 # Lesson 8.7 Multiplication of Fractions and Whole Numbers

```Objective
To provide experience finding the product of a whole
number and a fraction.
1
materials
Teaching the Lesson
Key Activities
Students use area models and a fraction multiplication algorithm to find the products of
whole numbers and fractions.
Key Concepts and Skills
Щ— Math Journal 2, pp. 268вЂ“270
б­њ
Щ— slates
вЂў Use given denominators to rename numbers as fractions.
[Number and Numeration Goal 5]
вЂў Find fractions of a set.
[Number and Numeration Goal 5]
вЂў Use an area model and a fraction multiplication algorithm to find fraction-by-wholenumber products.
[Operations and Computation Goal 5]
2
materials
Ongoing Learning & Practice
Students practice using order of operations by playing Name That Number.
Students practice and maintain skills through Math Boxes and Study Link activities.
Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1.
[Number and Numeration Goal 5]
3
materials
Differentiation Options
Students rename whole
numbers as fractions and
find common denominators.
ENRICHMENT
Students explore the use of
the commutative property to
simplify finding the product
of fractions.
Щ— Math Journal 2, p. 271
Щ— Student Reference Book, p. 325
Щ— Study Link Master (Math Masters,
p. 235)
Щ— number cards 0вЂ“9 (4 of each from
the Everything Math Deck, if
available)
Щ— calculator (optional)
EXTRA PRACTICE
Students practice
multiplying fractions and
whole numbers.
Щ— Teaching Master (Math Masters,
p. 236)
Щ— 5-Minute Math, pp. 23 and 185
Щ— slates
Technology
Assessment Management System
Math Boxes, Problem 1
See the iTLG.
654
Unit 8 Fractions and Ratios
Getting Started
Mental Math and Reflexes
Math Message
Have students solve fraction-of problems. Remind them to think of
when multiplying fractions.
Complete journal page 268.
1
1 1
бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ
5
2 10
1
1 1
бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ
8
2 16
2 1
1
бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ
14 14
2
3
1 1
бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ бЋЏбЋЏ
14 3 14
5
2
бЋЏбЋЏ of 12 6бЋЏбЋЏ
9
3
6 2
бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ 1
4 3
2
3 1
бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ
3
4 2
4
4 16
бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ
5
5 25
8 2 16
бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ бЋЏбЋЏ
9 3 27
б­њ
share their solution strategies for Problems 10
and 11.
1 Teaching the Lesson
б­¤ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 268)
Ask students whether they remember problems of this type from
earlier grades. Point out that in the fifth grade they still solve the
problems, but they also write the number models for calculations
that solve the problems.
Review the problems by having volunteers rewrite each problem in
the form бЋЏabбЋЏ вЂ« ШЎвЂ¬бЋЏdcбЋЏ on the board and then write the product.
1.
1
бЋЏбЋЏ
6
2. a.
b.
c.
3. a.
b.
of 1 бЋЏ16бЋЏ вЂ« ШЎвЂ¬бЋЏ11бЋЏ П­ бЋЏ16бЋЏ
3
бЋЏбЋЏ
4
2
бЋЏбЋЏ
3
2
бЋЏбЋЏ
2
1
бЋЏбЋЏ
4
1
бЋЏбЋЏ
8
4. a.
1
of 1 бЋЏ34бЋЏ вЂ« ШЎвЂ¬бЋЏ1бЋЏ П­ бЋЏ34бЋЏ
b.
of 1 бЋЏ23бЋЏ вЂ« ШЎвЂ¬бЋЏ11бЋЏ П­ бЋЏ23бЋЏ
c.
of 1 бЋЏ22бЋЏ вЂ« ШЎвЂ¬бЋЏ11бЋЏ П­ бЋЏ22бЋЏ П­ 1
5.
5
бЋЏбЋЏ
6
1
бЋЏбЋЏ
2
1
бЋЏбЋЏ
8
1
бЋЏбЋЏ
2
of бЋЏ14бЋЏ
of бЋЏ12бЋЏ
of бЋЏ18бЋЏ
1
бЋЏбЋЏ
2
1
бЋЏбЋЏ
8
1
бЋЏбЋЏ
2
вЂ« ШЎвЂ¬бЋЏ14бЋЏ П­ бЋЏ18бЋЏ
вЂ« ШЎвЂ¬бЋЏ12бЋЏ П­ бЋЏ11бЋЏ
6
вЂ« ШЎвЂ¬бЋЏ18бЋЏ П­ бЋЏ11бЋЏ
6
of 12 бЋЏ56бЋЏ вЂ« ШЎвЂ¬бЋЏ112бЋЏ П­ бЋЏ660бЋЏ П­ 10
6
of 16 бЋЏ14бЋЏ вЂ« ШЎвЂ¬бЋЏ116бЋЏ П­ бЋЏ14бЋЏ
П­4
of 16
1
бЋЏбЋЏ
8
вЂ«ШЎвЂ¬
16
бЋЏбЋЏ
1
П­
16
бЋЏбЋЏ
8
Student Page
Date
Time
LESSON
A Blast from the Past
8 7
б­њ
1. From Kindergarten Everyday Mathematics:
П­2
This slice of pizza is what
fraction of the whole pizza?
1
бЋЏбЋЏ
6
2. From First Grade Everyday Mathematics:
б­¤ Using an Area Model to
Write a fraction in each part of the diagrams below. Then color the figures as directed.
WHOLE-CLASS
ACTIVITY
Represent the Product of a
Fraction and a Whole Number
a.
b.
1
4
1
4
1
4
1
4
c.
1
3
3
4
1
3
1
3
2
3
Color бЋЏбЋЏ.
1 1
2 2
2
2
Color бЋЏбЋЏ.
Color бЋЏбЋЏ.
3. From Second Grade Everyday Mathematics:
a.
b.
(Math Journal 2, p. 269)
Ask students to solve the following problem on their slates:
1
2
бЋЏбЋЏ
3
вЂ«ШЎвЂ¬2П­?
Ask volunteers to show on the board or Class Data Pad how an
area model might be used to represent this problem. The basic
idea is that there are several wholes, each of which is divided into
fractional parts. Summarize studentsвЂ™ presentations using the
steps on the next page:
1
4. From Third Grade Everyday Mathematics:
1
1
a. бЋЏбЋЏ of бЋЏбЋЏ П­
2
4
1
бЋЏбЋЏ
8
1
1
b. бЋЏбЋЏ of бЋЏбЋЏ П­
8
2
1
бЋЏбЋЏ
16
1
1
c. бЋЏбЋЏ of бЋЏбЋЏ П­
2
8
1
бЋЏбЋЏ
16
5. From Fourth Grade Everyday Mathematics:
5
6
Cross out бЋЏбЋЏ of the dimes.
268
Math Journal 2, p. 268
Lesson 8 7
б­њ
655
Student Page
Date
Time
LESSON
Area Models
8 7
б­њ
Draw an area model for each product. Then write the product as a fraction or as a mixed number.
2
3
Example: бЋЏбЋЏ вЂ« ШЎвЂ¬2 П­
4
бЋЏбЋЏ
1
1. бЋЏбЋЏ вЂ« ШЎвЂ¬4 П­ 3
3
4
бЋЏбЋЏ,
3
or 1бЋЏ13бЋЏ
, or 1бЋЏ13бЋЏ
2. Note that the denominator of the fraction is 3. Divide both
rectangles into thirds.
3
бЋЏбЋЏ
4
1
2. бЋЏбЋЏ вЂ« ШЎвЂ¬3 П­
4
бЋЏбЋЏ
3
3. 2 вЂ« ШЎвЂ¬бЋЏбЋЏ П­ 5
5
6
, or 1бЋЏ15бЋЏ
9
, or 1бЋЏ18бЋЏ
бЋЏбЋЏ
3
4. бЋЏбЋЏ вЂ« ШЎвЂ¬3 П­ 8
8
1. Draw a number of rectangles equal to the whole number. In
this example, the whole is 2.
3. Note that the numerator of the fraction is 2. Shade бЋЏ23бЋЏ of each
rectangle.
269
Math Journal 2, p. 269
In each rectangle, there are 3 parts; 2 of them are shaded.
In the 2 rectangles, there are 4 shaded thirds altogether.
So бЋЏ23бЋЏ вЂ« ШЎвЂ¬2 П­ бЋЏ43бЋЏ or 1бЋЏ13бЋЏ.
Assign the journal page. When most students have finished,
bring the class together to discuss answers.
б­¤ Using an Algorithm to Multiply
a Fraction and a Whole Number
Student Page
Date
(Math Journal 2, p. 270)
Time
LESSON
Using the Fraction Multiplication Algorithm
8 7
б­њ
An Algorithm for Fraction Multiplication
a
бЋЏбЋЏ
b
c
d
aвЂ«ШЎвЂ¬c
bвЂ«ШЎвЂ¬d
вЂ« ШЎвЂ¬бЋЏбЋЏ П­ бЋЏбЋЏ
The denominator of the product is the product of the denominators, and
the numerator of the product is the product of the numerators.
2
3
Example: бЋЏбЋЏ вЂ« ШЎвЂ¬2
2
бЋЏбЋЏ
3
вЂ«ШЎвЂ¬2
2
2
3
1
2вЂ«ШЎвЂ¬2
3вЂ«ШЎвЂ¬1
4
1
бЋЏбЋЏ, or 1бЋЏбЋЏ
3
3
2
1
П­ бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏ бЋЏ
Think of 2 as бЋЏбЋЏ.
П­ бЋЏбЋЏ
Apply the algorithm.
П­
Calculate the numerator and denominator.
18
1
бЋЏбЋЏ, or 4бЋЏбЋЏ
4
2
15
1
бЋЏбЋЏ, or 1бЋЏбЋЏ
3
2
бЋЏбЋЏ вЂ« ШЎвЂ¬5 П­ 10
21
бЋЏбЋЏ ,
8
24
бЋЏбЋЏ ,
4
6 вЂ« ШЎвЂ¬бЋЏбЋЏ П­ 5
7
2. бЋЏбЋЏ вЂ« ШЎвЂ¬3 П­
8
3.
4.
10
5. Use the given rule to complete the table.
in ( )
Rule
П­
3
5
Вє бЋЏбЋЏ
1
бЋЏбЋЏ
2
2
4
бЋЏбЋЏ
5
3
бЋЏбЋЏ
4
3
5
2
or 2бЋЏ58бЋЏ
or 4бЋЏ45бЋЏ
Assign the journal page. Circulate and assist.
6. What is the rule for the table below?
out ( )
3
бЋЏбЋЏ
10
6
1
бЋЏбЋЏ, or 1бЋЏбЋЏ
5
5
12
бЋЏбЋЏ
25
9
бЋЏбЋЏ
20
9
4
бЋЏбЋЏ, or 1бЋЏбЋЏ
5
5
in ( ) out ( )
Rule
П­
Вє
1
бЋЏбЋЏ
2
270
Math Journal 2, p. 270
656
Refer students to the top of journal page 270, and ask how this
algorithm could be used to multiply a fraction and a whole number
such as бЋЏ23бЋЏ вЂ« ШЎвЂ¬2. Rewrite the whole number as a fraction. Remind
students that any number can be thought of as a fraction with a
denominator of 1.
Ask a volunteer to demonstrate using the algorithm to solve бЋЏ3бЋЏ вЂ« ШЎвЂ¬2.
2вЂ«ШЎвЂ¬2
2
2
4
1
бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ, or 1бЋЏбЋЏ
3
1
3
3
3вЂ«ШЎвЂ¬1
Use the fraction multiplication algorithm to calculate the following products.
3
1. бЋЏбЋЏ вЂ« ШЎвЂ¬6 П­
4
PARTNER
ACTIVITY
Unit 8 Fractions and Ratios
2
бЋЏбЋЏ
3
3
бЋЏбЋЏ
4
2
бЋЏбЋЏ
6
3
бЋЏбЋЏ
8
7
бЋЏбЋЏ
8
7
бЋЏбЋЏ
16
3
1бЋЏ1бЋЏ
2
When students have completed the page, ask what patterns they
notice about the numerators and denominators when multiplying
fractions by whole numbers. The denominators in the products are
always the same as the denominator of the fraction factor. The
numerator is the product of the whole number and the numerator
of the fraction factor.
Student Page
Emphasize that, when rewriting the whole number as a fraction,
the denominator is always 1. Ask students what true statement
they can make about multiplying by 1. Any number times 1 is
itself. Accordingly, the patterns for multiplying fractions by whole
aвЂ«ШЎвЂ¬c
бЋЏ
numbers can be represented as бЋЏabбЋЏ вЂ« ШЎвЂ¬c П­ бЋЏ
b .
Date
Time
LESSON
8 7
б­њ
а¬™
Math Boxes
2. Write true or false for each number sentence.
1. Complete.
1
a. бЋЏбЋЏ П­
5
2
b. бЋЏбЋЏ П­
3
5
c. бЋЏбЋЏ П­
8
4
d. бЋЏбЋЏ П­
7
4
20
6
15
15
0
false
1
5
1
1
1
5
2
6
3
3
2
6
40
d. 16 ПЄ (4 П© 8 ПЄ 2) / 2 П­ 3
32
П­
1
П­ 2,160
25
П­
24
42
b. (2 вЂ« ШЎвЂ¬10 ) П© (1 вЂ« ШЎвЂ¬10 ) П© (6 вЂ« ШЎвЂ¬10 )
10
П­
true
a. 5 вЂ«( ШЎвЂ¬6 П© 3) П­ (5 вЂ« ШЎвЂ¬6) П© (5 вЂ« ШЎвЂ¬3)
30
2
9
24
6
П­
108 109
56
e. 10 П­ 1 billion
6
true
false
false
222 223
3. On the grid, draw each animal whose
6
location is given below.
2 Ongoing Learning & Practice
Lake
5
a. A bird in C2.
4
b. A fish in D6.
3
c. A turtle in E3.
2
d. A snake in F1.
б­¤ Playing Name That Number
1
PARTNER
ACTIVITY
e. A frog in F4.
208
A
4. Draw an isosceles triangle.
(Student Reference Book, p. 325)
B
C
D
E
F
5. The shapes below represent geometric
solids. Name the solids.
Write a definition of an
isosceles triangle.
An isosceles
triangle has
two sides and
two angles
that are
equal.
Students play Name That Number to practice writing number
sentences using order of operations. Encourage them to find
number sentences that use all five numbers. Students can use
numbers as exponents or to form fractions.
a.
cone
b.
triangular
prism
147вЂ“149
144
271
Math Journal 2, p. 271
б­¤ Math Boxes 8 7
INDEPENDENT
ACTIVITY
б­њ
(Math Journal 2, p. 271)
Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lesson 8-5. The skills in Problems 4 and 5
preview Unit 9 content.
Ongoing Assessment:
Recognizing Student Achievement
Math Boxes
Problem 1
а¬™
Use Math Boxes, Problem 1 to assess studentsвЂ™ understanding of converting
fractions to decimals and percents. Have students write a response to the
following: Convert the fractions in Math Boxes, Problem 1 to the decimal and the
percent equivalents, and explain your solution strategy. Students are making
adequate progress if their conversions are correct and their writing demonstrates
an understanding of the role of the numerator and the denominator. Some
students might refer to the fact that the fractions are equivalent; therefore, they
have the same decimal and percent equivalents.
NOTE Students may use calculators. If they do, remind them to explain the part
of the conversion process the calculator is performing.
[Number and Numeration Goal 5]
Name
87
б­њ
Date
Use the fraction multiplication algorithm to calculate the following products.
45
бЋЏбЋЏ
3
,
or 15
5
1. бЋЏбЋЏ
3
Вє9П­
1
3. бЋЏбЋЏ
8
Вє5П­
5
бЋЏбЋЏ
8
70
бЋЏбЋЏ,
Вє 14 П­ 6
or 11бЋЏ23бЋЏ
5
5. бЋЏбЋЏ
6
7.
3
2. бЋЏбЋЏ
8
in (б®Ђ)
б­ќП­б®ЂВє4
2
бЋЏбЋЏ
3
4
бЋЏбЋЏ
5
б­њ
8
бЋЏбЋЏ
9
INDEPENDENT
ACTIVITY
(Math Masters, p. 235)
5
бЋЏбЋЏ
4
7
бЋЏбЋЏ
3
8.
1
б­ќ П­ б®Ђ Вє бЋЏбЋЏ
4
9.
4.
20 Вє бЋЏ3бЋЏ П­
6.
27 Вє бЋЏ2бЋЏ П­
4
9
36
1
бЋЏбЋЏ or 4 бЋЏбЋЏ
2
8
60
бЋЏбЋЏ, or 15
4
54
бЋЏбЋЏ, or 6
9
,
73
out (б­ќ)
8
бЋЏбЋЏ
3
16
бЋЏбЋЏ
5
, or 2бЋЏ23бЋЏ
, or 3 бЋЏ15бЋЏ
32
бЋЏбЋЏ, or 3бЋЏ5бЋЏ
9
9
20
бЋЏбЋЏ, or 5
4
28
бЋЏбЋЏ, or 9бЋЏ1бЋЏ
3
3
What is the rule for the table below?
Rule
Home Connection Students solve problems to find a
fraction of a whole number and a fraction of a fraction.
They solve вЂњWhatвЂ™s My Rule?вЂќ problems and make a
function table for fraction multiplication.
Вє 12 П­
Use the given rule to complete the table.
Rule
Time
Multiplying Fractions and Whole Numbers
in (б®Ђ)
out (б­ќ)
2
1
бЋЏбЋЏ
2
3
3
бЋЏбЋЏ
4
5
бЋЏбЋЏ
6
5
бЋЏбЋЏ
24
2
бЋЏбЋЏ
3
1
бЋЏбЋЏ
6
Make and complete your own вЂњWhatвЂ™s My Rule?вЂќ table on the back of
Math Masters, p. 235
Lesson 8 7
б­њ
657
3 Differentiation Options
б­¤ Writing Whole Numbers
SMALL-GROUP
ACTIVITY
15вЂ“30 Min
as Fractions
To reinforce studentsвЂ™ understanding of whole numbers written as
fractions, guide them through the following activity:
б­џ Remind students that any number can be thought of as a
fraction with a denominator of 1. Write the examples on the
236
0.5
бЋЏ
бЋЏбЋЏ
Examples: 3 П­ бЋЏ31бЋЏ, 236 П­ бЋЏ
1 , and 0.5 П­ 1
Ask students why this is true. The denominator represents how
many parts it takes to make a whole. If it takes only 1 part, then
the numerator represents wholes.
б­џ When applying a multiplication algorithm to problems of the
form бЋЏabбЋЏ вЂ« ШЎвЂ¬n, where one factor is a fraction and the other factor is
a whole number, think of the whole number as бЋЏn1бЋЏ.
б­џ Ask students to write each number as a fraction on their slates.
Then repeat the numbers, and ask students to rename each as
a fraction with a denominator of 2.
5 бЋЏ51бЋЏ; бЋЏ12бЋЏ0
3.5 7
бЋЏ бЋЏбЋЏ
3.5 бЋЏ
1 ; 2
0 20
2 вЂ« ШЎвЂ¬5 бЋЏ11бЋЏ
; бЋЏ2бЋЏ
7 бЋЏ71бЋЏ; бЋЏ12бЋЏ4
1 бЋЏ11бЋЏ; бЋЏ22бЋЏ
100% бЋЏ11бЋЏ; бЋЏ22бЋЏ
140 280
бЋЏ бЋЏбЋЏ
140 бЋЏ
1 ; 2
0.5 1
бЋЏ бЋЏбЋЏ
0.5 бЋЏ
1 ; 2
23 бЋЏ81бЋЏ; бЋЏ12бЋЏ6
ENRICHMENT
Teaching Master
Name
Date
LESSON
87
б­њ
Time
An Algorithm for Fraction Multiplication
c
d
aВєc
bВєd
Вє бЋЏбЋЏ П­ бЋЏбЋЏ
The denominator of the product is the product of the factor denominators, and
the numerator of the product is the product of the factor numerators.
aВєc
bВєd
cВєa
dВєb
The commutative property lets us write бЋЏбЋЏ as бЋЏбЋЏ. Study the examples.
6
112 112
8
14
2
7 Вє 16
П­бЋЏ
бЋЏ П­ бЋЏбЋЏ; бЋЏбЋЏ П¬ бЋЏбЋЏ П­ бЋЏбЋЏ, or бЋЏбЋЏ
Example 1: бЋЏ7бЋЏ Вє бЋЏ1бЋЏ
8 Вє 21
8
21
168 168
8
21
3
6
16
2
1
2
7
7 Вє 16
2Вє1
Example 2: бЋЏ7бЋЏ Вє бЋЏ1бЋЏ
П­бЋЏ
бЋЏ П­ бЋЏбЋЏ Вє бЋЏбЋЏ П­ бЋЏбЋЏ Вє бЋЏбЋЏ П­ бЋЏбЋЏ П­ бЋЏбЋЏ
8
1.
8 Вє 21
21
8
21
1
3
1Вє3
3
1
2
Example 3: 78 Вє 16
21
1
1Вє2
1Вє3
2
3
Describe the similarities and differences between Examples 2 and 3.
Use what you have discovered to solve the following problems. Show your work.
14
3. бЋЏбЋЏ
60
2
2
бЋЏ
Вє бЋЏ1бЋЏ
П­ бЋЏ
15
21
EXTRA PRACTICE
б­¤ 5-Minute Math
SMALL-GROUP
ACTIVITY
5вЂ“15 Min
3
Both examples have the same factors and
products. Example 3 has fewer steps than
Example 2 because the fractions are
reduced without rearranging them first.
36
4. бЋЏбЋЏ
88
3
3
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16
72
25
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54
Math Masters, p. 236
658
To extend studentsвЂ™ understanding of fraction multiplication and
lowest terms, have students explore the process of reducing factors
in fraction multiplication problems. When students have completed
the Math Masters page, discuss any difficulties or curiosities
they encountered.
Describe the similarities and differences between Examples 1 and 2.
Both examples have the same factors and
products. Example 1 is renamed in simplest
form after multiplying. Example 2 is renamed
in simplest form before multiplying.
2.
15вЂ“30 Min
(Math Masters, p. 236)
Simplifying Fraction Factors
a
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b
б­¤ Simplifying Fraction Factors
PARTNER
ACTIVITY
Unit 8 Fractions and Ratios
10
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27
45
To offer students more experience with fractions and whole
numbers, see 5-Minute Math, pages 23 and 185.
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