Objective
1
To introduce multiplication with mixed numbers.
materials
Teaching the Lesson
Key Activities
Students review conversions from mixed numbers to fractions and from fractions to mixed
numbers. Then they multiply mixed numbers by applying the conversions and by using the
partial-products method.
Key Concepts and Skills
• Convert between fractions and mixed numbers. [Number and Numeration Goal 5]
• Multiply mixed numbers. [Operations and Computation Goal 5]
• Use the partial-products algorithm to multiply whole numbers, fractions, and mixed numbers.
ٗ Math Journal 2, pp. 272–274
Щ— Study Link 8 7
б­њ
Щ— Teaching Aid Master (Math Masters,
414; optional)
Щ— slates
See Advance Preparation
[Operations and Computation Goal 5]
Ongoing Assessment: Informing Instruction See page 661.
Ongoing Assessment: Recognizing Student Achievement Use journal page 273.
[Operations and Computation Goal 5]
2
Ongoing Learning & Practice
Students practice using unit fractions to find a fraction of a number.
Students practice and maintain skills through Math Boxes and Study Link activities.
3
materials
Щ— Math Journal 2, pp. 275 and 276
Щ— Study Link Master (Math Masters,
p. 237)
materials
Differentiation Options
READINESS
EXTRA PRACTICE
Students compare and order
improper fractions.
Students practice converting between
fractions, decimals, and percents by playing
Frac-Tac-Toe.
Additional Information
Advance Preparation For Part 1, draw several blank “What’s My Rule?” rule boxes and
tables on the board to use with the Study Link 8 7 Follow-up.
б­њ
Щ— Student Reference Book,
pp. 309–311
Щ— Game Masters (Math Masters,
pp. 472–484)
ٗ number cards 0–10 (4 of each from
the Everything Math Deck, if
available); counters; slates
Щ— calculator (optional)
Technology
Assessment Management System
Journal page 273
See the iTLG.
Lesson 8 8
б­њ
659
Getting Started
Mental Math and Reflexes
Math Message
Have students write each mixed number
as a fraction. Suggestions:
Complete journal
page 272.
2 5
1бЋЏ3бЋЏ бЋЏ3бЋЏ
7 34
3бЋЏ9бЋЏ бЋЏ9бЋЏ
4 46
6бЋЏ7бЋЏ бЋЏ7бЋЏ
7 87
8бЋЏ10бЋЏ бЋЏ10бЋЏ
4 29
5бЋЏ5бЋЏ бЋЏ5бЋЏ
5 29
3бЋЏ8бЋЏ бЋЏ8бЋЏ
Study Link 8 7
Follow-Up
б­њ
Have partners compare answers and
resolve differences. Ask volunteers to
write the incomplete version of their
“What’s My Rule?” table for the class
to solve.
1 Teaching the Lesson
б­¤ Math Message Follow-Up
WHOLE-CLASS
DISCUSSION
(Math Journal 2, p. 272)
Ask students why the hexagons in the last row of the example on
the journal page are divided into sixths. Sample answer: To add
the 3 in 3 П© бЋЏ56бЋЏ, you need a common denominator. A simple way is
to think of each whole as бЋЏ66бЋЏ. Ask volunteers to share their solution
strategies for Problems 1–8.
б­¤ Multiplying with
WHOLE-CLASS
ACTIVITY
Mixed Numbers
Ask students how they would use the partial-products method to
calculate 6 ‫ ء‬4ᎏ35ᎏ. Discuss student responses as you summarize the
following strategy:
Student Page
Date
Time
LESSON
Review Converting Fractions to Mixed Numbers
8 8
б­њ
1. Think of 4бЋЏ35бЋЏ as 4 П© бЋЏ35бЋЏ.
Math Message
You know that fractions larger than 1 can be written in several ways.
Whole
Rule
Example:
hexagon
If a
is worth 1,
what is
worth?
5
5
5
The mixed-number name is 3 бЋЏ6бЋЏ (3 бЋЏ6бЋЏ means 3 П© бЋЏ6бЋЏ).
23
The fraction name is бЋЏ6бЋЏ. Think sixths:
5
5
23
3 бЋЏ6бЋЏ, 3 П© бЋЏ6бЋЏ, and бЋЏ6бЋЏ are different names for the same number.
Write the following mixed numbers as fractions.
13
бЋЏ
бЋЏ
5
бЋЏ5бЋЏ
3
3
1. 2 бЋЏ5бЋЏ П­
2
3. 1 бЋЏ3бЋЏ П­
3бЋЏ
9
бЋЏ
8
7
2. 4 бЋЏ8бЋЏ П­
6
4. 3 бЋЏ4бЋЏ П­
18
бЋЏ
бЋЏ
4
, or
9
бЋЏбЋЏ
2
Write the following fractions as mixed or whole numbers.
1
2 бЋЏ3бЋЏ
7
5. бЋЏ3бЋЏ П­
18
7. бЋЏ4бЋЏ
П­
6
6. бЋЏ1бЋЏ П­
бЋЏ2бЋЏ
4
4 , or 4
бЋЏ1бЋЏ
2
9
8. бЋЏ3бЋЏ
П­
6
3
Add.
7
7
9. 2 П© бЋЏ8бЋЏ П­
3
11. 3 П© бЋЏ5бЋЏ П­
2 бЋЏ8бЋЏ
3
бЋЏ3бЋЏ
5
3
3
10. 1 П© бЋЏ4бЋЏ П­
1
12. 6 П© 2 бЋЏ3бЋЏ П­
1бЋЏ4бЋЏ
1
8 бЋЏ3бЋЏ
272
Math Journal 2, p. 272
660
б­џ Calculate the partial products and add.
Unit 8 Fractions and Ratios
6 ‫ ء‬4ᎏ35ᎏ ϭ 6 ‫( ء‬4 ϩ ᎏ35ᎏ)
2. Write the problem as the sum of
partial products.
ϭ (6 ‫ ء‬4) ϩ (6 ‫ ء‬ᎏ35ᎏ)
3. Calculate the partial products.
П­ 24 П© бЋЏ15бЋЏ8
4. Convert бЋЏ15бЋЏ8 to a mixed number.
П­ 24 П© 3бЋЏ35бЋЏ
5. Add.
П­ 27бЋЏ35бЋЏ
Ongoing Assessment: Informing Instruction
Watch for students who have difficulty organizing partial products when
multiplying two mixed numbers. Show them the diagram below, and have them
write the partial products in a column.
3
4
3
2‫ء‬3 ϭ 6 ϭ 6
2в€—3
2
2в€—
2
бЋЏбЋЏ
3
3
4
2 ‫ ء‬ᎏ34ᎏ ϭ
2
бЋЏбЋЏ
3
2
3
2
3
2
3
в€—3
в€—
‫ء‬3 ϭ
‫ ء‬ᎏ43ᎏ ϭ
6
бЋЏбЋЏ
3
6
бЋЏбЋЏ
4
6
бЋЏбЋЏ
12
П­ 2
П­ 1бЋЏ12бЋЏ
П­
3
4
1
бЋЏбЋЏ
2
9бЋЏ22бЋЏ,
or 10
Ask students how they might use improper fractions to solve the
problem. Again, discuss student responses as you summarize the
following strategy:
б­џ Convert whole numbers and mixed numbers to improper
fractions.
1. Think of 6 as бЋЏ61бЋЏ and 4бЋЏ35бЋЏ as бЋЏ25бЋЏ3.
6 ‫ ء‬4ᎏ35ᎏ
2. Rewrite the problem as fraction
multiplication.
ϭ ᎏ61ᎏ ‫ ء‬ᎏ253ᎏ
6‫ء‬23
3. Use a fraction multiplication
algorithm.
бЋЏ
П­бЋЏ
1‫ء‬5
4. Multiply.
138
бЋЏ
П­бЋЏ
5
5. Simplify the answer by converting
138
бЋЏбЋЏ to a mixed number.
5
П­ 27бЋЏ35бЋЏ
Ask students to suggest advantages and disadvantages for each
method. Expect a variety of responses. Sample answers: The
partial-products method lets you work with smaller numbers but
has more calculations. The improper-fraction method lets you
multiply fractions where one of the denominators will be one, but
you have a larger number to divide to simplify the answer.
Ask students to work through two or three additional examples,
using either of the above strategies or others of their own
choosing. After each problem, ask volunteers to share their
solution strategies. Suggestions:
в—Џ
4 ‫ ء‬ᎏ35ᎏ 2ᎏ25ᎏ
в—Џ
2ᎏ14ᎏ ‫ ء‬ᎏ23ᎏ 1ᎏ12ᎏ
в—Џ
2ᎏ23ᎏ ‫ ء‬3 8
Student Page
Date
Time
LESSON
Multiplying Fractions and Mixed Numbers
8 8
б­њ
Using Partial Products
Example 2:
Example 1:
1
1
1
1
2 ᎏ3ᎏ ‫ ء‬2 ᎏ2ᎏ ϭ (2 ϩ ᎏ3ᎏ) ‫( ء‬2 ϩ ᎏ2ᎏ)
1
2
1
2
3 ᎏ4ᎏ ‫ ء‬ᎏ5ᎏ ϭ.(3 ϩ ᎏ4ᎏ) ‫ ء‬ᎏ5ᎏ
2
6
2‫ء‬2ϭ
4
3 ‫ ء‬ᎏ5ᎏ ϭ.ᎏ5ᎏ ϭ
2 ‫ ء‬ᎏ2ᎏ ϭ
1
1
1
бЋЏбЋЏ
4
1
бЋЏбЋЏ
3
‫ء‬2ϭ
2
бЋЏбЋЏ
3
1
бЋЏбЋЏ
3
1
бЋЏбЋЏ
2
‫ء‬
П­
1
1 бЋЏ5бЋЏ
2
1
2
‫ ء‬ᎏ5ᎏ ϭ.ᎏ20ᎏ ϭ ϩ ᎏ1ᎏ
0
3
1 бЋЏ10бЋЏ
1
П© бЋЏ6бЋЏ
5
5 бЋЏ6бЋЏ
Converting Mixed Numbers to Fractions
Example 3:
1
1
Example 4:
7
5
2 ᎏ3ᎏ ‫ ء‬2 ᎏ2ᎏ ϭ ᎏ3ᎏ ‫ ء‬ᎏ2ᎏ
1
2
13
2
3 ᎏ4ᎏ ‫ ء‬ᎏ5ᎏ ϭ ᎏ4ᎏ ‫ ء‬ᎏ5ᎏ
35
5
П­ бЋЏ6бЋЏ П­ 5 бЋЏ6бЋЏ
6
3
26
П­ бЋЏ20бЋЏ П­ 1 бЋЏ20бЋЏ П­ 1 бЋЏ10бЋЏ
Solve the following fraction and mixed-number multiplication problems.
7
77
43
бЋЏбЋЏ, or 7 бЋЏбЋЏ
бЋЏбЋЏ, or
1
1
3
1
10
10
8
1. 3 ᎏ2ᎏ ‫ ء‬2 ᎏ5ᎏ ϭ
2. 10 ᎏ4ᎏ ‫ ء‬ᎏ2ᎏ ϭ
3. The back face of a
calculator has an
area of about
11
16 бЋЏ6бЋЏ4
in2.
4. The area of this
sheet of notebook
paper is about
5
ACMECALC INC.
Model# JETSciCalc
Serial# 143
58 "
84
ଙ
1
10 2 "
in2.
7
28 "
8"
5. The area of this
computer disk
is about
5
13 бЋЏ1бЋЏ2
in2.
3
5бЋЏ8бЋЏ
6. The area of this
5
36 "
1
32 "
flag is about
8бЋЏ165бЋЏ, or 8бЋЏ25бЋЏ yd2.
7. Is the flag’s area greater or less than that of your desk?
1
2 3 yd
3
3 5 yd
Answers vary.
273
Math Journal 2, p. 273
Lesson 8 8
б­њ
661
Student Page
Date
Time
LESSON
Track Records on the Moon and the Planets
8 8
б­њ
б­¤ Multiplying Fractions and
PARTNER
ACTIVITY
Mixed Numbers
Every moon and planet in our solar system pulls objects toward it with a force
called gravity.
2
In a recent Olympic games, the winning high jump was 7 feet 8 inches, or 7бЋЏ3бЋЏ feet. The
winning pole vault was 19 feet. Suppose that the Olympics were held on Earth’s Moon,
or on Jupiter, Mars, or Venus. What height might we expect for a winning high jump
or a winning pole vault?
(Math Journal 2, pp. 273 and 274)
Assign both journal pages. Encourage students to consider the
numbers in each problem and then to use the method that is most
efficient for that problem. Circulate and assist.
1. On the Moon, one could jump about 6 times as high as on Earth.
What would be the height of the winning …
46
high jump? About
feet
114
pole vault? About
feet
3
2. On Jupiter, one could jump about бЋЏ8бЋЏ as high as on Earth.
What would be the height of the winning …
69
7
бЋЏбЋЏ
бЋЏбЋЏ
high jump? About 24 , or 2 8 feet
pole vault? About
3. On Mars, one could jump about
57
бЋЏбЋЏ,
8
or 7бЋЏ18бЋЏ
feet
2
2 бЋЏ3бЋЏ
times as high as on Earth.
What would be the height of the winning …
152
184
4
бЋЏбЋЏ,
бЋЏбЋЏ, or 20 бЋЏбЋЏ
9 feet
high jump? About 9
pole vault? About 3
or 50 бЋЏ23бЋЏ feet
Ongoing Assessment:
Recognizing Student Achievement
1
4. On Venus, one could jump about 1 бЋЏ7бЋЏ times as high as on Earth.
What would be the height of the winning …
152
184
16
бЋЏбЋЏ,
бЋЏбЋЏ, or 8 бЋЏбЋЏ
21 feet
high jump? About 21
pole vault? About 7
or 21 бЋЏ57бЋЏ feet
Journal
Page 273
Problem 5
ଙ
Use journal page 273, Problem 5 to assess students’ understanding
of multiplication with mixed numbers. Have students complete an Exit Slip (Math
Masters, 414) for the following: Explain how you solved Problem 5 on journal
page 273. Students are making adequate progress if they correctly reference
using partial products, improper fractions, or a method of their own.
5. Is Jupiter’s pull of gravity stronger or weaker than Earth’s? Explain your reasoning.
Sample answer: Because you can’t jump as high on
Jupiter as you can on Earth, the gravity pulling you
back on Jupiter must be stronger.
Try This
6. The winning pole-vault height given above was rounded to the nearest whole
1
number. The actual winning height was 19 feet бЋЏ4бЋЏ inch. If you used this actual
[Operations and Computation Goal 5]
measurement, about how many feet high would the winning jump be …
17
бЋЏ
114бЋЏ18бЋЏ
7бЋЏ
128
on the Moon?
on Jupiter?
50 бЋЏ11бЋЏ38
21бЋЏ3412бЋЏ
on Mars?
on Venus?
274
Math Journal 2, p. 274
2 Ongoing Learning & Practice
б­¤ Using Unit Fractions to Find
INDEPENDENT
ACTIVITY
a Fraction of a Number
(Math Journal 2, p. 275)
Students practice using unit fractions to find the fraction
of a number.
б­¤ Math Boxes 8 8
Student Page
Date
б­њ
Time
LESSON
(Math Journal 2, p. 276)
Finding Fractions of a Number
8 8
б­њ
One way to find a fraction of a number is to use a unit fraction. A unit fraction is a fraction
with 1 in the numerator. You can also use a diagram to help you understand the problem.
7
1
бЋЏбЋЏ
8
of 32 is 4. So
Mixed Practice Math Boxes in this lesson are paired
with Math Boxes in Lesson 8-6. The skill in Problem 5
previews Unit 9 content.
32
Example: What is бЋЏ8бЋЏ of 32?
7
бЋЏбЋЏ
8
of 32 is 7 ‫ ء‬4 ϭ 28.
?
Solve.
1
1. бЋЏ5бЋЏ of 75 П­
1
4. бЋЏ8бЋЏ of 120 П­
15
15
2
2. бЋЏ5бЋЏ of 75 П­
3
5. бЋЏ8бЋЏ of 120 П­
30
45
4
3. бЋЏ5бЋЏ of 75 П­
5
6. бЋЏ8бЋЏ of 120 П­
60
75
Solve Problems 7–18. They come from a math book that was published in 1904.
1
2
First think of бЋЏ3бЋЏ of each of these numbers, and then state бЋЏ3бЋЏ of each.
7. 9
10. 3
6
2
8. 6
11. 21
4
14
9. 12
12. 30
8
20
1
3
First think of бЋЏ4бЋЏ of each of these numbers, and then state бЋЏ4бЋЏ of each.
13. 32
16. 24
24
18
14. 40
17. 20
30
15
15.
12
18.
28
9
21
19. Lydia has 7 pages of a 12-page song memorized. Has she memorized more
2
than бЋЏ3бЋЏ of the song?
No
1
20. A CD that normally sells for $15 is on sale for бЋЏ3бЋЏ off. What is the sale price?
$10
1
21. Christine bought a coat for бЋЏ4бЋЏ off the regular price. She saved $20. What did she
pay for the coat?
$60
22. Seri bought 12 avocados on sale for $8.28. What is the unit price, the cost for 1 avocado?
$0.69
275
Math Journal 2, p. 275
662
INDEPENDENT
ACTIVITY
Unit 8 Fractions and Ratios
Writing/Reasoning Have students write a response to the
following: Explain how to use the division rule for finding
equivalent fractions to solve Problem 4b. Sample answer:
The division rule states that you can rename a fraction by dividing
the numerator and the denominator by the same number. I
divided the numerator and the denominator by 2 to rename the
4 4П¬ 2
2
бЋЏ
бЋЏбЋЏ
fraction бЋЏ50бЋЏ; бЋЏ
50П¬2 П­ 25
Student Page
б­¤ Study Link 8 8
б­њ
INDEPENDENT
ACTIVITY
(Math Masters, p. 237)
Date
Time
LESSON
Math Boxes
8 8
б­њ
1. a. Write a 7-digit numeral that has
5 in the ten-thousands place,
6 in the tens place,
9 in the ones place,
7 in the hundreds place,
3 in the hundredths place, and
2 in all the other places.
b. Write this numeral in expanded notation.
Home Connection Students practice multiplying
fractions and mixed numbers. They find the areas of
rectangles, triangles, and parallelograms.
52,769.23
50,000 П© 2,000 П© 700 П© 60 П© 9 П© 0.2 П© 0.03
4
1
3. Ellen played her guitar 2бЋЏбЋЏ hours on
3
Saturday and 1бЋЏ1бЋЏ hours on Sunday. How
4
2. Write 3 equivalent fractions for
each number. Sample
4
6
8
бЋЏбЋЏ бЋЏбЋЏ бЋЏбЋЏ
14 21 28
6
9
12
3
бЋЏбЋЏ бЋЏбЋЏ бЋЏбЋЏ
b. бЋЏбЋЏ
10 15 20
5
10 15 20
5
бЋЏбЋЏ бЋЏбЋЏ бЋЏбЋЏ
c. бЋЏбЋЏ
16 24 32
8
2 4 6
20
бЋЏбЋЏ бЋЏбЋЏ бЋЏбЋЏ
d. бЋЏбЋЏ
3 6 9
30
5
1
10
25
бЋЏбЋЏ бЋЏбЋЏ бЋЏбЋЏ
e. бЋЏбЋЏ
2 10 20
50
,
,
,
,
,
2
a. бЋЏбЋЏ
7
3 Differentiation Options
answers:
much longer did she play on Saturday?
2 бЋЏ13бЋЏ ПЄ 1бЋЏ14бЋЏ П­ h
1
бЋЏбЋЏ hours
h
П­
1
12
Answer:
,
,
,
,
,
Number model:
59
READINESS
б­¤ Ordering Improper Fractions
SMALL-GROUP
ACTIVITY
5
2
4
b. бЋЏбЋЏ П­
50
6
c. бЋЏбЋЏ
20
scalene triangle.
2
8
a. бЋЏбЋЏ П­
20
15–30 Min
To review converting between fractions and mixed numbers and
finding common denominators, have students order a set of
improper fractions. Write the following fractions on the board:
7 4 7
11
бЋЏбЋЏ, бЋЏбЋЏ, бЋЏбЋЏ, and бЋЏбЋЏ. Ask students to suggest how to order the fractions
2 1 3
6
from least to greatest. Expect that students will suggest the same
strategies they used with proper fractions, such as putting the
numbers in order and then comparing them to a reference. Use
their responses to discuss and demonstrate the following methods:
71
5. Use your Geometry Template to draw a
4. Complete.
25
3
П­
How does the scalene triangle differ
from other triangles on the
Geometry Template?
10
1
2
d. бЋЏбЋЏ П­
18
9
None of the sides are
the same length.
108 109
134
276
Math Journal 2, p. 276
б­џ Rename each improper fraction as an equivalent fraction with a
common denominator. Ask students what common denominator
to use. Sample answer: Use 6 because the other denominators
are all factors of 6. Have volunteers rename the fractions and
write the equivalent fractions on the board underneath the first
list of fractions. бЋЏ26бЋЏ1, бЋЏ26бЋЏ4, бЋЏ16бЋЏ4, бЋЏ16бЋЏ1
б­џ Write each fraction as a mixed number. Ask volunteers to write
the mixed numbers on the board underneath the second list.
3бЋЏ12бЋЏ, 4, 2бЋЏ13бЋЏ, 1бЋЏ56бЋЏ
Ask students to order the 3 lists on their slates. бЋЏ16бЋЏ1, бЋЏ73бЋЏ, бЋЏ72бЋЏ, бЋЏ41бЋЏ;
5
11 14 21 24
1
1
бЋЏбЋЏ, бЋЏбЋЏ, бЋЏбЋЏ, бЋЏбЋЏ; and 1бЋЏбЋЏ, 2бЋЏбЋЏ, 3бЋЏбЋЏ, 4 Discuss any difficulties or curiosities
6 6 6 6
3
2
6
that students encountered.
Study Link Master
Name
Date
STUDY LINK
88
1.
Multiply.
46
бЋЏбЋЏ,
5 бЋЏ3бЋЏ Вє бЋЏ2бЋЏ П­ 24
or 1бЋЏ112бЋЏ1
85
бЋЏбЋЏ,
5
1
c. 4 бЋЏбЋЏ Вє бЋЏбЋЏ П­ 24
3бЋЏ12бЋЏ34
a.
4
6
or
6
4
296
бЋЏбЋЏ,
1
3
e. 3 бЋЏбЋЏ Вє 1 бЋЏбЋЏ П­ 60
12
EXTRA PRACTICE
б­¤ Playing Frac-Tac-Toe
PARTNER
ACTIVITY
2.
Students play a favorite version of Frac-Tac-Toe to practice
converting between fractions, decimals, and percents.
or
5
5
b. бЋЏбЋЏ
8
Вє бЋЏ2бЋЏ П­
5
10
бЋЏбЋЏ,
40
1
бЋЏбЋЏ
4
or
76–78
7
7 бЋЏ2бЋЏ4
175
бЋЏбЋЏ,
1
1
d. 2 бЋЏбЋЏ Вє 3 бЋЏбЋЏ П­ 24
or
364
бЋЏбЋЏ,
4
2
f. 2 бЋЏбЋЏ Вє 3 бЋЏбЋЏ П­ 40
or 9бЋЏ11бЋЏ0
3
4бЋЏ11бЋЏ45
8
5
8
Find the area of each figure below.
Area of a Rectangle
Area of a Triangle
AП­bВєh
A П­ бЋЏ2бЋЏ Вє b Вє h
15–30 Min
Area of a Parallelogram
1
a.
(Student Reference Book, pp. 309–311;
Math Masters, pp. 472–484)
Time
Multiplying Fractions and Mixed Numbers
б­њ
b.
AП­bВєh
c.
5
6
1
2 3 yd
ft
3
2 4 ft
1
2 2 ft
2
3 3 yd
Area П­
3.
8бЋЏ59бЋЏ
4 ft
yd2
Area П­
5бЋЏ12бЋЏ
ft2
2бЋЏ112бЋЏ
Area П­
ft2
The dimensions of a large doghouse
are 2 бЋЏ1бЋЏ times the dimensions of
2
a small doghouse.
a.
If the width of the small doghouse
is 2 feet, what is the width of the
large doghouse?
5
b.
feet
If the length of the small doghouse
is 2 бЋЏ1бЋЏ feet, what is the length of the
4
large doghouse?
5
бЋЏбЋЏ
8
feet
5
2 ft
1
2 4 ft
Math Masters, p. 237
Lesson 8 8
б­њ
663

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