2014
Common Core
Mathematics Teacher Resource Book 5
Table of Contents
Ready® Common Core Program Overview
A6
Supporting the Implementation of the Common Core
A7
Answering the Demands of the Common Core with ReadyA8
The Standards for Mathematical Practice
A9
Depth of Knowledge Level 3 Items in Ready Common CoreA10
Cognitive Rigor Matrix
A11
Using Ready Common CoreA12
Teaching with Ready Common Core Instruction
Content Emphasis in the Common Core Standards
Connecting with the Ready® Teacher Toolbox
Using i-Ready® Diagnostic with Ready Common Core
Features of Ready Common Core Instruction
Supporting Research
A14
A16
A18
A20
A22
A38
Correlation Charts
Common Core State Standards Coverage by Ready Instruction
Interim Assessment Correlations
A42
A46
Lesson Plans (with Answers)
CCSS Emphasis
Unit 1: Number and Operations in Base Ten
Lesson 1
Understand Place Value
1
5
M
13
M
21
M
31
M
41
M
49
M
CCSS Focus - 5.NBT.A.1 Embedded SMPs - 1–8
Lesson 2
Understand Powers of Ten
CCSS Focus - 5.NBT.A.2 Embedded SMPs - 2–8
Lesson 3
Read and Write Decimals
CCSS Focus - 5.NBT.A.3a Embedded SMPs - 2, 4–7
Lesson 4
Compare and Round Decimals
CCSS Focus - 5.NBT.A.3b, 4 Embedded SMPs - 1, 2, 4–7
Lesson 5
Multiply Whole Numbers
CCSS Focus - 5.NBT.B.5 Embedded SMPs - 1–8
Lesson 6
Divide Whole Numbers
CCSS Focus - 5.NBT.B.6 Embedded SMPs - 1–5, 7
M = Lessons that have a major emphasis in the Common Core Standards
S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
Unit 1: Number and Operations in Base Ten (continued)
Lesson 7
Add and Subtract Decimals
CCSS Emphasis
57
M
67
M
77
M
CCSS Focus - 5.NBT.B.7 Embedded SMPs - 2–7
Lesson 8
Multiply Decimals
CCSS Focus - 5.NBT.B.7 Embedded SMPs - 1–5, 7
Lesson 9
Divide Decimals
CCSS Focus - 5.NBT.B.7 Embedded SMPs - 1–5, 7
Unit 1 Interim Assessment
89
Unit 2: Number and Operations—Fractions
92
Lesson 10 Add and Subtract Fractions
96
M
106
M
114
M
122
M
130
M
140
M
148
M
158
M
166
M
CCSS Focus - 5.NF.A.1 Embedded SMPs - 1, 2, 4, 7
Lesson 11 Add and Subtract Fractions in Word Problems
CCSS Focus - 5.NF.A.2 Embedded SMPs - 1–8
Lesson 12 Fractions as Division
CCSS Focus - 5.NF.B.3 Embedded SMPs - 1–5, 7
Lesson 13 Understand Products of Fractions
CCSS Focus - 5.NF.B.4a Embedded SMPs - 1–8
Lesson 14 Multiply Fractions Using an Area Model
CCSS Focus - 5.NF.B.4b Embedded SMPs - 1–8
Lesson 15 Understand Multiplication as Scaling
CCSS Focus - 5.NF.B.5a, 5b Embedded SMPs - 1, 2, 4, 6, 7
Lesson 16 Multiply Fractions in Word Problems
CCSS Focus - 5.NF.B.6 Embedded SMPs - 1–8
Lesson 17 Understand Division With Unit Fractions
CCSS Focus - 5.NF.B.7a, 7b Embedded SMPs - 1–8
Lesson 18 Divide Unit Fractions in Word Problems
CCSS Focus - 5.NF.B.7c Embedded SMPs - 1–8
Unit 2 Interim Assessment
M = Lessons that have a major emphasis in the Common Core Standards
S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
177
Unit 3: Operations and Algebraic Thinking
Lesson 19 Evaluate and Write Expressions
CCSS Emphasis
180
183
S/A
193
S/A
CCSS Focus - 5.OA.A.1, 2 Embedded SMPs - 1, 2, 5, 7, 8
Lesson 20 Analyze Patterns and Relationships
CCSS Focus - 5.OA.B.3 Embedded SMPs - 1, 2, 7, 8
Unit 3 Interim Assessment
203
Unit 4: Measurement and Data
206
Lesson 21 Convert Measurement Units
208
S/A
218
S/A
228
S/A
238
M
246
M
254
M
262
M
CCSS Focus - 5.MD.A.1 Embedded SMPs - 1, 2, 5–7
Lesson 22 Solve Word Problems Involving Conversions
CCSS Focus - 5.MD.A.1 Embedded SMPs - 1, 2, 5–7
Lesson 23 Make Line Plots and Interpret Data
CCSS Focus - 5.MD.B.2 Embedded SMPs - 1, 2, 4–7
Lesson 24 Understand Volume
CCSS Focus - 5.MD.C.3a, 3b Embedded SMPs - 2, 4–7
Lesson 25 Find Volume Using Unit Cubes
CCSS Focus - 5.MD.C.4 Embedded SMPs - 2, 4–7
Lesson 26 Find Volume Using Formulas
CCSS Focus - 5.MD.C.5a, 5b Embedded SMPs - 1–8
Lesson 27 Find Volume of Composite Figures
CCSS Focus - 5.MD.C.5c Embedded SMPs - 1–8
Unit 4 Interim Assessment
Unit 5: Geometry
Lesson 28 Understand the Coordinate Plane
271
274
277
S/A
285
S/A
295
S/A
303
S/A
CCSS Focus - 5.G.A.1 Embedded SMPs - 4, 6, 7
Lesson 29 Graph Points in the Coordinate Plane
CCSS Focus - 5.G.A.2 Embedded SMPs - 1, 2, 4–7
Lesson 30 Classify Two-Dimensional Figures
CCSS Focus - 5.G.B.4 Embedded SMPs - 2, 3, 5–7
Lesson 31 Understand Properties of Two-Dimensional Figures
CCSS Focus - 5.G.B.3 Embedded SMPs - 2, 6, 7
Unit 5 Interim Assessment
M = Lessons that have a major emphasis in the Common Core Standards
S/A = Lessons that have supporting/additional emphasis in the Common Core Standards
311
Focus on Math Concepts
Lesson 15
(Student Book pages 128–133)
Understand Multiplication as Scaling
LESSON OBJECTIVES
THE LEARNING PROGRESSION
•Understand that when one of the factors in a
multiplication problem increases or decreases, the
product also increases or decreases.
In previous grades, students understood multiplication
as equal groups.
•Understand that multiplying a number times a
number greater than 1 results in a product greater
than the original number.
Students already know that multiplying a number •Understand that multiplying a number times a
number less than 1 results in a product less than the
original number.
This lesson introduces multiplication as scaling.
by 1 results in the original number. Now they learn
that multiplying a number by a whole number greater
than 1 results in a product greater than the original
number, and multiplying a number by a fraction less
•Understand that multiplying a number less than 1
times another number less than 1 results in a
product less than either fraction.
than 1 results in a product less than the original
PREREQUISITE SKILLS
recognize 4 3 5 as 4 times as big as 5 and ​ 1 ​ 3 5 as •Multiply whole numbers.
2
··
number. Students connect these concepts to gain a
better understanding of multiplication as scaling. They
2
··
​ 1 ​ as big as 5 without performing computation.
•Multiply with fractions.
•Use visual models to multiply.
VOCABULARY
There is no new vocabulary. Review the following key
terms.
factor: a number that is multiplied by another number
Students use the concept of multiplication as scaling in mathematical contexts such as dilations and in
real-world contexts such as using scale models.
Understanding multiplication as scaling prepares
students for Grade 6 work with ratios and proportional
reasoning.
Teacher Toolbox
multiply: to find the total number of items in equalsized groups
Teacher-Toolbox.com
Prerequisite
Skills
✓
Ready Lessons
product: the result of multiplication
Tools for Instruction
5.NF.B.5a
5.NF.B.5b
✓✓
Interactive Tutorials
CCSS Focus
5.NF.B.5 Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the
indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number
(recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given
number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction
equivalence ​ a ​5 ​ (n 3 a) ​  
to the effect of multiplying ​ a ​by 1.
b
·
(n 3 b)
······
b
·
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1, 2, 4, 6, 7 (see page A9 for full text)
140
L15: Understand Multiplication as Scaling
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Part 1: Introduction
Lesson 15
AT A GLANCE
Students explore the concept of scaling—stretching or
shrinking a quantity. They consider verbal descriptions,
mathematical representations, and visual models.
Multiplication is introduced as a way to scale
quantities.
Focus on Math Concepts
Lesson 15
Part 1: Introduction
CCSS
5.NF.B.5a
5.NF.B.5b
Understand Multiplication as Scaling
What does scaling mean?
Think of how you use words and phrases such as “double,” “triple,” “half of,” or “take one
tenth.” These words and phrases describe changing the size of a quantity, or scaling.
Stretching and shrinking are two different ways to scale a quantity.
STEP BY STEP
The table below shows some ways that a quantity of 6 can be scaled.
•Introduce the Question at the top of the page.
stretching
•Read the text with students and discuss the table.
shrinking
Words
Symbols
6 doubled is 12.
2 3 6 5 12
6 tripled is 18.
3 3 6 5 18
Half of 6 is 3.
13653
2
··
1 365 6
10
10
··
··
A tenth of 6 is 6 .
10
··
•Read the Think section with students. Point out that
these are all area models.
Think How can you use models to show what scaling means?
Here is a rectangle with an area of 6 square units.
•Have students describe the models and relate them to
the multiplication expressions.
Circle the numbers
that describe how the
rectangle is being
stretched or shrunk.
The model for 2 3 6 has an area that is double the size of the original rectangle.
ELL Support
Students may be familiar with scales on maps or
with making a scale model of a car or a building.
Relate these uses of the word “scale” to the
discussion of scaling, or increasing and decreasing a quantity by a given factor.
Concept Extension
Deepen students’ understanding of scaling
by comparing stretching to addition.
•Ask students, Is there another way to scale a
quantity?
•Copy the headings and first row of the table onto
the board. Make three additional rows.
•In the second row, write, “6 added to 6 is 12” and
the symbols, “6 1 6 5 12.” Ask, Do you think that
adding 6 is the same as scaling? Why or why not?
Let students share their ideas.
•Say, Let’s experiment. In the next row, write “7 doubled is 14” and the equation “2 3 7 5 14.”
In the next row, write, “6 added to 7 is 13” and
the equation “7 1 6 5 13.” Have students explain
whether addition is the same as scaling. [It is not,
because adding the same number does not scale
the original quantity. Multiplying by the same
number always scales the original quantity in the
same way.]
L15: Understand Multiplication as Scaling
©Curriculum Associates, LLC Copying is not permitted.
The model for 1 3 6 has an area that is half the size of the original rectangle.
2
··
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L15: Understand Multiplication as Scaling
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Mathematical Discourse
•What do you notice about the models?
The 2 3 6 model is twice as big. The ​ 1 ​ 3 6
2
··
model is half as big. The 2 3 6 model has twice as many squares; if you re-arrange the ​ 1 ​ 3 6 model, it contains half as many squares.
2
··
•Why do you think multiplication is used to scale a
quantity?
No other operation allows for shrinking and
stretching the way that multiplication does.
While addition increases and subtraction
decreases, these operations do not address
doubling, tripling, taking a fraction of a
quantity, etc.
141
Part 1: Introduction
Lesson 15
AT A GLANCE
Part 1: Introduction
Students explore the effects of various scale factors on
products. They use a table to investigate the result of
multiplying a whole number by factors less than, equal
to, and greater than 1.
Lesson 15
Think How does the size of the factors affect the product?
Products aren’t always greater than their factors. The table below
shows products of different factors times 6.
3
6
STEP BY STEP
10
··
1
3
··
2
··
6
··
5
1
6
2
3
5
6
10
··
1
1
4
2
21
3
8
12
15
18
3
··
2
··
Notice that the products are sometimes less than 6,
sometimes greater than 6, and sometimes equal to 6.
•Read the Think section as a class.
Look at the products
that are less than 6, then
look at those that are
greater than 6.
What do you notice
about the factors?
What do the products that are less than 6 have in common?
The other factor is less than 1. If you multiply 6 by a factor
less than 1, the product will be less than 6.
•Discuss the table and the questions with students.
What do the products that are greater than 6 have in common? The other factor is
greater than 1. If you multiply 6 by a factor greater than 1, the product will be greater
than 6.
SMP Tip: Guide students to understand the
The product of a factor times 6 is equal to 6 when the other factor equals 1 or a number
that is equivalent to 1.
structure and pattern (SMP 7) shown in the table.
Consider having students fill in another row, using 3 or 5 as the given number. Exploring
additional examples will help students generalize
their learning.
Reflect
1 Describe the products you can get if you multiply 8 by a factor less than 1. Describe
the products you can get if you multiply 8 by a factor greater than 1. Give some
examples that justify your answers.
Possible answer: If you multiply 8 by a factor less than 1, you get a product
that is less than 8. If you multiply 8 by a factor greater than 1, you get a
product that is greater than 8. 1 3 8 5 4, 1 3 8 5 1; 2 3 8 5 16, 3 3 8 5 12 .
2
··
8
··
2
··
•Have students read and reply to the Reflect directive.
L15: Understand Multiplication as Scaling
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129
Mathematical Discourse
•Suppose you had a photo with an area of 10 square
inches. Suppose you want to scan the photo, and your
scanner lets you set the scaling to make the photo a
different size. How could you multiply to make the
photo’s area 5 square inches, or 30 square inches?
To make the photo smaller, scale using a factor
that is less than 1. To make the photo larger,
scale using a factor that is greater than 1. ​ 1 ​ 3 10 5 5, and 3 3 10 5 30.
2
··
142
L15: Understand Multiplication as Scaling
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Part 2: Guided Instruction
Lesson 15
AT A GLANCE
Part 2: Guided Instruction
Students use a number line to explore what happens
when a fraction is multiplied by a factor less than 1.
Explore It
A number line can help you see what happens when a fraction is multiplied by a
factor less than 1.
STEP BY STEP
2 You can show 1 3 3 on a number line. If you break up 3 into 3 equal parts, each part
3 ··
4
4
··
··
is 1 .
4
··
•Tell students that they will have time to work
individually on the Explore It problems on this page
and then share their responses in groups. You may
choose to work through the first problem together as a class.
0
may still believe that multiplication always makes a
quantity “bigger.” Talk with students about finding
“half of” or “one third of” a quantity. Demonstrate
this on the number line with a simple example, such
as “one third of 6.” ​1 ​ 1 ​ 3 6 5 2 2​ Then demonstrate
3
··
1
“one third of 1.” ​1 ​   ​ 3 1 5 ​ 1 ​ 2​ . Finally, work problem
3
3
··
··
2 with students ​1 ​ 1 ​ 3 ​ 3 ​ 5 ​ 1 ​ 2​ .
3
4
4
··
··
··
3
4
1
2
1
1335
4
··
4
··
Is the product less than, greater than, or equal to 3 ?
less than
4
··
3 Show 2 3 3 on a number line. If you break up 3 into 3 equal parts, each part is 1 .
3 ··
4
4
4
··
··
··
Since you multiply by 2 , you need 2 of those parts. Shade and label 2 of 3 .
3
3 ··
4
··
··
0
2
4
2335
4
··
•Make sure students understand how to represent
fraction multiplication on the number line: break up
one factor into the number of equal parts specified in
the second factor’s denominator. Then use the second
factor’s numerator to guide how many parts to shade.
STUDENT MISCONCEPTION ALERT: Students
1
4
3
··
•As students work individually, circulate among them.
This is an opportunity to assess student
understanding and address student misconceptions.
Use the Mathematical Discourse questions to engage
student thinking.
•Take note of students who are still having difficulty
and wait to see if their understanding progresses as
they work in their groups during the next part of the lesson.
Lesson 15
3
··
3
3
4
4
1
2
2 or 1
2
··
4
··
Is the product less than, greater than, or equal to 3 ?
less than
4
··
4 You multiplied 3 by two different factors. What is true about both of those factors?
4
··
What happens when you multiply a given fraction by a factor less than 1?
Possible answer: Both of the factors are less than 1. When you multiply a given
fraction by a factor less than 1, the product will be less than the given fraction.
130
L15: Understand Multiplication as Scaling
©CurriculumAssociates,LLC Copyingisnotpermitted.
Concept Extension
Connect number line and area models
for scaling.
•Use the ​ 1 ​ 3 ​ 1 ​ area model from page 120.
2
··
4
··
•Draw a number line. Mark and label 0, ​ 1 ​ , and 1.
2
··
•Ask, How many parts will I break the halves into?
[Break each half into 4 parts, for a total of
8 parts.] Mark eighths on the number line; do not label them.
•Have students discuss how the area model and
number line model are alike and different.
•Ask, How does the area model show ​ 1 ​ 3 ​ 1 ​ ? ​3 By shading ​ 1 ​ of ​ 1 ​ , which is ​ 1 ​ . 4​
4
··
2
··
4
··
2
··
8
··
•Ask a student to shade the number line to show 3 
​ 1 ​ 3 ​ 1 ​ . ​ Shade ​ 1 ​ .  ​
4
2
8
··
··
··
•Point out that ​ 1 ​ is ​ 1 ​ as big as ​ 1 ​ .
8 ··
4
2
··
··
L15: Understand Multiplication as Scaling
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4
143
Part 2: Guided Instruction
Lesson 15
AT A GLANCE
Part 2: Guided Instruction
Students use a number line to explore what happens
when a fraction is multiplied by a factor greater than 1.
Talk About It
A number line can also help you see what happens when a fraction is multiplied by
a factor greater than 1.
STEP BY STEP
5 Shade the number line to show 4 3 3 .
3 ··
4
··
•Organize students into pairs or groups. You may
choose to work through the first Talk About It
problem together as a class.
0
1
greater than
4
··
6 Shade and label the number line to show 7 3 3 .
3 ··
4
··
0
7335
3 ··
4
··
1
3
4
7
4
··
2
7
4
Is the product less than, greater than, or equal to 3 ?
greater than
4
··
7 Think about how each of your answers compared to 3 . What can you say about the
4
··
product of a given fraction and a factor greater than 1?
factor, ​ 3 ​ , on the number line. Then identify what 4
··
​ 1 ​ of this quantity is and use that information to
3
··
Possible answer: The product will be greater than the given fraction.
Try It Another Way
Explore multiplying 3 by a fraction using an area model.
4
··
The model to the right represents 3 .
shade the product.
3
4
4
··
8 Show 1 3 3 using the area model.
2 ··
4
··
3
9 1335
8
··
2 ··
4
··
SMP Tip: As students grapple with these concepts,
encourage them to justify their arguments and ideas
(SMP 3) with mathematical concepts and
terminology. Ask students, Do you agree? Why or
why not? How can you show that . . .
•Direct the group’s attention to Try It Another Way.
Have a volunteer from each group come to the board
to draw and explain the group’s solutions to the
problems.
•How does it help you to know what ​ 1 ​ of ​ 3 ​ is?
3
··
4
··
​ 1 ​ 3 ​ 3 ​ 5 ​ 1 ​ . So, ​ 4 ​ of ​ 3 ​ , for example, is 3
··
10 Is the product less than, greater than, or equal to 3 ?
4
··
1
2
less than
11 Could you have answered problem 11 without drawing a model? Explain.
Possible answer: Yes, 1 is less than 1, so I know that 1 3 3 will be less than 3 .
2
2 ··
4
4
··
··
··
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L15: Understand Multiplication as Scaling
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Hands-On Activity
Model fraction multiplication on the
number line.
Mathematical Discourse
4
··
2
Is the product less than, greater than, or equal to 3 ?
•Encourage students to start by shading the given
4
··
1
3
4
4335
3 ··
4
··
•Walk around to each group, listen to, and join in on
discussions at different points. Use the Mathematical
Discourse questions to help support or extend
students’ thinking.
3
··
Lesson 15
4
··
4 “groups” of ​ 1 ​ , or 1 whole.
4
··
•How do your answers to these problems relate to
stretching, or scaling?
When we multiply a fraction by a factor greater
than 1, the total distance covered on the
number line “stretches” (i.e., gets larger) by the
factor we are using.
Materials: 1-inch strips of paper, number lines with
10 marks, each 1 inch apart, pencils
•Explain that the number line is marked in fifths. Have students mark 0, 1, and 2 on the
number line.
•Tell students they will model ​ 8 ​ 3 ​ 3 ​ on 3
5
··
··
the number line.
•Students use 1-inch strips to cover ​ 3 ​ on 5
··
the number line.
•Ask, What is ​ 1 ​ of ​ 3 ​ ? ​3 ​ 1 ​ 4​ 3
··
5
··
5
··
•Students use 1-inch strips to model ​ 8 ​ 3 ​ 3 ​ [Cover ​ 8 ​ on the number line.]
3
··
144
5
··
5
··
L15: Understand Multiplication as Scaling
©Curriculum Associates, LLC Copying is not permitted.
Part 3: Guided Practice
Lesson 15
AT A GLANCE
Part 3: Guided Practice
Students demonstrate their understanding of
multiplication as scaling. They use reasoning to answer
questions about scaling and represent fraction
multiplication using a model.
Lesson 15
Connect It
Talk through these problems as a class, then write your answers below.
12 Analyze: Use reasoning to order the following expressions from least to greatest.
Don’t calculate any of the products. Explain your reasoning.
7 3 348,980
12 3 348,980
50 3 348,980
9
11
50
··
··
··
7 3 348,980, 50 3 348,980; 12 3 348,980; Possible reasoning: 7 , 1 so
9
50
11
9
··
···
···
··
7 3 348,980 will be less than 348,980, 50 5 1, so 50 3 348,980 will be equal to
9
50
50
··
···
···
348,890; and 12 . 1, so 12 3 348,980 will be greater than 348,980.
11
11
···
···
STEP BY STEP
•Discuss each Connect It problem as a class using the
discussion points outlined below.
13 Explain: Gillian said that the product of a given number and a fraction is always less
than the given number. Explain what is wrong with Gillian’s statement and give an
example that does not follow her rule.
Possible answer: Multiplying a number by a fraction less than 1 gives a
Analyze:
product less than the number you started with. If the fraction you are
multiplying by is greater than 1, the product will be greater than the number
•You may choose to have students work in pairs to
encourage sharing ideas.
you started with. 3 3 4 5 6. 6 . 4.
2
··
14 Compare: Represent the expression 4 3 8 with a model. Write a sentence comparing
4 ··
5
··
the product with 8 . Explain your reasoning.
5
··
•Use the following to lead the class discussion:
What do you notice about the whole numbers in this
problem? [They are equal.]
What do you notice about the fractions? [One is , 1, one is 5 1, and one is . 1.]
How can you use this information to solve
the problem?
SMP Tip: Students who are confronted with the
Possible model:
0
1
8
5
2
Possible answer: If you break up 8 into 4 equal parts, each part is 2 . Since you
5
··
5
··
are multiplying by 4 , you need 4 of those parts. 4 3 8 is equal to 8 .
4
··
132
4
··
5
··
5
··
L15: Understand Multiplication as Scaling
©CurriculumAssociates,LLC Copyingisnotpermitted.
complexity of the Analyze problem can make sense
of the problem (SMP 1) by looking for meaningful
Compare:
entry points to the solution. They may wish to
•This discussion gives students an opportunity to
draw a model to represent a multiplication
expression involving fractions.
consider simpler problems that are analogous to the
given problem, such as ​ 7 ​ 3 3, ​ 12 ​  3 3, and ​ 50 ​ 3 3.
9
··
11
··
50
··
Looking at simpler whole numbers can help
students focus on the differences among the
fractions, which leads to an elegant solution.
•Ask, What is the size of one equal part? ​3 ​ 2 ​ 4​ What does
5
··
it mean to multiply by ​ 4 ​ ? [You need all 4 of the 4
··
4 equal parts. Some students may recognize that this
means you are multiplying by 1.]
Explain:
•The second problem focuses on the significance of a
fraction being greater than 1.
•If students think that fractions are always less than
1, they will have difficulty explaining this problem.
Ask, Can you think of a fraction that wouldn’t work
with Gillian’s rule? [any fraction that is . 1]
L15: Understand Multiplication as Scaling
©Curriculum Associates, LLC Copying is not permitted.
145
Part 4: Common Core Performance Task
Lesson 15
AT A GLANCE
Part 4: Common Core Performance Task
Students complete a multi-part performance task to
demonstrate their understanding of multiplication as
scaling and the relationship of the size of the factors to
the size of the product.
Lesson 15
Put It Together
15 You can compare the size of a product to the size of the factors in a multiplication
equation if you know whether the factors are greater than, less than, or equal to 1.
A Write a multiplication equation (different from one in this lesson) where the
product is greater than both of the factors. Draw a model to support your answer.
Answers will vary, but both factors should be greater than 1. Models could
STEP BY STEP
include number lines or area models.
•Direct students to complete the Put It Together task
on their own.
•Explain that this task requires them to think and use
the math they have learned in this lesson.
B Write a multiplication equation (different from one in this lesson) where the
factors are both fractions and the product is less than both of the factors. Draw a
model to support your answer.
•As students work on their own, walk around to
assess their progress and understanding, to answer
their questions, and to give additional support, if needed.
Answers will vary, but both fractions should be less than 1. Models could
include number lines or area models.
•If time permits, have students share their answers
and discuss.
C Write a multiplication sentence (different from one in this lesson) where the
product is equal to one of the factors.
SCORING RUBRICS
Answers will vary, but one of the factors should be 1 or a fraction
equivalent to 1.
See student facsimile page for possible student answers.
A
B
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L15: Understand Multiplication as Scaling
Points Expectations
2
Both factors are greater than 1, the product
is correct, and the model correctly
represents the product.
1
Both factors are greater than 1, and the
product is correct. The model may have
some errors but is recognizable.
0
One or both factors are less than or equal to
1. The product may or may not be correct.
The model may or may not have errors.
Points Expectations
2
Both factors are less than 1, the product is
correct, and the model correctly represents
the product.
1
Both factors are less than 1 and the product
is correct. The model may have some errors
but is recognizable.
0
One or both factors are greater than or
equal to 1. The product may or may not be
correct. The model may or may not have
errors.
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133
C
Points Expectations
2
One or both factors are 1 or a fraction that is
equivalent to 1, and the product is correct.
1
One factor is 1 or a fraction that is
equivalent to 1, and the other factor is not
equal to 1. The product is equal to 1.
0
Neither factor is 1, and neither factor is
equal to 1. The product may or may not be
correct.
L15: Understand Multiplication as Scaling
©Curriculum Associates, LLC Copying is not permitted.
Differentiated Instruction
Lesson 15
Intervention Activity
On-Level Activity
Use paper-folding to explore scaling area
when multiplying a fraction by a fraction.
Write word problems for scaling with
fractions.
Materials: pieces of paper, markers or crayons
Discuss everyday examples of shrinking and
stretching, such as scale models, enlarging or
shrinking a picture, doubling or halving a recipe, etc.
Have students write a multiplication word problem
for a scaling situation. Students should identify
whether the problem involves shrinking or
stretching. Have students swap problems with a partner and solve.
Give students a piece of paper. Have them fold it
vertically into fourths. Have them shade ​ 1 ​ . Say, Let’s
4
··
find out what half of this fourth is. Have students fold
the paper horizontally in half, so that the fold line
cuts each fourth in half. Ask, Is this fourth split in half
now? [Yes.] Have students shade half of the fourth
with a different color. Have students write the
equation “​ 1 ​ 3 ​ 1 ​ 5 ?” somewhere on the paper. Have
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4
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them count the parts and name them. [eighths] Ask,
Are the eighths smaller or larger than the fourths?
[smaller] Why? [because we split the fourths in half]
Ask, What is half of ​ 1 ​ ? ​3 ​ 1 ​ 4​ Give each student a new
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piece of paper and repeat the activity using the
expression ​ 1 ​ 3 ​ 3 ​ .
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4
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Challenge Activity
Draw more difficult number line models for multiplying fractions.
Have students draw number line models to multiply fractions in which the numerator of one fraction is not the denominator of the other. For example, students could model ​ 3 ​ 3 ​ 2 ​ . Allow students to develop their 4
··
5
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own methods. One way to solve this is to find a common denominator and use it to determine the size of the
equal parts (20ths, in this case). Then they can shade ​ 2 ​ as ​ 8  ​ and determine that, since there are 8 twentieths, 5
··
20
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​ 1 ​ of that is 2 twentieths. ​3 ​ 2  ​ 4​ Therefore, ​ 3 ​ 3 ​ 2 ​ 5 3 times ​ 2  ​ , or ​ 6  ​ . Have students explain how they got their
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20
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4
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5
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20
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20
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answer. Repeat the activity using a fraction that is greater than 1, such as ​ 7 ​ 3 ​ 3 ​ . ​3 ​ 21 ​  4​
2
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L15: Understand Multiplication as Scaling
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Develop Skills and Strategies
Lesson 16
(Student Book pages 134–143)
Multiply Fractions in Word Problems
LESSON OBJECTIVES
THE LEARNING PROGRESSION
•Represent real-world problems involving
multiplication of fractions using visual models and
equations.
In Grade 4, students solved word problems involving
multiplying a fraction by a whole number by using
visual models and equations.
•Solve real world problems involving multiplication of
fractions using visual models and equations.
Earlier in Grade 5, students learned to multiply
fractions by fractions, fractions by mixed numbers, and
mixed numbers by mixed numbers and developed
understanding of multiplication as scaling. Now,
students solve real-world problems involving
multiplication of fractions and mixed numbers using
visual models and equations. Students should be
comfortable using both fractions and mixed numbers,
and, when necessary, converting between the two.
PREREQUISITE SKILLS
•Multiply with fractions and mixed numbers.
•Use visual models to represent problem situations.
VOCABULARY
There is no new vocabulary. Review the following key terms.
product: the result of multiplication
In later grades, students extend these understandings
to solve multiplication problems involving rational
numbers; the new understanding will involve
multiplying negative numbers.
factor: a number that is multiplied by another number
Teacher Toolbox
multiply: to find the total number of items in equal-sized groups
fraction: a number that compares a part, the numerator, to a whole, the denominator
equation: a mathematical sentence that uses an equal sign (5) to show that two expressions have the same value
Teacher-Toolbox.com
Prerequisite
Skills
Ready Lessons
Tools for Instruction
Interactive Tutorials
✓✓
5.NF.B.6
✓
✓✓
✓✓
CCSS Focus
5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or
equations to represent the problem.
STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1–8 (see page A9 for full text)
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L16: Multiply Fractions in Word Problems
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Part 1: Introduction
Lesson 16
AT A GLANCE
Develop Skills and Strategies
Students explore a real-world problem involving
multiplication of fractions. They use a visual model of a
number line to model the problem situation and use
reasoning to solve the problem.
Lesson 16
Part 1: Introduction
CCSS
5.NF.B.6
Multiply Fractions in Word Problems
Now that you have learned how to multiply fractions, take a look at this problem.
Graysonlives4milefromthepark.Hehasalreadywalked3ofthewaythere.
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STEP BY STEP
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HowfarhasGraysonwalked?
•Tell students that this page models a way to solve a
real-world problem that uses the math they have
recently been studying.
Explore It
Use the math you already know to solve the problem.
You can draw a model to help you solve the problem. Locate a point on the number
line below to show how far Grayson lives from the park.
•Have students read the problem at the top of the page.
0
•Work through Explore It as a class.
1
5
1
4
5
Label the point to show the distance to the park.
Shade the segment that shows one fourth of the way to the park. One fourth of the
way to the park is
SMP Tip: Help students make sense of the
1
5
··
of a mile. Label this distance.
Two fourths of the way to the park is
Three fourths of the way to the park is
problem situation (SMP 1) and persevere in solving
2
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3
5
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of a mile.
of a mile.
Explain how you can use the model to show how far Grayson has already walked.
Possible answer: You can see that each 1 of the way is 1 of a mile so 3 of the
it. Some students may be tempted to simply subtract
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5
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way is 3 of a mile.
1​ e.g., find the difference between ​ ··45 ​ and ​ ··34 ​ 2​ . This is
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an incorrect interpretation of the problem. Use a
simpler problem to help students see that
multiplication is needed. For instance, what if he
lived 4 miles from the park? Then each mile is ​ 1 ​ of
the distance and ​ 3 ​ of the distance is 3 miles.
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L16: Multiply Fractions in Word Problems
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•Have students describe where they placed Grayson’s
house on the number line and how they labeled it.
•Emphasize the importance of knowing how much ​ 1 ​ of the way to the park is.
Mathematical Discourse
•How are the fourths represented on the number line?
The four sections of the number line between 0 and ​ 4 ​ represent Grayson’s walk broken into
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4
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•Ask student pairs or groups to explain their answers
for the last bullet. Ask questions to help students
develop their thinking about using ​ 1 ​ of the way
fourths. Each fourth of his walk covers ​ 1 ​ of a
mile, because he lives ​ 4 ​ from the park.
5
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5
··
4
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(e.g., ​ 1 ​ of a mile) to determine how far ​ 3 ​ of the 5
··
way is.
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L16: Multiply Fractions in Word Problems
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Part 1: Introduction
Lesson 16
AT A GLANCE
Students find out more about the problem they solved
on the previous page. They analyze the problem as a
multiplication situation, write an equation to solve the
problem, and check the reasonableness of the result.
Part 1: Introduction
Lesson 16
Find Out More
The distance you need to find is a fraction of a fraction: 3 of 4 mile.
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5
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Finding 3 of a number is the same as multiplying the number by 3 .
4
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4
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3 of 4 means 3 3 4
4 ··
5
4 ··
5
··
··
STEP BY STEP
To multiply two fractions, multiply the numerators to get the numerator of the product,
and then multiply the denominators to get the denominator of the product.
•Read Find Out More as a class.
•Discuss the meaning of finding ​ 3 ​ “of” a number.
4
··
•Review the procedure for multiplying two fractions.
•Say, Let’s compare this result to the result we got when
we reasoned with the number line, ​ 3 ​ . Then walk
5
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through the process for finding equivalent fractions
and compare the result to the previous answer.
•Point out that it is always good to check whether the answer is reasonable. Read and discuss with the class.
•Have students complete Reflect individually. Discuss
students’ opinions and explanations. Encourage
students to accept a variety of approaches.
3 3 4 5 3 3 4 5 12
5 ·····
4 3 5 ··
20
··
4
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The fraction 12 is equivalent to 3 . To find equivalent fractions, multiply or divide the
20
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5
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numerator and denominator of the fraction by the same number.
12 5 12 4 4 5 3
20 4 4 ··
5
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20
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Both methods give you the same answer: Grayson has walked 3 mile.
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Your answer is reasonable, since it is less than 4 . When you multiply 4 by a factor less
5
5
··
··
than 1, the product should be less than 4 .
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Reflect
1 Which strategy, drawing a model or writing an equation, made more sense to you for
solving this problem? Why?
Answers will vary. Students may say drawing a model helps them
understand, or they may say writing an equation is faster.
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L16: Multiply Fractions in Word Problems
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Real-World Connection
Encourage students to think about everyday places
or situations where people might need to multiply
fractions.
Examples: preparing ​ 1 ​ of a recipe that includes
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fractional measurements, watching ​ 2 ​ of a half-hour
3
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television show, cutting ​ 3 ​ of a ​ 5 ​ -yard piece of fabric
for a craft project
150
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8
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L16: Multiply Fractions in Word Problems
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Part 2: Modeled Instruction
Lesson 16
AT A GLANCE
Part 2: Modeled Instruction
Students use a picture to understand a problem
situation involving finding a fraction of a fractional
quantity. Then, they model the situation with an
equation.
Lesson 16
Read the problem below. Then explore different ways to understand how to find a
fraction of a fraction.
Brandon’smotherleft3ofapizzaonthecounter.IfBrandoneats2ofit,howmuch
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3
··
oftheoriginalwholepizzadidBrandoneat?
STEP BY STEP
Picture It
You can draw a picture to help you understand the problem.
•Read the problem at the top of the page as a class.
Show 3 of a pizza.
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•Read Picture It as a class. Guide students to
recognize that each fourth of the original pizza is ​ 1 ​ of the remaining pizza on the counter.
Since Brandon eats 2 of what is left, shade in 2 of the 3 pieces that are left. You can see
3
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3
··
from the shaded parts how much of the original whole pizza Brandon ate.
•Read Model It as a class. Remind students that, to
multiply fractions, you multiply the numerators and
multiply the denominators. Explain that one way to
write this is to show the multiplication in the
numerator and show the multiplication in the
denominator.
Model It
You can write an equation to help you understand the problem.
You need to find a fraction of a fraction: 2 of 3 of a pizza.
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2335233
4 ·····
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3
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SMP Tip: When students hear “model,” they tend
to think of illustrating a problem, acting it out, or
drawing a diagram. Students should learn to see an
equation as a mathematical model (SMP 4) of a
real-world situation. Explain that many professions
such as the sciences, economics, and sports, use
mathematics to model real-world situations.
Hands-On Activity
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2 of 3 means 2 3 3
4
3 ··
4
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··
3
··
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L16: Multiply Fractions in Word Problems
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Mathematical Discourse
•How would you explain this problem to a friend?
The original pizza is 1 whole. It is cut into 4ths.
Act out the problem.
There are ​ 3 ​ left, so each piece is ​ 1 ​ of what’s left.
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Materials: fraction circles (fourths)
•Have students use fraction circles to act out the
problem situation. First, have students set ​ 3 ​ of a
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circle on their desks.
•Tell students they can think of this as “the whole
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Brandon eats ​ 2 ​ of what is left. Each of these 3
··
​ 2 ​ pieces was a fourth of the original pizza.
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•How are the pictures and the equation model alike?
How are they different?
They both show ​ 3 ​ of the original whole pizza.
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amount that is left.” Brandon ate ​ 2 ​ of what is left.
Then the picture shades ​ 2 ​ of what is left, and
There are 3 equal-sized pieces left, so each piece
the equation multiplies what is left by ​ 2 ​ . The
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is ​ 1 ​ of “the whole amount that is left.”
3
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3
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3
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picture illustrates the problem using pictures of
•Have students remove ​ 2 ​ of what is left (2 pieces).
real objects. The equation only uses numbers
•Have students take the pieces they removed and
and does not represent the pizza at all.
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place them over a whole fraction circle. Students
should describe the amount as ​ 1 ​ of a whole circle.
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L16: Multiply Fractions in Word Problems
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151
Part 2: Guided Instruction
Lesson 16
AT A GLANCE
Students revisit the problem on page 136 and compare
the two strategies that were presented for solving the
problem—drawing a picture and creating an equation
to model the situation. Then, they apply these strategies
to solve similar problems.
STEP BY STEP
•Explain that Connect It refers to the pizza problem
on page 136.
Part 2: Guided Instruction
Lesson 16
Connect It
Now you will solve the problem from the previous page comparing both strategies.
2 Look at the picture. Why did you shade 2 of the 3 parts of the pizza?
Possible answer: The problem says that Brandon eats 2 of what was left.
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3 How much of the whole pizza did Brandon eat? Explain your reasoning.
Possible answer: 2 1 or 1 2 The whole pizza is divided into 4 equal parts.
4
2
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··
Brandon ate 2 of the 4 parts of the whole pizza.
4 Look at the model. How do you know that you should multiply 2 3 3 ?
3 ··
4
··
Possible answer: You need to find 2 of 3 of a pizza. To find a fraction of a
3
4
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··
number, you multiply.
6
•After students have completed all of the problems,
discuss their answers as a class.
5 What is 2 3 3 ?
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•Have students explain their thinking for problems 4 and 5. Ask, Why do you think it’s helpful to know
how to find equivalent fractions? [They help you see if your solution is reasonable.]
6 What are some strategies you can use to solve a word problem that involves
•Use problem 6 to summarize the two strategies.
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···
Is this answer the same as your answer to problem 3 above? Explain.
Possible answer: Yes; 6 is equivalent to 2 1 or 1 2 .
12
···
4
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2
··
multiplying fractions? Possible answer: You can draw a picture to represent
the problem and reason to a solution, or you can write an equation that
represents the problem and solve the equation.
Try It
Use what you just learned about finding products of fractions to solve these
problems. Show your work on a separate sheet of paper.
7 Lewis rode his bike 10 miles. He stopped for a break 2 of the way into his ride. How
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many miles did Lewis ride before he stopped for a break?
SMP Tip: Selecting appropriate strategies is a
crucial skill that helps students persevere in solving
problems (SMP 1). If students try one strategy and get
stuck, explain that there are usually many ways to
work on a problem and encourage them to try a
different strategy. Once students find a solution, have
them use a different strategy to check their work.
•Have students solve the Try It problems individually.
Have students explain which strategies they used
and why.
ELL Support
Students are accustomed to comparing quantities.
Explain that, in general, to compare means to see
how things are the same or how they are different.
When we compare numbers, we see if they are equal
or not. On this page, we compare strategies to see
how they are alike or different. This helps us
understand when it is best to use each strategy.
4 miles
8 Jamie worked 5 hour filing papers for her mother. She listened to music for 2 of the
6
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··
time she spent filing. How much time did Jamie spend listening to music?
2 or 1 hour
1 2
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3
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L16: Multiply Fractions in Word Problems
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TRY IT SOLUTIONS
7Solution: 4 miles; Students may draw a picture or
write an equation ​1 ​ 2 ​ 3 10 5 4 2​ to solve the
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problem. Or, they may reason that ​ 1  ​  of 10 miles is
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1 mile and that ​ 2 ​ 5 ​ 4  ​  , so Lewis rode 4 miles.
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8Solution: ​ 2 ​ ​1 or ​ 1 ​ 2​ hour; Students may illustrate the
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3
··
problem using a picture of an analog clock. They
may also use an equation ​1 ​ 2 ​ 3 ​ 5 ​ 5 ​ 2 ​ 5 ​ 1 ​ 2​ .
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6
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3
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Either ​ 2 ​ hour or ​ 1 ​ hour is acceptable as the product.
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3
··
ERROR ALERT: Students who wrote ​ 300  ​ hours or
30
···
10 hours (or another fraction equivalent to 10) may
have rewritten each fraction as an equivalent
fraction with the common denominator 30 and then
multiplied the resulting numerators.
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L16: Multiply Fractions in Word Problems
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Part 3: Modeled Instruction
Lesson 16
AT A GLANCE
Part 3: Modeled Instruction
Students use a picture to understand a problem
situation involving finding a fraction of a mixed
number. Then they model the situation with an
equation.
Lesson 16
Read the problem below. Then explore different ways to understand multiplying
fractions and mixed numbers.
Janiehas23yardsofyellowfabric.Sheuses1 ofthefabrictomakeablanketfor
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hernewbabycousin.HowmanyyardsoffabricdidJanieusefortheblanket?
STEP BY STEP
Picture It
You can use an area model to help you understand the problem.
•Read the problem at the top of the page as a class.
The darker shading of the area model shows half of 2 3 .
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3
24
•Have a volunteer explain the meaning of the
problem. ​3 Janie has some fabric and uses half of it.
1
2
So, multiply ​ 1 ​ times the amount of fabric she has. 4​
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•Discuss Picture It. Use the Visual Model and
Mathematical Discourse to guide students to
understand the details of this diagram so that they
can draw similar visual models to solve problems.
1 yard
Model It
You can write an equation to help you understand the problem.
You can write 2 3 as a fraction.
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23 5 2 1 3
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•Work through Model It. Review how to write a
mixed number as a fraction. Then show how this
fraction is used in the model of the problem
situation.
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5813
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5 11
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You need to find a fraction of a fraction: 1 of 11 yards of fabric.
2
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1 of 11 means 1 3 11
4
2 ··
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5 1 3 11
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2
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L16: Multiply Fractions in Word Problems
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Mathematical Discourse
Visual Model
Take apart the area model to understand it.
•Draw the area model. Show all shading as light
green. Ask, How many wholes does the model
show? [3]
•Say, Let’s take this model apart and look at each
piece. Make a copy of the area model, split apart
into its three wholes.
•Say, Janie uses half of the fabric she has. Let’s take
half of this first whole. Label the top half “​ 1 ​” and
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draw a horizontal line through the middle of the
square. Shade the top part dark green.
•Have a volunteer take half of the next whole.
•Point to the last whole. Say, Janie does not have
3 whole yards. This one has only ​ 3 ​yard of fabric.
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•Why do you think the area model shows 3 wholes?
It is better to show a fractional amount within a
whole. Since Janie has 2 yards plus a fraction of
another yard, we show three wholes.
•How would you explain this area model to a family
member?
The total area is 3 wholes. 1 whole is split into
fourths so we can show ​ 3 ​ . Across the top, we
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show the fabric that Janie has, 2 ​ 3 ​ yards. We
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shaded this in light green. Then, along the side,
we show ​ 1 ​ because she only used half of the
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fabric she had. We shade this in light green and
we shade the overlapping area in dark green to
show that this represents half of 2 ​ 3 ​ .
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Let’s cut it in half. Draw the line and shade the top
part dark green.
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Part 3: Guided Instruction
Lesson 16
AT A GLANCE
Students revisit the problem on page 138 and compare
two strategies for solving the problem—drawing a
picture and writing an equation to model the situation.
Then, they apply these strategies to another problem.
STEP BY STEP
•Explain that Connect It refers to the problem on page 138.
•For problems 9 and 10, discuss with students how to
use reasoning along with the picture to find the
solution.
•Problems 12 and 13 show another way to solve the
equation ​ 1 ​ 3 2 ​ 3 ​ : find half of 2, find half of ​ 3 ​ , and
2
4
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··
··
add the partial products.
•Ask students to describe the strategies they used for
problem 14.
•Have students solve the Try It problem individually.
Have students explain which strategies they used
and why.
Part 3: Guided Instruction
Lesson 16
Connect It
Now you will solve the problem from the previous page comparing the two
strategies.
9 Does Janie use more or less than 2 3 yards of fabric for the blanket? Explain.
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Possible answer: Janie uses 1 of the fabric she has. That will be less than
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3
2 yards.
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3
1
Explain how you can use the
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Possible answer: The picture shows that Janie
10 How many yards of fabric did Janie use?
picture to answer the question.
used 1 1 1 1 3 yards of fabric.
2
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2
··
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Possible answer:
11 How do you know that you should multiply to solve the problem?
You need to find 1 of 2 3 . To find 1 of something, you can multiply by 1 .
2
4
2
2
··
··
··
··
Possible answer: You can find 1 of 2 and 1 of 3 ,
2
2
4
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··
··
12 How can you multiply 2 3 by 1 ?
4
2
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and then add the products.
13 What is 1 3 2?
2
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1
Add the two products.
3
What is 1 3 3 ?
2
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1
1
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3
8
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5
13
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Yes
Is this answer the same as your answer to question 10 above?
14 Suppose Janie had 2 1 yards of fabric. Explain how you could find how many yards of
4
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Possible answer: I could multiply 2 by 1 and
2
fabric she used for the blanket.
··
1 by 1 , and then add the products. I could also draw a model that shows 1 of 2 1 .
4
2
2
4
··
··
··
··
Try It
Use what you just learned about multiplying mixed numbers to solve this
problem. Show your work on a separate sheet of paper.
15 Izzy has a length of 3 1 yards of sidewalk to decorate for her school festival. She
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decides to decorate 3 of her sidewalk space with a drawing of the school. How many
5
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21 or 2 1 yards
10 1
10 2
···
···
meters of space does Izzy use to draw the school?
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L16: Multiply Fractions in Word Problems
SMP Tip: Students can use their knowledge of the
structure (SMP 7) of fractions and mixed numbers
to solve problems. Some students will break apart
the mixed number into a whole number part and a
fractional part and find the fraction of each. Other
students may prefer to work with a fraction greater
than 1, using a two-step process: first write the
mixed number as a fraction and then multiply the
fractions.
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TRY IT SOLUTIONS
15 Solution: ​ 21 ​ ​
 1 or 2 ​ 1  ​  2​ yards; Students may draw a
10
··
10
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picture modeling the sidewalk. They may find
that ​ 3 ​ 3 3 5 ​ 9 ​ and ​ 3 ​ 3 ​ 1 ​ 5 ​ 3  ​  . Rewriting ​ 9 ​ as
5
··
5
··
5
··
2
··
10
··
5
··
tenths, they have ​ 18 ​  . Adding the partial products
10
··
yields ​ 21 ​  . Alternatively, students may rewrite 3 ​ 1 ​ as
10
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Concept Extension
the fraction ​ 7 ​ and multiply this by ​ 3 ​ to get ​ 21 ​  .
2
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Deepen students’ understanding of the area model.
5
··
10
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ERROR ALERT: Students who wrote ​ 12 ​  or 1 ​ 2  ​  may
10
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•Say, On page 138, we rewrote 2 ​ 3 ​as a fraction.
2
··
10
··
have added 3 to the 1 in the numerator of ​ 1 ​ , instead
4
··
2
··
Let’s show this in our area model.
of writing 3 as ​ 6 ​ and adding that to ​ 1 ​ . Then, they
2
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•Draw the area model.
2
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may have multiplied ​ 4 ​ by ​ 3 ​ to get ​ 12 ​  .
•Draw vertical lines to show 4 fourths in each
2
··
5
··
10
··
whole. Ask, How many fourths do you see in 2 ​ 3 ​ ?
[11 fourths]
4
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•Say, Look at the model. How much fabric did Janie
use? ​3 ​ 11  ​ yards 4​
8
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L16: Multiply Fractions in Word Problems
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Part 4: Guided Practice
Lesson 16
Part 4: Guided Practice
Lesson 16
Study the solution below. Then solve problems 16–18.
Student Model
The student wrote and
solved an equation to
solve the problem.
Part 4: Guided Practice
Lesson 16
17 A field is in the shape of a rectangle 5 mile long and 3 mile wide.
6
4
··
··
What is the area of the field?
Chrisis41feettall.Hismomis11 timesastall.Howtallis
Show your work.
Chris’smom?
Possible student work using an area model:
4
··
2
··
What model can I use to
help understand this
problem?
3
4
Look at how you can solve this problem using an equation.
41 3 1 5 41
4
··
4
··
41 3 1 5 4 3 1 1 1 3 1 5 2 1 1
4
··
Pair/Share
How does the answer
compare to 4 1 feet?
Solution:
4
··
How do I know what
operation to use to solve
this problem?
2
··
2
··
4
··
2
··
5
6
8
··
41 1 2 1 1 5 61 1 1 5 62 1 1 5 63
6 3 feet
4
··
8
··
4
··
8
··
8
··
8
··
8
··
Pair/Share
8
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16 Josh exercises at the gym 3 3 hours a week. He spends 2 of his time
4
5
··
··
lifting weights. How many hours a week does Josh spend lifting
weights at the gym?
18 Ari had 3 of a bag of popcorn. His friends ate 1 of his popcorn. What
4
2
··
··
fraction of the whole bag of popcorn did Ari’s friends eat? Circle the
Show your work.
What equation can I write
to solve this problem?
letter of the correct answer.
Possible student work using an equation:
3321332561 6
5
··
4
··
5
··
5
··
20
···
5 24 1 6
20
···
20
···
5 30
Pair/Share
What is a reasonable
estimate for the number
of hours Josh lifts
weights each week?
Can you solve this
problem in another way?
15 or 5 square mile
1 ··8 2
24
Solution: ···
20
···
5 3 or 1 1
2
··
2
··
3 or 1 1 hours
22
··
21
Solution: ··
A
B
C
D
1
4
··
3
8
··
5
4
··
3
2
··
Kayla chose A as the correct answer. How did she get that answer?
Possible answer: Kayla subtracted 1 from 3 instead of
2
··
4
··
multiplying 1 by 3.
2
··
4
··
Pair/Share
Does Kayla’s answer
make sense?
140
L16: Multiply Fractions in Word Problems
141
L16: Multiply Fractions in Word Problems
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©CurriculumAssociates,LLC Copyingisnotpermitted.
AT A GLANCE
SOLUTIONS
Students study a model of a word problem involving
multiplying two mixed numbers. Then, they solve
several word problems involving fraction
multiplication.
Ex An equation is shown as one way to solve the
problem. Students could also solve the problem by
drawing a model.
16 Solution: ​ 3 ​ or 1 ​ 1 ​ hours; Students could solve the
2
··
STEP BY STEP
•Ask students to solve the problems individually.
Circulate to monitor and provide support.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
•Use the Pair/Share prompts to encourage students’
attention to problem solving. The prompts guide
students to make estimates, look for different entry
points and strategies for solving the problems, write
equations, and evaluate whether their approaches
and solutions make sense.
L16: Multiply Fractions in Word Problems
©Curriculum Associates, LLC Copying is not permitted.
2
··
problem by using the equation 3 3 ​ 2 ​ 1 ​ 3 ​ 3 ​ 2 ​ or
5
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the equation ​ 15  ​ 3 ​ 2 ​ . (DOK 1)
4
··
4
··
5
··
5
··
17 Solution: ​ 15  ​ or ​ 5 ​ square mile; Students could solve
24
··
8
··
the problem by drawing an area model. (DOK 1)
18 Solution: B; Write the equation ​ 3 ​ 3 ​ 1 ​ or draw an
4
··
2
··
area model. ​ 3 ​ 3 ​ 1 ​ 5 ​ 3 3 1  
​ 5 ​ 3 ​
4
2
432
8
··
··
·····
··
Explain to students why the other two answer
choices are not correct:
C is not correct because it is the sum of ​ 3 ​ 1 ​ 1 ​ .
4
2
··
··
D is not correct because it is the product of 3 3 ​ 1 ​ .
2
··
(DOK 3)
155
Part 5: Common Core Practice
Part 5: Common Core Practice
Lesson 16
Part 5: Common Core Practice
3
Solve the problems.
1
Lesson 16
Lesson 16
Look at the rectangle below.
2 2 in.
On Sunday, Kristen bought a carton of 24 bottles of water.
5
4 in.
48
What is the area of the rectangle?
1 of the bottles in the carton.
• On Monday, Kristen drank }
6
1
• On Tuesday, Kristen drank } of the bottles that remained in the carton after Monday.
4
4
5
···
93
5
··
square inches
Lily designed the letters of her name on the computer and printed them on paper. The table
below shows the width and height of the printed letters.
Which picture represents the number of bottles of water remaining in the carton after Kristen
drank the water on Tuesday?
Letter
Width
Height
L
1"
2}
2
3"
1}
4
2"
1}
3
4"
I
A
or
C
Y
4"
4"
3 . Make a table to show the
She used a copier to change the size of the letters by a factor of }
4
new dimensions of each letter.
Show your work.
B
D
Possible student work using equations:
2 3 3 1 1 3 3 5 6 1 3 5 12 1 3 5 15
4
··
2
Milo’s pancake recipe makes 9 servings. It calls for 3 cup milk. Milo wants to make 6 servings.
4
··
How much milk will he need?
18 or 1
2 cup
··
2 ··
4 ··
4 ··
8 ···
8
8 ···
8
··
··
1 3 3 1 3 3 3 5 3 1 9 5 12 1 9 5 21
4 ··
4 ··
4 ··
4 ···
16 ···
16 ···
16 ···
16
··
1 3 3 1 2 3 3 5 3 1 6 5 9 1 6 5 15
4 ··
3 ··
4 ··
4 ···
12 ···
12 ···
12 ···
12
··
4 3 3 5 12
4 ···
4
··
4 3 3 5 12 or 3
4
···
Letter
Width
Height
L
7"
1 ··
8
3”
I
5"
1 ···
16
3”
Y
1"
1 ··
4
3”
134
·····
36
···
Self Check Go back and see what you can check off on the Self Check on page 85.
142
L16: Multiply Fractions in Word Problems
143
L16: Multiply Fractions in Word Problems
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©CurriculumAssociates,LLC Copyingisnotpermitted.
3Solution: ​ 48  ​ or 9​ 3 ​ square inches; Multiply 2​ 2 ​ by 4.
AT A GLANCE
Students multiply fractions to solve word problems that
might appear on a mathematics test.
(DOK 1)
5
··
5
··
5
··
4Solution: See table above. Students multiply both the
width and height of each letter by ​ 3 ​ . (DOK 1)
4
··
SOLUTIONS
1Solution: B; First, find ​ 1 ​ of 24, which is 4. Subtract 4
6
··
from 24 to find the number of bottles remaining
after Monday. Then, find ​ 1 ​ of 20, which is 5.
4
··
Subtract 5 from 20 to find the number of bottles
remaining after Tuesday. (DOK 2)
2Solution: ​ 18 ​ or ​ 1 ​ cup; Since he’s making only
36
··
2
··
6 servings, multiply ​ 6 ​ by ​ 3 ​. (DOK 2)
9
··
156
4
··
L16: Multiply Fractions in Word Problems
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Differentiated Instruction
Lesson 16
Assessment and Remediation
•Ask students to solve the following problem and show their work: Delonte practices piano for ​ 3 ​ hour every
4
··
day. He spends ​ 1 ​ of that time practicing scales. How long does he practice scales? ​3 ​ 1 ​ 3 ​ 3 ​ 5 ​ 3  ​ hour 4​
5
··
5
··
4
··
20
··
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Ask students to solve the following problem and show their work: Greta is making soup. Her recipe uses ​ 2 ​ cup onions. She wants to make 1 ​ 1 ​ times as
3
··
2
··
much soup. How many cups of onions does she need? ​3 1 ​ 1 ​ 3 ​ 2 ​ 5 1 3 ​ 2 ​ 1 ​ 1 ​ 3 ​ 2 ​ 5 ​ 2 ​ 1 ​ 2 ​ 5 1 cup 4​
2
··
3
··
3
··
2
··
3
··
3
··
6
··
•If a student is still having difficulty, use Ready Instruction, Level 5, Lessons 13 and 14.
If the error is . . .
​ 19 ​ hour
20
··
Students may . . .
To remediate . . .
have added the
fractions.
Discuss the meaning of the problem with students. Reinforce the
idea that the word “of” in a problem indicates multiplication.
Hands-On Activity
1 ​ 2 ​ and ​ 2 ​ should look darker than the rest of the
Use tracing paper to model fraction
multiplication.
over the other two (use tape if necessary to hold
Materials: tracing paper, regular paper, ruler, pencil,
colored pencils or crayons, tape (optional)
5
··
3
··
shading. Have students place the last tracing paper
them in place) and trace the entire model onto this
sheet of tracing paper, shading the section that
models ​ 2 ​ 3 1 ​ 2 ​ in a darker shade.
3
··
Organize students into pairs or groups of three. Give each group three pieces of tracing paper and the problem: Amanda played with her friends for 5
··
Challenge Activity
1 ​ 2 ​ hours. They spent ​ 2 ​ of that time outdoors. How
Write and solve two-step problems involving
fraction multiplication.
On regular paper, have students draw two squares
Give students a sample problem involving two steps,
5
··
3
··
long did they play outdoors? ​3 ​ 14 ​  hour 4​
15
··
that share a common side. Have them copy this onto
the three pieces of tracing paper. On one tracing
paper, have students model 1 ​ 2 ​ , making vertical lines
5
··
in the right-hand square to represent fifths and
shading the whole and 2 of the fifths. On a second
tracing paper, have students draw ​ 2 ​ horizontally, 3
··
by dividing both squares into 3 equal parts with
horizontal lines and then shading the top two rows.
Have students place the 1 ​ 2 ​ paper on top of the
5
··
one of which is fraction multiplication. For example:
Timothy made a snack mix with ​ 3 ​cup of raisins
4
··
and ​ 5 ​ cup of nuts. He and his friends ate ​ 1 ​of the snack
8
··
5
··
mix. How many cups of the mix did they eat? ​3 ​ 3 ​ 1 ​ 5 ​ 5 ​ 11  ​ and ​ 1 ​ 3 ​ 11  ​ 5 ​ 11 ​ cup 4​
4
··
8
··
8
··
5
··
8
··
40
··
Have students solve the problem. Then, have them
make up their own problems that involve two steps,
one of which is fraction multiplication. Have
students swap problems with a partner, draw models,
and solve.
original plain paper. Then, have students place the ​ 2 ​ paper on top of the 1 ​ 2 ​ paper. The intersection of
3
··
5
··
L16: Multiply Fractions in Word Problems
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157