
$6.50 on each of his prizes, while Maggie spent $5.50 on
each of the prizes she purchased. To find the total amount
spent on prizes, Tom wrote down the following expression on
a sheet of paper: 4 3 $6.50 1 4 3 $5.50. Maggie’s math
class has been learning how to factor, and she noticed that
Tom could use factoring to help him simplify this expression.
factoring
writing an expression
as a product
can Tom use factoring to simplify the following
? How
expression: 4 3 $6.50 1 4 3 $5.50?
A. Remember the distributive property states that for all
numbers a, b, and c, a(b 1 c) 5 a 3 b 1 a 3 c. If this is
true, then we know that the opposite must also be true.
Complete the following statement, using the distributive
property in reverse: a 3 b 1 a 3 c 5 ?
B. Fill in the numbers from the problem for the variables in
this expression: a 3 b 1 a 3 c.
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6B The Distributive Property in Reverse
1
C. Now, rewrite the expression in this form: a(b 1 c), filling
in the numbers from the problem for each variable.
D. Perform the addition inside the brackets, followed by
multiplication, to determine the total spent.
You have just simplified Tom’s expression, using the
distributive property in reverse.
Reflecting
1. The distributive property in reverse with addition states
that for all numbers a, b, and c, a 3 b 1 a 3 c 5 ?
2. Write the statement for the distributive property in reverse
with subtraction.
3. When might you choose to factor an expression using the
distributive property in reverse?
Work with the Math
Example 1: Using the distributive property
in reverse with addition
Use the distributive property in reverse to evaluate the expression
8 3 2 1 8 3 9.
Kurt’s Solution:
8321839
2
This expression is in the same form as
a 3 b 1 a 3 c. Since 8 is a factor in each term of
the sum, I realize that it can be factored using the
distributive property in reverse.
5 8(2 1 9)
According to the distributive property, I can factor
out the 8 and rewrite the expression in the form
a(b 1 c), where a 5 8, b 5 2, and c 5 9.
5 8(11)
5 88
Now I simplify the expression inside the brackets
and then multiply.
Nelson Mathematics Elementary Year Two, Cycle One
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Copyright © 2009 by Nelson Education Ltd.
Example 2: Using the distributive property
in reverse with subtraction
Use the distributive property in reverse to simplify the expression 6 3 4 2 6 3 2.
Chandra’s Solution:
6342632
A
This expression is in the same form as
a 3 b 2 a 3 c. Since 6 is a factor in each term
of the difference, I realize that it can be factored
using the distributive property in reverse.
5 6(4 2 2)
According to the distributive property, I can factor
out the 6 and rewrite the expression in the form
a(b 2 c), where a 5 6, b 5 4, and c 5 2.
5 6(2)
5 12
Now I simplify the expression inside the brackets
and then multiply.
Checking
4. Use the distributive property in reverse
to simplify the following expression:
26 3 4 1 3 3 26.
5. Use the distributive property in reverse
to factor the following expression:
7 3 8 2 8 3 5.
B
Practising
6. Identify if the distributive property in
reverse was used on each of the
following equations.
3 3 8 1 3 3 3 5 3 (8 1 3)
7 2 4 3 7 2 2 5 7 (4 2 2)
9 3 4 1 3 3 5 5 9 (4 1 5)
22 3 4 1 7 3 (22) 5
22 (4 1 7)
e) 3 3 (25) 2 2 3 (25) 5
25 (3 2 2)
a)
b)
c)
d)
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Reproduction permitted for classrooms
7. Factor each expression, using the
distributive property in reverse.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
4361433
2382234
(22) 3 7 1 (22) 3 6
8352832
(27) 3 4 2 (27) 3 3
9 3 (24) 1 9 3 3
6341632
7382237
6352533
10 3 3 2 10 3 2
22 3 11 1 22 3 9
42 3 3 1 42 3 2
36 3 3 2 1 3 36
30 3 8 1 30 3 9
65 3 40 1 10 3 65
2 3 21 1 21 3 8
6B The Distributive Property in Reverse
3
8. Factor each expression, using the
distributive property in reverse. Then
evaluate.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
o)
p)
7321739
2361237
4332734
6352632
9361339
10 3 9 2 10 3 4
15 3 2 1 15 3 3
32 3 10 2 32 3 9
9371239
81 3 11 2 81 3 4
44 3 8 2 6 3 44
4351533
14 3 2 1 14 3 6
61 3 2 1 4 3 61
12 3 8 2 8 3 3
25 3 6 1 25 3 7
Nelson Mathematics Elementary Year Two, Cycle One
Extending
10. Factor each expression, using the
distributive property in reverse. Then
evaluate.
a)
b)
c)
d)
e)
93619341932
11 3 8 2 11 3 5 2 11 3 1
16 3 4 1 16 3 9 2 16 3 2
80 3 10 1 8 3 80 2 80 3 9
100 3 35 2 100 3 19 2 2 3 100
11. Determine if the two expressions
below have the same value. If so, state
the value. If they do not, state the
value of each expression.
a) 4 3 6 1 5 3 4 2 4 3 2
b) 6 3 7 2 5 3 6 1 3 3 2 1 6 3 3
9. Mr. Malloy cashes a check at the bank
and gets 10 of each of the following:
$1 coins, $2 coins, $5 bills, $10 bills,
and $50 bills. Write two expressions
for the amount of money that
Mr. Malloy received and then
determine the amount.
4
C
12. Use the distributive property in reverse
to factor the expression
a3 1 ab 1 ac2 1 a2d.
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