Download Operations with Fractions

Operations with
Fractions
?
ESSENTIAL QUESTION
How can you use operations
with fractions to solve
real-world problems?
4
MODULE
You can represent real-world
quantities as fractions, and
then solve the problems using
the appropriate operation(s).
LESSON 4.1
Applying GCF and
LCM to Fraction
Operations
COMMON
CORE
6.NS.4
LESSON 4.2
Dividing Fractions
COMMON
CORE
6.NS.1
LESSON 4.3
Dividing Mixed
Numbers
COMMON
CORE
6.NS.1
LESSON 4.4
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Solving Multistep
Problems with
Fractions and Mixed
Numbers
COMMON
CORE
Real-World Video
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75
Module 4
6.NS.1
To find your average rate of speed, divide the distance
you traveled by the time you traveled. If you ride in a
taxi and drive _12 mile in _14 hour, your rate was 2 mi/h
which may mean you were in heavy traffic.
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Interactively explore
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75
Are You Ready?
Are YOU Ready?
Assess Readiness
Complete these exercises to review skills you will need
for this module.
Use the assessment on this page to determine if students need
intensive or strategic intervention for the module’s prerequisite skills.
Write an Improper Fraction
as a Mixed Number
2
1
Response to
Intervention
Intervention
Enrichment
Online Assessment
and Intervention
1. _94
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17
5. __
5
Differentiated Instruction
• Skill 21 Write an Improper
Fraction as a Mixed
Number
• Challenge worksheets
• Skill 36 Multiplication Facts
Write as a sum using names for one plus a
proper fraction.
Write each name for one as one.
Add the ones.
Write the mixed number.
2_14
2. _83
3_25
15
6. __
8
2_23
23
3. __
6
1_78
33
7. __
10
3_56
11
4. __
2
3
3__
10
29
8. __
12
5_12
5
2__
12
Multiplication Facts
Online and Print Resources
Skills Intervention worksheets
Online
Assessment and
Intervention
Write each improper fraction as a mixed number.
Access Are You Ready? assessment online, and receive
instant scoring, feedback, and customized intervention or
enrichment.
Personal
Math Trainer
13 _
__
= 55 + _55 + _35
5
= 1 + 1 + _35
= 2 + _35
= 2_35
EXAMPLE
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7×6=
EXAMPLE
7 × 6 = 42
PRE-AP
Use a related fact you know.
6 × 6 = 36
Think: 7 × 6 = (6 × 6) + 6
= 36 + 6
= 42
Multiply.
Extend the Math PRE-AP
Lesson Activities in TE
9. 6 × 5
13. 9 × 7
• Skill 38 Division Facts
30
63
10. 8 × 9
14. 8 × 6
72
48
11. 10 × 11
15. 9 × 11
110
99
12. 7 × 8
16. 11 × 12
56
132
Division Facts
EXAMPLE
63 ÷ 7 =
Think:
63 ÷ 7 = 9
So, 63 ÷ 7 = 9.
7 times what number equals 63?
7 × 9 = 63
Divide.
17. 35 ÷ 7
21. 36 ÷ 4
76
5
9
18. 56 ÷ 8
22. 45 ÷ 9
7
5
19. 28 ÷ 7
23. 72 ÷ 8
4
9
20. 48 ÷ 8
24. 40 ÷ 5
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3
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6
8
Unit 2
PROFESSIONAL DEVELOPMENT VIDEO
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Author Juli Dixon models successful
teaching practices as she explores
multiplying and dividing fractions in an
actual sixth-grade classroom.
Online Teacher Edition
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resources online—plan, present,
and manage classes and
assignments.
Professional
Development
ePlanner
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access all your resources online.
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Interactive Answers and
Solutions
Customize answer keys to print
or display in the classroom.
Choose to include answers only
or full solutions to all lesson
exercises.
Interactive Whiteboards
Engage students with interactive
whiteboard-ready lessons and
activities.
Personal Math Trainer: Online
Assessment and Intervention
Assign automatically graded
homework, quizzes, tests,
and intervention activities.
Prepare your students with
updated practice tests aligned
with Common Core.
Operations with Fractions
76
Reading Start-Up
Reading Start-Up
Have students complete the activities on this page by working alone
or with others.
Visualize Vocabulary
Use the ✔ words to complete the triangle. Write the review word
that fits the description in each section of the triangle.
Visualize Vocabulary
The summary triangle helps students review vocabulary associated
with fractions and operations. In or next to each section of the triangle,
students should write the review word that fits the description.
part
of a whole
fraction
top number
of a fraction
Understand Vocabulary
numerator
Use the following explanation to help students learn the preview
words.
To help you remember the word reciprocal, think about
how things go together. For example, visualize two
building blocks that fit together. Together they make one
whole.
Vocabulary
Review Words
area (área)
✔ denominator
(denominador)
✔ fraction (fracción)
greatest common factor
(GCF) (máximo común
divisor (MCD))
least common multiple
(LCM) (mínimo común
múltiplo (m.c.m.))
length (longitud)
✔ numerator (numerador)
product (producto)
width (ancho)
Preview Words
bottom number of a fraction
mixed number (número
mixto)
order of operations (orden
de las operaciones)
reciprocals (recíprocos)
denominator
Understand Vocabulary
In each grouping, select the choice that is described by the given
vocabulary word.
1. reciprocals
A 1:15
3 _
1
_
B 4÷6
C
Integrating Language Arts
2. mixed number
1 1
A _-_
3 5
1
B 3_
2
C -5
3. order of operations
A 5-3+2=0
B 5-3+2=4
C 5-3+2=6
Students can use these reading and note-taking strategies to help
them organize and understand new concepts and vocabulary.
COMMON
CORE
ELA-Literacy.RST.6-8.7 Integrate quantitative or technical information
expressed in words in a text with a version of that information expressed visually (e.g., in a
flowchart, diagram, model, graph, or table).
Additional Resources
Differentiated Instruction
• Reading Strategies ELL
© Houghton Mifflin Harcourt Publishing Company
Active Reading
3
_
and _53
5
Active Reading
Layered Book Before beginning the module,
create a layered book to help you learn the
concepts in this module. Label each flap with
lesson titles. As you study each lesson, write
important ideas, such as vocabulary and
processes, under the appropriate flap. Refer
to your finished layered book as you work on
exercises from this module.
Module 4
Before
Students understand multiplication
and division:
• multiply whole numbers
• divide whole numbers
77
Module 4
In this module
Students learn to multiply and divide positive rational
numbers fluently:
• multiply fractions
• multiply mixed numbers
• divide fractions
• divide mixed numbers
After
Students will connect rational
numbers and integers:
• multiply rational numbers fluently
• divide rational numbers fluently
77
MODULE 4
Unpacking the Standards
Unpacking the Standards
Understanding the standards and the vocabulary terms in the
standards will help you know exactly what you are expected to
learn in this module.
Use the examples on the page to help students know exactly what
they are expected to learn in this module.
6.NS.1
COMMON
CORE
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions by
fractions, e.g., by using visual
fraction models and equations
to represent the problem.
Common Core Standards
Content Areas
The Number System—6.NS
Apply and extend previous understandings of multiplication and division to divide fractions
by fractions.
COMMON
CORE
Key Vocabulary
The Number System—6.NS
Compute fluently with multi-digit numbers and find common factors and multiples.
COMMON
CORE
What It Means to You
You will learn how to divide two fractions. You will also understand
the relationship between multiplication and division.
UNPACKING EXAMPLE 6.NS.1
Zachary is making vegetable soup. The recipe makes 6_34 cups of
soup. How many 1_12 -cup servings will the recipe make?
6_34 ÷ 1_12
quotient (cociente)
The result when one number is
divided by another.
27 _
= __
÷ 32
4
fraction (fracción)
A number in the form _ba , where
b ≠ 0.
= _92
27 _
× 23
= __
4
= 4_12
The recipe will make 4_12 servings.
6.NS.4
COMMON
CORE
Find the greatest common
factor of two whole numbers
less than or equal to 100 and
the least common multiple of
two whole numbers less than or
equal to 12. Use the distributive
property to express a sum of
two whole numbers 1–100 with
a common factor as a multiple
of a sum of two whole numbers
with no common factor.
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to see all the
Common Core
Standards
unpacked.
What It Means to You
You can use greatest common factors and least common multiples
to simplify answers when you calculate with fractions.
UNPACKING EXAMPLE 6.NS.4
Add. Write the answer in simplest form.
Use the LCM of 3 and 6 as
a common denominator.
1 _
_
+ 16 = _26 + _16
3
2+1
= _____
6
Add the numerators.
= _36
3÷3
= ____
6÷3
Simplify by dividing by the GCF.
The GCF of 3 and 6 is 3.
= _12
Write the answer in simplest form.
© Houghton Mifflin Harcourt Publishing Company
Go online to see
a complete
unpacking of the
Common Core
Standards.
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78
Common Core Standards
Lesson
4.1
6.NS.1 Interpret and compute quotients of fractions and
solve word problems involving division of fractions by
fractions, …
6.NS.4 Find the greatest common factor of two whole numbers
and the least common multiple of two whole numbers … Use the
Distributive Property to express the sum of two whole numbers
1–100 with a common factor as a multiple of a sum of two whole
numbers.
Unit 2
Lesson
4.2
Lesson
4.3
Lesson
4.4
COMMON
CORE
COMMON
CORE
COMMON
CORE
COMMON
CORE
Operations with Fractions
78
LESSON
4.1
Applying GCF and LCM to
Fraction Operations
Common Core Standards
The student is expected to:
COMMON
CORE
ESSENTIAL QUESTION
The Number System—6.NS.4
Find the greatest common factor of two whole numbers
less than or equal to 100 and the least common multiple
of two whole numbers less than or equal to 12. Use the
distributive property to express a sum of two whole
numbers 1–100 with a common factor as a multiple of a
sum of two whole numbers with no common factor.
Mathematical Practices
COMMON
CORE
Engage
How do you use the GCF and LCM when adding, subtracting, and multiplying fractions?
You use the GCF to simplify fractions when you find sums, differences, and products of
fractions. You use the LCM of the denominators of fractions to add and subtract fractions.
Motivate the Lesson
Ask: Suppose you want to make half a batch of cookies and the recipe calls for __34 cup of
pecans. How many pecans should you use? Begin the Explore Activity to find out.
Explore
MP.5 Using Tools
Connect Multiple Representations
Explain to students that there are two ways to multiply fractions.
• Method A: Simplify after you multiply by dividing the numerator and denominator of the
product by the GCF.
• Method B: Simplify before multiplying by dividing a numerator and a denominator by a
common factor.
Explain
ADDITIONAL EXAMPLE 1
Multiply. Write the product in simplest
form.
A) __89 × __12 __49
3
B) __34 × __25 __
10
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Interactive example available online
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EXAMPLE 1
Questioning Strategies
CC Mathematical Practices
• Does the order in which you multiply two fractions matter? Explain. No. The Commutative
Property of Multiplication states that changing the order of the factors does not change
the product.
• How can you tell when it would be easier to simplify before multiplying than to simplify
the product? When one of the factors in the numerator is the same as one of the factors in
the denominator, it is easier to simplify before multiplying.
Focus on Communication
Remind students that the multiplication sign means “of.” So, __12 × __14 means “What is __12 of __14 ?”
Use a diagram to help them see that __12 of __14 is __18 .
YOUR TURN
ADDITIONAL EXAMPLE 2
The gas tank on Kayla’s car holds
16 gallons of gas. The fuel gauge of
her car shows that the gas tank is __34 full.
How many gallons of gas are in the gas
tank? 12 gallons
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79
Lesson 4.1
Avoid Common Errors
Students often forget to simplify the product after multiplying. Remind students that their
final answer should always be in simplest form.
EXAMPLE 2
Questioning Strategies
CC Mathematical Practices
18
__
• Why can you write 18 as ? 18 is a rational number, and you can write every rational
1
number as a quotient of two integers.
Focus on Reasoning
CC Mathematical Practices
Have students relate the Reflect question to the similar question about multiplying
two fractions. Remind them that the multiplication sign means “of.” So, __59 × 18 means
“What is __59 of 18?”; the answer must be less than 18.
4.1
?
Applying GCF and
LCM to Fraction
Operations
ESSENTIAL QUESTION
COMMON
CORE
6.NS.4
Find the greatest common
factor…and the least
common multiple of two
whole numbers…
YOUR TURN
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Math Trainer
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and Intervention
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Multiplying Fractions
numerator × numerator
numerator
_____________________
= __________
denominator × denominator
denominator
Math On the Spot
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The resulting product may need to be written in simplest form. To write
a fraction in simplest form, you can divide both the numerator and the
denominator by their greatest common factor.
Example 1 shows two methods for making sure that the product of two
fractions is in simplest form.
Multiply. Write the product in simplest form.
A _13 × _35
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1× 3
1 _
_
× 35 = _____
3
3× 5
Write the problem as a single fraction.
3
= __
15
Multiply numerators. Multiply denominators.
3÷ 3
= ______
15 ÷ 3
Simplify by dividing by the GCF.
The GCF of 3 and 15 is 3.
= _15
Write the answer in simplest form.
6 in the numerator and 3 in the
denominator have a common factor
other than one. Divide by the GCF 3.
B _67 × _23
6 _
6× 2
_
× 23 = _____
7
7× 3
3. _37 × _23
Math On the Spot
Simplify before multiplying using the GCF.
2× 2
= _____
7× 1
Multiply numerators. Multiply denominators.
1
= _47
4. _45 × _27
6. _67 × _16
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6.NS.4
In (A), you
simplify the
final product
using the
GCF of the
numerator and
denominator;
in (B) you
simplify before
multiplying.
You can use
this method if
the numerator
of one fraction
and the
denominator
of the other
share a GCF.
To multiply a fraction by a whole number, you rewrite the whole number as a
fraction and multiply the two fractions. Remember to use the GCF to write the
product in simplest form.
EXAMPLE 2
Math Talk
Mathematical Practices
Compare the methods in
the Example. How do you
know if you can use the
method in B ?
Lesson 4.1
79
COMMON
CORE
6.NS.4
A class has 18 students. The teacher asks how many students in the class have
pets and finds _59 of the students have pets. How many students have pets?
STEP 1
Estimate the product. Multiply the whole number by the nearest
benchmark fraction.
5
_
is close to _12 , so multiply _12 times 18.
9
1
_
× 18 = 9
2
STEP 2
Multiply. Write the product
in simplest form.
5
_
× 18
9
Math Talk
Mathematical Practices
How can you check
to see if the answer is
correct?
Write the problem as a single fraction.
2
6× 2
= ______
7× 3
7
__
12
8
__
35
1
_
7
2. _34 × _79
Multiplying Fractions and
Whole Numbers
To multiply two fractions you first multiply the numerators and then multiply
the denominators.
EXAMPL 1
EXAMPLE
1
__
10
2
_
7
4
__
15
1. _16 × _35
8
7
× __
5. __
10
21
How do you use the GCF and LCM when adding, subtracting,
and multiplying fractions?
COMMON
CORE
Multiply. Write each product in simplest form.
Compare the
answer to the
estimate. The
answer, 10, is
close to the
estimate, 9.
80
5
18
_
× 18 = _59 × __
1
9
5
times 18
You can write __
9
three ways.
5
__
×18
9
5
__
· 18
9
5
__
(18)
9
Rewrite 18 as a fraction.
× 182
= 5______
19 × 1
Simplify before multiplying using the GCF.
5×2
= ____
1×1
Multiply numerators. Multiply denominators.
10
= 10
= __
1
Simplify by writing as a whole number.
10 students have pets.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Life on
white/Alamy
LESSON
Unit 2
PROFESSIONAL DEVELOPMENT
CC
Integrate Mathematical
Practices MP.5
This lesson provides an opportunity to address
the Mathematical Practices standard that
calls for students to use appropriate tools
strategically. In Example 1, students use an
algorithm to multiply two fractions. Then, in
Example 2, students estimate and then use the
algorithm to multiply a fraction and a whole
number by rewriting the whole number as a
fraction. Finally, in Example 3, students use
the LCM to rewrite fractions so they have the
same denominator to simplify expressions.
Math Background
Although students may have little difficulty
applying the multiplication algorithm, they may
not understand the underlying rationale. They
should recognize that any product m · n, for
positive real numbers m and n, can be defined
as the area of a rectangle with sides of length
m and n.
ac
Thus, __ab · __dc = __
can be defined as the area
bd
of a rectangle with side lengths __ab and __dc . This
may be easier for sixth-graders to understand
when presented on a labeled model than when
presented algebraically as above.
Applying GCF and LCM to Fraction Operations
80
YOUR TURN
Focus on Models
CC Mathematical Practices
Remind students to estimate the product first. They may find it helpful to draw a 0 to 1
number line and label it in fourths to decide which benchmark fraction to use as a
multiplier.
ADDITIONAL EXAMPLE 3
2
Add __23 + __
. Write the sum in simplest
4 15
__
form. 5
Interactive Whiteboard
Interactive example available online
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EXAMPLE 3
Questioning Strategies
CC Mathematical Practices
16
5
8
8
__
• How do you know that 30 + __
is equivalent to __
+ __16 ? __
is multiplied by __22 , or 1. So
15
1 30
5 15
__
its value does not change. is multiplied by , or 1, so its value does not change.
__
6
5
Because the fractions being added are equivalent, the sums are equivalent.
YOUR TURN
Avoid Common Errors
Some students may forget to find a common denominator for the fractions and add or
subtract both numerators and denominators. Remind students that you must find a
common denominator before adding and subtracting fractions.
Elaborate
Talk About It
Summarize the Lesson
Ask: How are adding and subtracting two fractions and multiplying two fractions
similar and different? In both types of problems, you can use the GCF to write the
answer in simplest form. But when adding or subtracting fractions, you first use the LCM of
the denominators to rewrite the fractions so they have the same denominator.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercise 1, have two students come up to the whiteboard. Instruct one student
to simplify before multiplying and the other student to simplify after multiplying.
Have students compare the two methods and discuss their reasons for preferring one
method over the other.
Avoid Common Errors
Exercises 2–10 Some students may not write the product in simplest form. Remind them
to check each product to be certain that the numerator and denominator do not have a
common factor.
Exercises 11–16 Some students may forget to find a common denominator for the
fractions and add or subtract both numerators and denominators. Remind students that
you must find a common denominator before adding and subtracting fractions.
81
Lesson 4.1
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Reflect
7.
YOUR TURN
Analyze Relationships Is the product of a fraction less than 1 and a
whole number greater than or less than the whole number? Explain.
Add or subtract. Write each sum or difference in simplest form.
Personal
Math Trainer
Less than; a fraction of a whole, if the fraction is less
than 1, is always less than the whole.
11
__
21
1
__
24
13
6 __
15
5
15. __
+ _16
14
Online Assessment
and Intervention
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5
17. __
- _38
12
19. _23 + 6 _15
4
__
15
11
1 __
20
1
3 __
42
5
3
16. __
- __
12 20
3
18. 1 __
+ _14
10
20. 3 _16 - _17
YOUR TURN
Multiply. Write each product in simplest form.
5
_
× 24
8
10.
1
_
× 8
3
12.
7
× 7
3 __
10
15
8
_
3
9
25 __
10
9.
3
_
× 20
5
11.
1
_
× 14
4
13.
3
× 10
2 __
10
Personal
Math Trainer
12
Online Assessment
and Intervention
7
_
2
Guided Practice
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Multiply. Write each product in simplest form. (Example 1)
23
1. _12 × _58
5
__
16
38
4. 2 _38 × 16
Adding and Subtracting Fractions
COMMON
CORE
© Houghton Mifflin Harcourt Publishing Company
STEP 1
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My Notes
Rewrite the fractions as equivalent fractions. Use the
LCM of the denominators as the new denominator.
8
8×2
16
__
→ _____
→ __
15
15 × 2
30
1
×5
5
1
_
____
__
→ 6 × 5 → 30
6
STEP 2
Math On the Spot
6.NS.4
8
1
_
Add __
15 + 6 . Write the sum in simplest form.
7. _14 of 12 bottles of water =
3
9. _35 of $40 restaurant bill = $
24
5
11. _38 + __
24
?
?
6
2
× 5
6. 1 __
10
bottles
8. _23 of 24 bananas =
10. _56 of 18 pencils =
16
15
bananas
pencils
7
__
12
24
__
35
5
1
+ __
12. __
20
12
5
15. 3 _38 + __
12
7
__
15
19
3 __
24
9
- _1
13. __
20 4
7
5
- __
16. 5 __
10 18
1
_
5
19
5 __
45
ESSENTIAL QUESTION CHECK-IN
17. How can knowing the GCF and LCM help you when you add, subtract,
and multiply fractions?
Simplify by dividing by the GCF.
The GCF of 21 and 30 is 3.
Knowing the GCF helps to simplify products, sums,
and differences as well as fractions before multiplying.
Knowing the LCM is necessary to add and subtract
Reflect
14.
3
__
20
3. _38 × _25
Add or subtract. Write each sum or difference in simplest form.
9
3
- __
14. __
10 14
The LCM of 15 and 6 is 30.
Add the numerators of the equivalent fractions. Then simplify.
16 __
5
21
__
+ 30
= __
30
30
÷3
_____
= 21
30 ÷ 3
7
= __
10
5
5. 1 _45 × __
12
1
_
3
3
_
4
Find each amount. (Example 2)
You have learned that to add or subtract two fractions, you can rewrite the
fractions so they have the same denominator. You can use the least common
multiple of the denominators of the fractions to rewrite the fractions.
EXAMPL 3
EXAMPLE
2. _35 × _59
© Houghton Mifflin Harcourt Publishing Company
8.
Can you also use the LCM of the denominators of the fractions to
8
_1
rewrite the difference __
15 - 6 ? What is the difference?
fractions.
11
yes; __
30
Lesson 4.1
6_MFLESE056695_U2M04L1.indd 81
81
13/11/13 1:05 PM
82
Unit 2
6_MFLESE056695_U2M04L1 82
23/02/13 1:47 AM
DIFFERENTIATE INSTRUCTION
Manipulatives
Modeling
Additional Resources
Give each student a piece of paper and two
colored pencils. Have students fold the paper to
find the product of two fractions, such as __12 × __34 .
Students can use graph paper to model
multiplying a fraction and a whole number when
the denominator of the fraction is a factor of the
whole number. To model __23 × 12:
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners ELL
• Reteach
• Challenge PRE-AP
• Fold the paper vertically in half, unfold it, and
color one column to represent __12 .
• Fold the paper horizontally into fourths, unfold
it, and color over three previously colored
sections with a different color.
• The double-colored sections represent the
product __38 .
• Draw 12 equal-sized squares.
• Divide the 12 squares into 3 equal groups by
drawing rings around the groups. (Point out
that there are 4 squares in each group.)
• The numerator of the fraction is 2. Shade the
squares in 2 of the groups.
• There are 8 shaded squares, so __23 × 12 = 8.
Applying GCF and LCM to Fraction Operations
82
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Online homework
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Evaluate
GUIDED AND INDEPENDENT PRACTICE
COMMON
CORE
6.NS.4
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4.1 LESSON QUIZ
COMMON
CORE
6.NS.4
In one school, __56 of the
sixth-graders take a foreign
language. Of these students,
2
__
take French.
5
Concepts & Skills
Practice
Example 1
Multiplying Fractions
Exercises 1–6, 18
Example 2
Multiplying Fractions and Whole Numbers
Exercises 7–10, 19, 21
Example 3
Adding and Subtracting Fractions
Exercises 11–16, 20, 22–23
1. What fraction of the sixth-graders
take French?
2. If there are 150 sixth-grade students,
how many take French?
Exercise
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
Multiply. Write each product in
simplest form.
18
2 Skills/Concepts
MP.4 Modeling
19
3 Strategic Thinking
MP.4 Modeling
3. __78 × __25
20
2 Skills/Concepts
MP.4 Modeling
4. __23 ×18
21
3 Strategic Thinking
MP.3 Reasoning
22
2 Skills/Concepts
MP.4 Modeling
23–24
3 Strategic Thinking
MP.4 Modeling
25
3 Strategic Thinking
MP.7 Using Structure
26
3 Strategic Thinking
MP.3 Reasoning
Lesson Quiz available online
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Answers
1. __13
2. 50
Additional Resources
7
3. __
20
Differentiated Instruction includes:
• Leveled Practice worksheets
4. 12
83
Lesson 4.1
Class
Date
4.1 Independent Practice
COMMON
CORE
6.NS.4
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Solve. Write each answer in simplest form.
18. Erin buys a bag of peanuts that weighs
_3 of a pound. Later that week, the bag is _2
4
3
full. How much does the bag of peanuts
weigh now? Show your work.
2
__
3
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Math Trainer
Online
Assessment and
Intervention
21. Marcial found a recipe for fruit salad that
he wanted to try to make for his birthday
party. He decided to triple the recipe.
×3
6
× __34 = 2______
= ___
= __12 ; __12 pound
3×4
12
19. Multistep Marianne buys 16 bags of
potting soil that comes in _58 -pound bags.
a. How many pounds of potting soil does
Marianne buy?
3 _78 quarts
23. Each of 15 students will give a 1_12 -minute speech in English class.
a. How long will it take to give the speeches?
3_12 cups thinly sliced rhubarb
Yes, there is enough time; 15 min for the teacher’s
15 seedless grapes, halved
introduction + 15 min for students to get ready +
22 _12 min for speeches = 52 _12 min
_1 orange, sectioned
2
10 fresh strawberries, halved
_3 apple, cored and diced
c. How much time is left on the digital camera?
5
FOCUS ON HIGHER ORDER THINKING
24. Represent Real-World Problems Kate wants to buy a new bicycle from
a sporting goods store. The bicycle she wants normally sells for $360. The
store has a sale where all bicycles cost _56 of the regular price. What is the
sale price of the bicycle?
_1 cup fresh blueberries
4
5 more bags
a. What are the new amounts for the
oranges, apples, blueberries, and
peaches?
$300.00
oranges = 1_12 , apples = 1_45,
20. Music Two fifths of the instruments in the
marching band are brass, one third are
percussion, and the rest are woodwinds.
a. What fraction of the band is
woodwinds?
4
__
15
b. One half of the woodwinds are
clarinets. What fraction of the band is
clarinets?
2
__
15
c. One eighth of the brass instruments
are tubas. If there are 240 instruments
in the band, how many are tubas?
12
PRE-AP
25. Error Analysis To find the product _37 × _49 , Cameron simplified _37 to _17 and
4
then multiplied the fractions _17 and _49 to find the product __
63. What is
Cameron’s error?
blueberries = _34 cup,
peaches = 2
Work Area
Cameron divided a factor in one of the numerators by the
b. Communicate Mathematical Ideas
The amount of rhubarb in the original
recipe is 3_12 cups. Using what you know
of whole numbers and what you know
of fractions, explain how you could
triple that mixed number.
Sample answer: If you triple 3,
it becomes 9. If you triple _12,
GCF but did not divide a factor in one of the denominators
by the GCF. He should have simplified _49 to _43 and
4
multiplied _17 and _43 to find the product __
.
21
26. Justify Reasoning To multiply a whole number by a fraction, you can
first write the whole number as a fraction by placing the whole number
in the numerator and 1 in the denominator. Does following this step
change the product? Explain.
No; the fraction bar represents division, and dividing the
it becomes 1_12. Add 9 + 1_12 =
whole number by 1 does not change its value, so the
10 _12
product is the same.
Lesson 4.1
EXTEND THE MATH
7_12 minutes
_2 peach, sliced
1 plum, pitted and sliced
b. If Marianne’s father calls and says he
needs 13 pounds of potting soil, how
many additional bags should she buy?
22_12 minutes
b. If the teacher begins recording on a digital camera with an hour
available, is there enough time to record everyone if she gives a
15-minute introduction at the beginning of class and every student
takes a minute to get ready? Explain.
Fruit Salad
3
10 pounds
© Houghton Mifflin Harcourt Publishing Company
22. One container holds 1 _87 quarts of water and a second container holds
5 _34 quarts of water. How many more quarts of water does the second
container hold than the first container?
© Houghton Mifflin Harcourt Publishing Company
Name
83
Activity available online
84
Unit 2
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Activity In the United States, we use the Fahrenheit scale to measure temperature.
Other countries use the Celsius temperature scale. The equation C = __59 × (F - 32) is
used to convert a temperature in °F to °C. Show students that if the temperature is
50 °F, the equivalent temperature in °C is __59 × (50 - 32) = __59 × 18 = 10, or 10 °C. Give
students other temperatures to convert and have students check each other’s
conversions. Start with temperatures that convert to whole numbers, such as 41 °F,
59 °F, 68 °F, 77 °F, 86 °F, and so on.
Applying GCF and LCM to Fraction Operations
84
LESSON
4.2 Dividing Fractions
Common Core Standards
The student is expected to:
COMMON
CORE
ESSENTIAL QUESTION
The Number System—6.NS.1
Interpret and compute quotients of fractions, and solve
word problems involving division of fractions by fractions,
e.g., by using visual fraction models and equations to
represent the problem.
Mathematical Practices
COMMON
CORE
Engage
MP.4 Modeling
How do you divide fractions? Multiply the dividend by the reciprocal of the divisor.
Motivate the Lesson
Ask: Suppose you have a 2 __12 -lb package of ground beef and you want to make __14 -lb
hamburger patties for dinner. How many hamburger patties can you make? Begin the
Explore Activity to find out more about dividing fractions.
Explore
EXPLORE ACTIVITY 1
Focus on Modeling
CC Mathematical Practices
In A, make sure that students realize that the bar model represents 1 whole. Point out that
the solid lines represent the division of the bar into 4 equal pieces. Next, point out that each
of the shaded sections was divided in half to make a total of 8 sections. Explain that this
level of subdivision is necessary, since each burrito requires __18 cup of salsa.
Explain
ADDITIONAL EXAMPLE 1
Find the reciprocal of each fraction.
5
A) __
12
12
__
B) __1
7
C) 10
1
__
7
EXAMPLE 1
Focus on Math Connections
CC Mathematical Practices
Remind students that two numbers are reciprocals if their product is 1. Note also that every
number except 0 has a reciprocal.
5
Questioning Strategies
10
Interactive Whiteboard
Interactive example available online
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CC Mathematical Practices
• If the value of a fraction is less than one, what will be true about its reciprocal? It will be a
fraction greater than 1.
• B shows that the reciprocal of the unit fraction __18 is the integer 8. Is the reciprocal of a unit
fraction always an integer? Justify your answer. Yes. Because a unit fraction has 1 in the
numerator, the reciprocal will have 1 in the denominator.
YOUR TURN
Talk About It
Check for Understanding
Ask: How do you find the reciprocal of a fraction? Switch the numerator and the
denominator.
85
Lesson 4.2
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LESSON
COMMON
CORE
4.2 Dividing Fractions
6.NS.1
Reciprocals
Interpret and compute
quotients of fractions, …,
e.g., by using visual fraction
models… .
Another way to divide fractions is to use reciprocals. Two numbers whose
product is 1 are reciprocals.
Math On the Spot
?
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ESSENTIAL QUESTION
3 _
12
_
× 43 = __
=1
4
12
_3 and _4 are reciprocals.
4
3
To find the reciprocal of a fraction, switch the numerator and denominator.
How do you divide fractions?
numerator · ___________
denominator = 1
____________
numerator
denominator
6.NS.1
EXAMPLE 1
Modeling Fraction Division
4
Math Talk
How can you check
that the reciprocal in A is
correct?
3
4
1
8
B Five people share _12 pound of cheese
equally. How much cheese does each
person receive?
© Houghton Mifflin Harcourt Publishing Company
Multiply the
reciprocal by
the fraction to
check that the
product is 1:
18
2 _
_
× 92 = __
=1
9
18
6
6
much is in each of
5
1
__
10
1
__
Each person will receive 10 pound.
2
Switch the numerator and denominator.
The reciprocal of _18 is _81, or 8.
C 5
5 = _51
Rewrite as a fraction.
5
_
1
1
_
5
Switch the numerator and the denominator.
The reciprocal of 5 is _15.
Reflect
2. Is any number its own reciprocal? If so, what number(s)? Justify your answer.
yes; 1; 1 × 1 = 1.
No; 0 does not have a reciprocal; It is not possible to
2
multiply 0 by any number to get a product of 1.
parts.
How much is in each part?
9
8
_
1
B _18
3. Communicate Mathematical Ideas Does every number have a
reciprocal? Explain.
1
To find how much cheese each person
receives, you need to determine how
Switch the numerator and denominator.
The reciprocal of _2 is _9.
Mathematical Practices
To find the number of burritos that can
be made, you need to determine how
many _18 -cup servings are in _43 cups.
Use the diagram. How many eighths
9
_
2
A _29
A You have _34 cup of salsa for making burritos. Each
burrito requires _18 cup of salsa. How many burritos
can you make?
You have enough salsa to make
burritos.
Prep for 6.NS.1
Find the reciprocal of each number.
In some division problems, you may know a number of groups and need to
find how many or how much are in each group. In other division problems,
you may know how many there are in each group, and need to find the
number of groups.
are there in _3 ?
COMMON
CORE
1
4. The reciprocal of a whole number is a fraction with
numerator.
1
10
in the
© Houghton Mifflin Harcourt Publishing Company
EXPLORE ACTIVITY 1
COMMON
CORE
YOUR TURN
Reflect
1. Write the division shown by each model.
Personal
Math Trainer
3 _
1
_
÷ 18 = 6; _12 ÷ 5 = __
4
10
Online Assessment
and Intervention
Find the reciprocal of each number.
7
_
5. 8
8
_
7
6. 9
1
_
9
1
__
7. 11
11
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Lesson 4.2
6_MFLESE056695_U2M04L2.indd 85
InCopy Notes
1. This is a list
86
85
02/04/13 7:52 PM
Unit 2
6_MFLESE056695_U2M04L2.indd 86
15/11/13 3:10 PM
InDesign Notes
1. This is a list
PROFESSIONAL DEVELOPMENT
CC
Integrate Mathematical
Practices MP.4
This lesson provides an opportunity to address
this Mathematical Practice standard. It calls for
students to communicate mathematical ideas
using multiple representations as appropriate. In
the Explore Activity, students use bar models to
model the division of fractions. Using the model,
they are able to see that dividing a fraction by a
fraction can result in a whole number quotient, a
concept that many students do not find intuitive.
Math Background
Being comfortable with dividing fractions and
finding the reciprocal will be very helpful for
students when it is time to simplify complex
__2
fractions such as ____31 . A simple fraction, such as __12 ,
4
can be thought of as a division problem: 1 ÷ 2.
So, the complex fraction can be simplified by
writing it as a division problem:
2
_
8
3 _
1 =_
2 ×_
4 =_
2
_
= 2 ÷_
= 2_
1
3 4 3 1 3
3
_
4
Dividing Fractions
86
EXPLORE ACTIVITY 2
Focus on Critical Thinking
CC Mathematical Practices
Be sure students understand that, when they divide either a fraction by a fraction or a whole
number by a fraction, the result will be an answer that is larger than the dividend. Explain
that this is because they are finding the number of smaller groups within a large group.
For example, in the first division in the table, __67 ÷ __27 , you are asking the question “How many
groups of two sevenths are in six sevenths?” The answer is 3 groups.
Engage with the Whiteboard
Have students sketch a bar model, like the one in Explore Activity 1, to represent
each division problem in the table.
ADDITIONAL EXAMPLE 2
Divide. __38 ÷ __12
EXAMPLE 2
Questioning Strategies
__3
4
Interactive Whiteboard
Interactive example available online
my.hrw.com
Animated Math
Dividing Fractions
Students explore fraction division using
dynamic fraction bars.
my.hrw.com
CC Mathematical Practices
1
__
• Are the two expressions ÷ __5 and 2 × __5 equivalent? Explain. No, they are not the same.
2
6
6
The wrong number has been replaced by its reciprocal. The reciprocal of the divisor, not
the dividend, should be used.
Connect Vocabulary
ELL
Stress proper mathematical language to avoid confusion regarding the change to
multiplication by the reciprocal. The dividend divided by the divisor becomes the dividend
multiplied by the reciprocal of the divisor.
YOUR TURN
Avoid Common Errors
If students are not sure which number is the divisor and incorrectly find the reciprocal of the
dividend, point out that ÷ means “divided by” and the number that follows it is the divisor. It
may help it they circle the divisors before they begin the exercises.
Elaborate
Talk About It
Summarize the Lesson
Ask: How is multiplication related to dividing fractions? You can divide a number
by a fraction by multiplying the dividend by the reciprocal of the divisor.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercises 4–6, have students sketch bar models, like the one in Explore Activity 1,
to check their answers.
Avoid Common Errors
Exercise 2 Some students may have difficulty writing the reciprocal of a unit fraction.
Remind students that the reciprocal of a unit fraction is always an integer.
87
Lesson 4.2
EXPLORE ACTIVITY 2
COMMON
CORE
6.NS.1
Using Reciprocals to Divide Fractions
Dividing by a fraction is equivalent to
multiplying by its reciprocal.
Using Reciprocals to Find
Equivalent Values
Math On the Spot
Division
Divide _59
Multiplication
6 _
_
÷2=3
7 7
6 _
_
×7=
7 2
5 _
_
÷ 3 = _5
8 8 3
5
5 _
_
×8=_
8 3 3
1 _
_
÷ 5 = _1
6 6 5
1
6 _
1× _
_
=
6 5
5
1 _
_
÷ 1 = _3
4 3 4
3
1 _
_
×3=_
4 1 4
4 _
1×_
_
= 45
1
5
EXAMPLE 2
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A Complete the table below.
1 _
_
÷ 14 = _45
5
÷
STEP 1
3
COMMON
CORE
Rewrite as multiplication, using the reciprocal of the divisor.
5 _
_
÷ 2 = _59 × _32
9 3
Animated
Math
STEP 2
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2 __
The reciprocal of __
is 32 .
3
Multiply and simplify.
5 _
15
_
× 32 = __
9
18
Multiply the numerators. Multiply the denominators
= _56
B How does each multiplication problem compare to its corresponding
division problem?
6.NS.1
2
_
. Write the quotient in simplest form.
3
Write the answer in simplest form.
15 ÷ 3 __
______
=5
18 ÷ 3 6
5 _
_
÷ 23 = _56
9
The first fraction is the same. The second fractions are
reciprocals of each other.
YOUR TURN
Divide.
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Math Trainer
C How does the answer to each multiplication problem compare to the
answer to its corresponding division problem?
The answers are the same.
2 _14
9
10. __
÷ _25 =
10
Online Assessment
and Intervention
9
11. __
÷ _35 =
10
1_12
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© Houghton Mifflin Harcourt Publishing Company
Find the reciprocal of each fraction. (Example 1)
8. Make a Conjecture Use the pattern in the table to make a conjecture
about how you can use multiplication to divide one fraction by another.
1. _25
5
_
2
2. _19
9
10
3. __
3
3
__
10
Multiply the first fraction by the reciprocal of the
Divide. (Explore 1, Explore 2, and Example 2)
second fraction.
4. _43 ÷ _53 =
?
?
9. Write a division problem and a corresponding multiplication problem like
those in the table. Assuming your conjecture in 8 is correct, what is the
answer to your division problem?
4
_
5
3
5. __
÷ _45 =
10
3
_
8
6. _12 ÷ _25 =
1_14
ESSENTIAL QUESTION CHECK-IN
© Houghton Mifflin Harcourt Publishing Company
Guided Practice
Reflect
7. How do you divide fractions?
20
.
Sample answer: _57 ÷ _34 ; _57 × _43 ; __
21
Multiply the dividend by the reciprocal of the divisor.
Lesson 4.2
87
88
Unit 2
DIFFERENTIATE INSTRUCTION
Manipulatives
Communicating Math
Use pattern blocks to model the division
__2 ÷ __1 = 4. Remind students that the division is
3
6
asking “How many one-sixths are in two-thirds?”
Point out that the blue rhombus = __13 and the
green triangle = __16 .
2 ÷_
1 =?
_
3 6
2 ÷_
1 =4
_
3 6
2
_
3
2 =_
4
_
3 6
On the board, write 8 ÷ 2 = 4 and 8 × __12 = 4.
Invite students to explain why the two
expressions have the same result. Guide
students to understand that dividing 8 by 2 is
finding the size of each group when 8 is divided
into 2 groups. Point out that this is really the
same as finding half of 8, or 8 × __12. So, dividing by
2 and multiplying by its reciprocal, __12 , gives the
same result.
Additional Resources
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners ELL
• Reteach
• Challenge PRE-AP
Dividing Fractions
88
Personal
Math Trainer
Online Assessment
and Intervention
Online homework
assignment available
Evaluate
GUIDED AND INDEPENDENT PRACTICE
COMMON
CORE
6.NS.1
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4.2 LESSON QUIZ
COMMON
CORE
6.NS.1
Find the reciprocal of each
fraction.
7
1. __
12
2. __17
16
3. __
9
4. __35
5
5. __
11
10
6. __
7
Concepts & Skills
Practice
Explore Activity 1
Modeling Fraction Division
Exercises 4–6
Example 1
Reciprocals
Exercises 1–6
Explore Activity 2
Using Reciprocals to Find Equivalent Values
Exercises 4–6
Example 2
Using Reciprocals to Divide Fractions
Exercises 4–6, 9–16
Divide.
3
7. __38 ÷ __
10
8. __12 ÷ __16
7
9. __
÷ __18
12
10. __58 ÷ __15
Exercise
11. Carlos has __58 pound of pumpkin
seeds. He puts the seeds evenly
into 10 envelopes. How much
pumpkin seed is in each
envelope?
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Answers
12
1. __
7
2. 7
9
3. __
16
4. __5
COMMON
CORE
Mathematical Practices
2 Skills/Concepts
MP.4 Modeling
16
3 Strategic Thinking
MP.3 Logic
17
3 Strategic Thinking
MP.7 Using Structure
18
2 Skills/Concepts
MP.4 Modeling
19
3 Strategic Thinking
MP.6 Precision
20
3 Strategic Thinking
MP.4 Modeling
21
3 Strategic Thinking
MP.3 Logic
8–15
Lesson Quiz available online
Depth of Knowledge (D.O.K.)
Additional Resources
Differentiated Instruction includes:
• Leveled Practice Worksheets
3
11
5. __
5
7
6. __
10
7. 1__14
8. 3
9. 4__23
10. 3__18
1 pound
11. __
16
89
Lesson 4.2
CC
Exercise 21 combines concepts from the Common Core cluster “Apply
and extend previous understandings of multiplication and division to
divide fractions by fractions.”
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Name
Class
Date
4.2 Independent Practice
6.NS.1
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8. Alison has _12 cup of yogurt for making fruit
parfaits. Each parfait requires _18 cup of
yogurt. How many parfaits can she make?
4 parfaits
_1 -mile
4
9. A team of runners is needed to run a
1
relay race. If each runner must run __
mile,
16
how many runners will be needed?
4 runners will be needed.
10. Trevor paints _16 of the fence surrounding his
farm each day. How many days will it take
him to paint _34 of the fence?
It will take him 4 _12 days.
13. Jackson wants to divide a _34 -pound box of
trail mix into small bags. Each of the bags
1
will hold __
pound of trail mix. How many
12
bags of trail mix can Jackson fill?
17. Interpret the Answer The length of a ribbon is _43 meter. Sun Yi needs
pieces measuring _13 meter for an art project. What is the greatest number
of pieces measuring _1 meter that can be cut from the ribbon? How much
3
ribbon will be left after Sun Yi cuts the ribbon? Explain your reasoning.
1
Greatest number of pieces: 2; Ribbon left: __
12 m; Sample
answer: _3 ÷ _1 = 2 _1 . The 2 represents 2 pieces that are
4
1
3
4
1
1
1
each _3 m long. The _4 represents _4 of a _3 m piece of ribbon,
9 bags
1
1
1
1
not _4 m. Sun Yi will have _4 × _3 = __
12 m of ribbon left.
_2 quart
3
14. A pitcher contains
of lemonade.
If an equal amount of lemonade is
poured into each of 6 glasses, how much
lemonade will each glass contain?
1
_
quart
9
9
18. Represent Real-World Problems Liam has __
gallon of paint for painting
10
1
the birdhouses he sells at the craft fair. Each birdhouse requires __
gallon
20
of paint. How many birdhouses can Liam paint? Show your work.
20
9
9
1
__
÷ ___
= __
× __
= 18
1
10
20
10
15. How many tenths are there in _45 ?
8
11. Six people share _35 pound of peanuts
16. You make a large bowl of salad to share
with your friends. Your brother eats _13 of it
before they come over.
equally. What fraction of a pound of
peanuts does each person receive?
1 pound
__
10
© Houghton Mifflin Harcourt Publishing Company • Image Credits: © Irochka/
Fotolia
Online
Assessment and
Intervention
Work Area
FOCUS ON HIGHER ORDER THINKING
1
12. Biology If one honeybee makes __
12
teaspoon of honey during its lifetime, how
many honeybees are needed to make _12
teaspoon of honey?
6 honeybees
19. Justify Reasoning When Kaitlin divided a fraction by _12, the result was
a mixed number. Was the original fraction less than or greater than _12 ?
Explain your reasoning.
Greater than _12 ; since the quotient was a mixed number,
a. You want to divide the leftover salad
evenly among six friends. What
expression describes the situation?
Explain.
2
_
÷ 6, because if the brother
3
ate _1, then there is _2 of the
3
3
the original fraction contained more than 1 unit of _12 .
20. Communicate Mathematical Ideas The reciprocal of a fraction less than
1 is always a fraction greater than 1. Why is this?
In a fraction less than 1, the numerator is less than the
salad left to split between the
denominator. In its reciprocal, the numerator is greater
six friends.
than the denominator, making it a fraction greater than 1.
b. What fractional portion of the original
bowl of salad does each friend receive?
2
2
__
÷ 6 = _23 ÷ _61 = _23 × _16 = __
= _19 ,
3
18
so each friend receives _19 of
1
21. Make a Prediction Susan divides the fraction _58 by __
. Her friend Robyn
16
1
divides _58 by __
. Predict which person will get the greater quotient. Explain
32
and check your prediction.
Robyn; Robyn is dividing _58 into smaller groups than
the original bowl of salad.
© Houghton Mifflin Harcourt Publishing Company
COMMON
CORE
Personal
Math Trainer
Susan; there are more thirty-seconds in _58 than there are
1
1
= 10; _58 ÷ __
= 20.
sixteenths. _58 ÷ __
16
32
Lesson 4.2
6_MFLBESE056695_U2M04L2.indd 89
Extend the Math 89
11/10/14 4:10 PM
PRE-AP
90
Unit 2
6_MFLESE056695_U2M04L2.indd 90
Activity available online
08/04/14 3:33 AM
my.hrw.com
Activity Have students follow the procedure
below to divide fractions.
Finally, multiply by another fraction that is equal
to 1 and results in a whole-number denominator.
Begin by dividing the first numerator by the
second numerator, and the first denominator by
the second denominator.
3
_
​   ​ 
3
÷
2
3
2
2
_
​   ​  ÷ _
​   ​  = _
​ 
 
 ​ = _
​   ​ 
5
5 3 5÷3 _
​   ​ 
3
Next, multiply by a fraction that is equal to 1 and
results in a whole-number numerator.
3
​ _  ​ 
3
_
​ 2  ​  × _
​ 2 ​  = _
​    ​  
5
10
2
_
_
​   ​ 
​   ​ 
3
3
3
3
9
​ _   ​ × _
​   ​  = _
​    ​ 
10
3 10
_
​   ​  
3
Invert and multiply to confirm that this product is
correct. Then repeat the procedure for another
pair of fractions.
Dividing Fractions 90
LESSON
4.3 Dividing Mixed Numbers
Common Core Standards
The student is expected to:
COMMON
CORE
ESSENTIAL QUESTION
The Number System—6.NS.1
Interpret and compute quotients of fractions, and solve
word problems involving division of fractions by fractions,
e.g., by using visual fraction models and equations to
represent the problem.
Mathematical Practices
COMMON
CORE
Engage
MP.4 Modeling
How do you divide mixed numbers? Rewrite the mixed numbers as fractions and multiply
the dividend by the reciprocal of the divisor. Write the product in simplest form.
Motivate the Lesson
Ask: You have 2__14 lb of trail mix and want to know how many __18 -lb servings you can make.
Begin the Explore Activity to find out more about dividing mixed numbers.
Explore
EXPLORE ACTIVITY 1
Engage with the Whiteboard
Have students use the whiteboard to explain how the model represents the given
situation by showing how the top and bottom rows relate to the question.
Explain
ADDITIONAL EXAMPLE 1
Ryan feeds his chicken 1__13 cups of
grain per day. The grain container has
6 __23 cups of grain left. For how many
days can Ryan feed the chicken?
5 days
Interactive Whiteboard
Interactive example available online
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EXAMPLE 1
Focus on Reasoning
CC Mathematical Practices
Make sure students understand the problem. Explain that an amount of cereal (given in
ounces) is going to be divided into smaller servings (also expressed in ounces). Since the
question is how many servings can be made, the answer is the number of servings.
Questioning Strategies
CC Mathematical Practices
35
1
__
__
• Why is 17 rewritten as ? Because you need to write the mixed number as a fraction
2
2
before multiplying: 17 × 2 = 34, and 34 + 1 = 35, so 35 is the numerator.
• What would happen if you did not simplify by using the GCF before multiplying? The
175
product would be ___
, which would still need to be simplified.
14
YOUR TURN
Focus on Critical Thinking
CC Mathematical Practices
Encourage students to estimate the answer before actually solving the problem. Each
container will hold more than 1 lb but less than 2 lb, so the number of containers should
be less than 10 but more than 5. Since 1__14 is closer to 1 than to 2, the answer will be closer
to 10 than to 5.
91
Lesson 4.3
Dividing Mixed
Numbers
4.3
?
COMMON
CORE
6.NS.1
Using Reciprocals to Divide
Mixed Numbers
Interpret and compute
quotients of fractions,
and solve word problems
involving division of fractions
by fractions… .
Math On the Spot
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ESSENTIAL QUESTION
How do you divide mixed numbers?
COMMON
CORE
EXPLORE ACTIVITY
My Notes
6.NS.1
1
4
1
4
1
4
1
4
B How many fourths are in 2_12?
1
4
1
4
10
STEP 3
STEP 4
Communicate Mathematical Ideas Which mathematical operation
could you use to find the number of sushi rolls that Antoine can make?
Explain.
7
5
__
The reciprocal of __
5 is 7 .
Multiply.
25
, or 12_12
= __
2
the amount needed for each sushi roll.
Simplify first using the GCF.
Multiply numerators. Multiply denominators.
Write the result as a mixed number.
There are 12_1 servings of cereal in the box.
2
Multiple Representations Write the division shown by the model.
Reflect
2_12 ÷ _14 = 10
3.
Rewrite the problem as multiplication using the reciprocal of
the divisor.
5 35 5
35 _
__
× 57 = __
× _7
2
2
1
5
×
5
= _____
2× 1
division; You are dividing the total amount of rice by
2.
Rewrite the mixed numbers as fractions greater than 1.
35 _
35 _
__
÷ 75 = __
× 57
2
2
sushi rolls.
Reflect
1.
You need to find how many
2
1
17__
groups of 1__
2.
5 are in
35 _
÷ 75
17_12 ÷ 1_25 = __
2
1
4
10
Antoine has enough rice to make
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Stockbyte /Getty
Images
STEP 2
1
2
1
1
4
6.NS.1
Write a division statement to represent the situation.
17_12 ÷ 1_25
A To find the number of sushi rolls that can be made, you
need to determine how many fourths are in 2 _12. Use fraction
pieces to represent 2_12 on the model below.
1
COMMON
CORE
One serving of Harold’s favorite cereal contains 1_25 ounces. How many
servings are in a 17_1 -ounce box?
STEP 1
Antoine is making sushi rolls. He has 2_12 cups of rice and will use _14 cup
of rice for each sushi roll. How many sushi rolls can he make?
1
4
EXAMPLE 1
2
Modeling Mixed Number Division
1
4
Dividing by a fraction is equivalent to multiplying by its reciprocal. You can
use this fact to divide mixed numbers. First rewrite the mixed numbers as
fractions greater than 1. Then multiply the dividend by the reciprocal of the
divisor.
4.
Analyze Relationships Explain how can you check the answer.
Multiplication is the inverse of division. Multiply
What If? Suppose Antoine instead uses _18 cup of rice for each sushi roll.
How would his model change? How many rolls can he make? Explain.
12_12 ×1_25 . Since 12 _12 ×1_25 = 17 _12 , the answer is correct.
5.
His fraction model would need to be divided in eighths.
The total number of rolls he can make would be 20,
11_23 servings; 17_12 ÷ 1_12 = 11_23 servings.
which is twice as many because _1 is twice as large as _1.
4
What If? Harold serves himself 1_12 -ounces servings of cereal each
morning. How many servings does he get from a box of his favorite
cereal? Show your work.
© Houghton Mifflin Harcourt Publishing Company
LESSON
8
Lesson 4.3
91
92
Unit 2
DIFFERENTIATE INSTRUCTION
CC
Integrate Mathematical
Practices MP.4
This lesson provides an opportunity to address
this Mathematical Practice standard. It calls for
students to apply mathematics to problems
arising in everyday life. In the Explore Activity,
Example 1 and Example 2, students divide mixed
numbers to solve real-world mathematical
problems involving measures of volume, weight,
and area. This helps students understand that the
division skills presented are applicable to
everyday life.
Math Background
The use of multiplication by the reciprocal to
divide by a fraction can be traced to Hindu and
Arab mathematicians in the early Middle Ages.
A second method is to first rewrite the dividend
and the divisor using a common denominator.
20
21
Thus, __56 ÷ __78 could be rewritten as __
÷ __
. Then
24
24
the quotient of the numerators is found:
20
20
21
__
÷ __
= __
.
24
24
21
This method has the advantage of fitting in nicely
with the algorithms learned for addition and
subtraction. Its disadvantage lies in the need for a
common denominator.
Dividing Mixed Numbers
92
ADDITIONAL EXAMPLE 2
The area of a rectangular herb garden is
39 __23 square feet. The width of the herb
garden is 4 __14 feet. What is the length?
The length of the herb garden is 9 __31
feet.
Interactive Whiteboard
Interactive example available online
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EXAMPLE 2
Questioning Strategies
CC Mathematical Practices
• Why do you use division to find the width? Because area is the product of the length times
the width, you can divide the area by the length to find the width.
• Suppose you had been given the width instead of the length. Could you use multiplication to find the length of the sandbox? No; you still need to use division, as area is the
product of length times width.
Avoid Common Errors
170
17
In Step 2, some students may want to rewrite the expression ___
÷ __
by dividing out the
3
2
GCF of 2 from the numerator of the dividend and the denominator of the divisor. Remind
students that they can simplify only after they have rewritten the division problem as
multiplication by the reciprocal.
YOUR TURN
Avoid Common Errors
Exercises 8–9 Students may neglect to include the units when reporting an answer.
Remind students that when they solve a problem involving measurements, they need
to check to see whether a unit should be included in the answer.
Elaborate
Talk About It
Summarize the Lesson
Ask: Complete the graphic organizer with students to discuss and summarize
lesson concepts.
To divide mixed numbers
fractions
Write both mixed numbers as _______.
reciprocal of the divisor.
Find the ________
Multiply the dividend times the reciprocal of the divisor.
_______
GUIDED PRACTICE
Engage with the Whiteboard
For Exercises 3–4, have students use the whiteboard to make models that represent
the division of mixed numbers. Then have students explain their reasoning.
Avoid Common Errors
Exercises 7–8 Students may neglect to include the units when reporting an answer.
Remind students that when they solve a problem involving measurements, they need to
check to see whether a unit should be included in the answer.
93
Lesson 4.3
YOUR TURN
YOUR TURN
Sheila has 10 _12 pounds of potato salad. She wants to divide the potato
salad into containers, each of which holds 1_14 pounds. How many containers
does she need? Explain.
9; 10 _12 ÷ 1 _14 = 8_25; She will need 9 containers.
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and Intervention
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Solving Problems Involving Area
6.NS.1
17
4
© Houghton Mifflin Harcourt Publishing Company
STEP 3
Math Talk
Explain how to find
the length of a rectangle
when you know the area
and the width.
Rewrite the problem as multiplication using the reciprocal of
the divisor.
20
, or 6 _23
= __
3
Multiply numerators. Multiply denominators.
1
3. 4 ÷ 1_18 =
5. 8_13 ÷ 2_12 =
Divide the
area by the
width to find
the length.
Simplify and write as a mixed number.
Reflect
7.
Check for Reasonableness How can you determine if your answer
is reasonable?
3
4
9
2
_
3
3_59
4. 3_15 ÷ 1_17 =
3_13
2_45
6. 15_13 ÷ 3_56 =
4
Write each situation as a division problem. Then solve. (Example 2)
7. A sandbox has an area of 26 square feet, and
the length is 5_12 feet. What is the width of the
sandbox?
?
?
Sample answer: Use compatible numbers to estimate.
4
2
8. Mr. Webster is buying carpet for an exercise
room in his basement. The room will have an
area of 230 square feet. The width of the room
is 12_12 feet. What is the length?
The width of the sandbox is 6 _23 feet.
9
2
_____ × _____ =
3
5_23
Mathematical Practices
Rewrite the mixed numbers as fractions greater than 1.
170 __
170 __
2
___
÷ 17
× 17
= ___
3
2
3
10 170 × 2
= ______
3 × 17
4
_____ × _____ =
4
3
_____ ÷ _____ =
4
17
Write the situation as a division problem.
170 __
÷ 17
56 _23 ÷ 8 _12 = ___
3
2
3_14 yards
2. 1_12 ÷ 2_14
3=
_____ ÷ __
56 _23 ÷ 8 _12
STEP 2
1
The area of a rectangular rug is 14 __
12 square yards.
The length of the rug is 4 _13 yards. What is the width?
1. 4_14 ÷ _34
The area of a rectangular sandbox is 56_23 square feet. The length of the
sandbox is 8_12 feet. What is the width?
STEP 1
9.
4_12 meters
Divide. Write each answer in simplest form. (Explore Activity and Example 1)
Math On the Spot
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EXAMPL 2
EXAMPLE
The area of a rectangular patio is 12_38 square meters.
The width of the patio is 2_34 meters. What is the length?
Guided Practice
Recall that to find the area of a rectangle, you multiply length × width. If you
know the area and only one dimension, you can divide the area by the known
dimension to find the other dimension.
COMMON
CORE
8.
8
feet.
26 ÷ 5 _12; The width is 4__
11
2
feet
230 ÷ 12 _12 ; 18__
5
ESSENTIAL QUESTION CHECK-IN
© Houghton Mifflin Harcourt Publishing Company
6.
9. How does dividing mixed numbers compare with dividing fractions?
Round the area to 56 square feet and the length to
8 feet. 56 ÷ 8 = 7. Since 6 _23 is close to 7, the answer
The process is the same except that you must first
change the mixed numbers to fractions.
is reasonable.
Lesson 4.3
93
94
Unit 2
DIFFERENTIATE INSTRUCTION
Number Sense
Critical Thinking
Additional Resources
Encourage students to always estimate the
quotients when dividing mixed numbers before
they actually solve the problem. Review
rounding and using compatible numbers for
division. For example, in 27__14 ÷ 3__15 students can
simply round to the nearest whole numbers to
find 27 ÷ 3. For 33__16 ÷ 5__89 , they could use
compatible numbers and find 30 ÷ 5. Remind
students that estimating is not exact, so there is
more than one right way to estimate.
Challenge students to find the missing numbers
in the following division problems. Have them
check their answers.
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners ELL
• Reteach
• Challenge PRE-AP
1. __34 ÷
2.
3. 2__12 ÷
4.
= 3__34 __15
÷ __56 = 1__45 1__12
= 1__37 1__34
1
= 1__23 3__12
÷ 2__
10
For an added challenge, ask students to explain
how they found their answers.
Dividing Mixed Numbers
94
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and Intervention
Online homework
assignment available
Evaluate
GUIDED AND INDEPENDENT PRACTICE
COMMON
CORE
6.NS.1
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4.3 LESSON QUIZ
COMMON
CORE
6.NS.1
Divide.
1. 9 __13 ÷ 1__59
3.
5.
4 __38 ÷ 1__25
16 __23 ÷ 4 _16
2. 2 __47 ÷ 3
4.
6.
15 ÷ 6 __12
23 __47 ÷ 3 __17
7. Melinda is baking Irish soda bread
for the PTA bake sale. She has
10__12 cups of flour. The recipe
1
for 1 loaf calls for 2__3 cups of flour.
This is enough flour for how many
loaves of bread?
8. The area of a rectangular tablecloth
is 4 __16 square meters. The width of
the tablecloth is 1__14 meters. What is
the length of the tablecloth?
Lesson Quiz available online
Concepts & Skills
Practice
Explore Activity
Modeling Mixed-Number Division
Exercises 10–11
Example 1
Using Reciprocals to Divide Mixed Numbers
Exercises 1–6, 12–13, 16
Example 2
Solving Problems Involving Area
Exercises 7–8, 14–15, 17
Exercise
Depth of Knowledge (D.O.K.)
2 Skills/Concepts
MP.2 Reasoning
12–13
2 Skills/Concepts
MP.4 Modeling
14–15
3 Strategic Thinking
MP.4 Modeling
16–17
2 Skills/Concepts
MP.4 Modeling
18
3 Strategic Thinking
MP.3 Logic
19
3 Strategic Thinking
MP.7 Using Structure
20
3 Strategic Thinking
MP.6 Precision
Additional Resources
1. 6
2. __67
3. 3__18
4
4. 2 __
13
5. 4
6. 7__12
7. 4 loaves
8. 3__13 meters
95
Lesson 4.3
Mathematical Practices
10–11
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Answers
COMMON
CORE
Differentiated Instruction includes:
• Leveled Practice Worksheets
Name
Class
Date
4.3 Independent Practice
Personal
Math Trainer
6.NS.1
COMMON
CORE
Online
Assessment and
Intervention
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10. Jeremy has 4 _12 cups of iced tea. He wants to divide the tea into _34 -cup
servings. Use the model to find the number of servings he can make.
1
1
4
1
4
1
1
4
1
4
1
4
1
4
1
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
1
4
1
4
1
4
1
4
1
4
5 _14 feet
11. A ribbon is 3 _23 yards long. Mae needs to cut the ribbon into pieces that are
_2 yard long. Use the model to find the number of pieces she can cut.
3
1
1
3
1
1
3
1
3
1
3
1
1
3
1
3
1
3
16. Ramon has a rope that is 25 _12 feet long. He wants to cut it into 6 pieces
that are equal in length. How long will each piece be?
17. Eleanor and Max used two rectangular wooden boards to make a set for
the school play. One board was 6 feet long, and the other was 5 _12 feet
long. The two boards had equal widths. The total area of the set was
60 _38 square feet. What was the width?
1
4
6 servings
1
3
yes because the height is 4 _14 feet
4 _14 feet
1
1
4
11
15. The area of a rectangular mirror is 11 __
16 square feet. The width of the
mirror is 2 _34 feet. If there is a 5 foot tall space on the wall to hang the
mirror, will it fit? Explain.
1
1
3
1
3
1
3
Work Area
FOCUS ON HIGHER ORDER THINKING
1
3
5 _12 pieces
2
18. Draw Conclusions Micah divided 11 _23 by 2 _56 and got 4 __
17 for an answer.
Does his answer seem reasonable? Explain your thinking. Then check
Micah’s answer.
12. Dao has 2 _38 pounds of hamburger meat. He is making _14 -pound hamburgers.
Does Dao have enough meat to make 10 hamburgers? Explain.
Sample answer: The answer seems reasonable because
11 _23 can be rounded to 12, and 2 _56 can be rounded to
no; He has only enough meat to make 9 _12 quarter pound
hamburgers.
2
2
. 4 __
× 2 _56 = 11_23 ,
3, and 12 ÷ 3 = 4, which is close to 4 __
17 17
13. Multistep Zoey made 5 _12 cups of trail mix for a camping trip. She wants
to divide the trail mix into _3 -cup servings.
so Micah’s answer is correct.
© Houghton Mifflin Harcourt Publishing Company
a. Ten people are going on the camping trip. Can Zoey make enough
_3 -cup servings so that each person on the trip has one serving?
4
19. Explain the Error To divide 14 _23 ÷ 2 _34 , Erik multiplied 14 _23 × _43 . Explain
Erik’s error.
No, it only makes 7 _13 servings.
He used the reciprocal of _34 instead of the reciprocal
b. What size would the servings need to be for everyone to have a
serving? Explain.
4
of 2 _34 . The reciprocal of 2 _34 is __
.
11
11
11 __
1
11
__
of a cup; 5 _12 cups ÷ 10 people = __
× 10
= __
20
2
20
c. If Zoey decides to use the _34 -cup servings, how much more trail mix
will she need? Explain.
20. Analyze Relationships Explain how you can find the missing number
= 2 _57 . Then find the missing number.
in 3 _45 ÷
She would need 7 _12 cups total, so she would need
Divide 3 _45 by 2 _57 , since 3 _45 is the product of 2 _57 and the
2 more cups of trail mix.
missing number. The missing number is 1_25 .
14. The area of a rectangular picture frame is 30 _13 square inches. The length of
the frame is 6 _12 inches. Find the width of the frame.
© Houghton Mifflin Harcourt Publishing Company
4
4 _23 in.
Lesson 4.3
EXTEND THE MATH
PRE-AP
Activity Adrian is trying to solve the problem
shown at the right. He thinks that more than
one answer is possible. Is he right? Justify your
answer.
Sample answer: Yes, 9 __13 ÷ 2 __23 = 3 __12 and
9 __15 ÷ 2 __45 = 3 __27 . Both meet the conditions of
the clues:
• 9 __13 + 2 __23 = 12 and 9 __15 + 2 __45 = 12
• Both quotients are >3 but <4.
• In __12 and __27 , 1, 2, and 7 are prime numbers <10.
95
96
Unit 2
Activity available online
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What is the mixed number division problem that
these clues describe?
A The sum of the dividend and the divisor
is 12.
B The quotient is greater than 3, but less than 4.
C The dividend and the divisor in the fractional
part of both quotients are prime numbers less
than 10.
Dividing Mixed Numbers
96
LESSON
4.4
Solving Multistep Problems with
Fractions and Mixed Numbers
Common Core Standards
The student is expected to:
COMMON
CORE
The Number System—6.NS.1
Interpret and compute quotients of fractions, and solve
word problems involving division of fractions by fractions,
e.g., by using visual fraction models and equations to
represent the problem.
Mathematical Practices
COMMON
CORE
Engage
ESSENTIAL QUESTION
How can you solve word problems involving more than one fraction operation? Determine
which operations to use; then follow the order of operations.
Motivate the Lesson
Ask: Have you ever used a recipe? Have you ever had to use fractions to measure the
ingredients? Begin Example 1 to see how to use fractions to find the total amount of an
ingredient needed in a recipe.
MP.1 Problem Solving
Explore
Engage with the Whiteboard
List the following fractions on the whiteboard. Then have students identify the pairs
of fractions with a product of 1. Remind students what a reciprocal is and how to use
it when dividing fractions.
2 _
_
, 2, _3, _3, _4 , _5, _5, _6
3 5 2 4 3 2 6 5
2
_
and _32, _25 and _52, _34 and _43, _56 and _65
3
Explain
ADDITIONAL EXAMPLE 1
Jillian bought __12 pound of turkey and
1__13 pounds of cheddar cheese at the
deli. Both items are on sale for $6 per
pound. How much does the turkey
and cheese cost? $11
Interactive Whiteboard
Interactive example available online
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EXAMPLE 1
Engage with the Whiteboard
In Step 1, under Analyze Information, have students draw bar models to represent
the amount of lentils needed, one bar for __34 and another for 1__12 . Then have students
divide the models into eighths to find the number of scoops needed.
Questioning Strategies
CC Mathematical Practices
• Does the order of the fractions in the parentheses matter? No. Addition is commutative.
YOUR TURN
Avoid Common Error
Some students will want to cancel common factors out of numerators and denominators in
a division problem. Remind students that they can only cross out common factors after they
have rewritten the division problem as multiplication by the reciprocal.
97
Lesson 4.4
LESSON
4.4
?
Solving Multistep Problems
with Fractions and Mixed
Numbers
COMMON
CORE
6.NS.1
Jon needs 2 _14 cups of dried lentils for both the soup and the salad.
...Solve word problems
involving division of fractions
by fractions...
To find how many _18 -cup scoops he needs, divide the total amount of dried
lentils into groups of _18 .
2_14 ÷ _18 = _94 ÷ _18
= _94 × _81
ESSENTIAL QUESTION
×8
= 9____
4×1
How can you solve word problems involving more than one
fraction operation?
1
Jon will need 18 scoops of dried lentils to have enough for both the lentil
barley soup and the lentil salad.
Sometimes more than one operation will be needed to solve a multistep
problem. You can use parentheses to group different operations. Recall
that according to the order of operations, you perform operations in
parentheses first.
Problem
Solving
Simplify before
multiplying using
the GCF.
18
= __
= 18
1
Solving Problems with
Rational Numbers
EXAMPL 1
EXAMPLE
2
COMMON
CORE
Justify and Evaluate
Math On the Spot
You added _34 and 1_12 first to find the total number of cups of lentils. Then
you divided the sum by _18 to find the number of _18 -cup scoops.
my.hrw.com
6.NS.1
YOUR TURN
Jon is cooking enough lentils for lentil barley soup and lentil salad. The
lentil barley soup recipe calls for _34 cup of dried lentils. The lentil salad
recipe calls for 1_12 cups of dried lentils. Jon has a _18 -cup scoop. How many
scoops of dried lentils will Jon need to have enough for the soup and
the salad?
Personal
Math Trainer
Online Assessment
and Intervention
1. Before conducting some experiments, a scientist mixes
_1 gram of Substance A with _3 gram of Substance B. If
4
2
the scientist uses _18 gram of the mixture for each
experiment, how many experiments can be conducted?
10
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Guided Practice
• Jon has a _18 -cup scoop.
• You need to find the total number of many scoops of lentils he needs.
Formulate a Plan
(
1. An art student uses a roll of wallpaper to decorate two gift boxes. The
student will use 1 _13 yards of paper for one box and _56 yard of paper for the
other box. The paper must be cut into pieces that are _16 yard long. How
many pieces will the student cut to use for the gift boxes? (Example 1)
)
You can use the expression _34 + 1_12 ÷ _18 to find the number of scoops of
dried lentils Jon will need for the soup and the salad.
?
?
Justify and Evaluate
Solve
Follow the order of operations. Perform the operations in parentheses first.
13
ESSENTIAL QUESTION CHECK-IN
2. How can you solve a multistep problem that involves fractions?
Decide which operations to use; follow the order of operations.
First add to find the total amount of dried lentils Jon will need.
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Analyze Information
Identify the important information.
• Jon needs _34 cup of dried lentils for soup and 1_12 cups for salad.
3
_
+ 1_12 = _34 + _32
4
= _34 + _64
= _94
John needs
1
2 __
cups of
4
lentils.
= 2_14
Lesson 4.4
97
98
Unit 2
DIFFERENTIATE INSTRUCTION
Critical Thinking
Challenge students to find the missing numbers
in the following problems. Have them check
their answers.
1. __34 ÷
2.
= 3__34
__1
÷ __56 = 1__45
1__12
3. 2__12 ÷
4.
8
= 1__37 1__34
1
÷ 2__
= 1__23 3__12
10
Multiple Representations
Show students two ways to evaluate __12 (0.7):
1
_
(0.7) = 0.5 · 0.7 = 0.35
2
1
7
7
_
(0.7) = _12 · __
= __
2
10
20
0.35
⎯
7
⟌
Show that __
=
0.35
by
dividing:
.
20
7.00
20
Have students find each of the following
products both ways and show that the
results are equal in each case.
3
= 0.15
1. __12 (0.3) __
20
2
2. __15 (0.4) __
= 0.08
25
3
= 0.075
3. __34 (0.1) __
40
3
4. 1__12 (0.2) __
= 0.3
10
Additional Resources
Differentiated Instruction includes:
• Reading Strategies
• Success for English Learners ELL
• Reteach
• Challenge PRE-AP
Solving Multistep Problems with Fractions and Mixed Numbers
98
Elaborate
Talk About It
Summarize the Lesson
Ask: How can you solve problems that involve both fractions and mixed numbers
when multiplication and/or division is required? Convert the mixed number to an
improper fraction before multiplying and/or dividing.
GUIDED PRACTICE
Engage with the Whiteboard
For Exercise 2, have students identify the important information and write an
expression to represent each situation on the whiteboard. Then have the students
solve the problems using the order of operations.
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4.4 LESSON QUIZ
COMMON
CORE
GUIDED AND INDEPENDENT PRACTICE
COMMON
CORE
6.NS.1
Concepts and Skills
Practice
Example 1
Solving Problems with Rational Numbers
Exercises 1, 3–10
6.NS.1
1. Liam has 8 __34 gallons of paint.
He wants to use __25 of the paint to
paint his living room. How many
gallons of paint will Liam use?
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Answer
1. 3 __12 gallons
99
Evaluate
Lesson 4.4
Exercise
Depth of Knowledge (D.O.K.)
COMMON
CORE
Mathematical Practices
3–10
2 Skills/Concepts
MP.1 Problem Solving
11–12
3 Strategic Thinking
MP.2 Reasoning
13
3 Strategic Thinking
MP.4 Modeling
Additional Resources
Differentiated Instruction includes:
• Leveled Practice worksheets
Class
Date
4.4 Independent Practice
COMMON
CORE
6.NS.1
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3. Naomi has earned $54 mowing lawns
the past two days. She worked 2 _12 hours
yesterday and 4 _14 hours today. If Naomi is
paid the same amount for every hour she
works, how much does she earn per hour
to mow lawns? (Example 2)
$8 per hour
4. An art teacher has 1_12 pounds of red clay
and _34 pound of yellow clay. The teacher
mixes the red clay and yellow clay
together. Each student in the class needs
_1 pound of the clay mixture to finish the
8
assigned art project for the class. How
many students can get enough clay to
finish the project?
18 students
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1
6. A picture framer has a thin board 10 __
12
feet long. The framer notices that 2 _38 feet
of the board is scratched and cannot be
used. The rest of the board will be used to
make small picture frames. Each picture
frame needs 1 _23 feet of the board. At most,
how many complete picture frames can be made?
19
3 __
hr
20
FOCUS ON HIGHER ORDER THINKING
Sample answer: A person taking a test averages _83 minute
to read a question and _65 minute to answer the question. It
7. Jim’s backyard is a rectangle that is 15 _56
yards long and 10 _25 yards wide. Jim buys sod
in pieces that are 1 _1 yards long and 1 _1 yards
3
takes the person 29 minutes to answer all of the questions.
How many questions are on the test? 24 questions
3
wide. How many pieces of sod will Jim need
to buy to cover his backyard with sod?
93 pieces
9. Mark wants to paint a mural. He has 1 _13
gallons of yellow paint, 1 _14 gallons of green
paint, and _78 gallon of blue paint. Mark
plans to use _34 gallon of each paint color.
How many gallons of paint will he have left
after painting the mural?
5
gallons
1 __
24
Work Area
11. Represent Real-World Problems Describe a real-world problem that
can be solved using the expression 29 ÷ ( _38 + _56 ). Find the answer in the
context of the situation.
4 picture frames
8. Eva wants to make two pieces of pottery.
She needs _35 pound of clay for one piece and
7
__
10 pound of clay for the other piece. She
has three bags of clay that weigh _45 pound
each. How many bags of clay will Eva need
to make both pieces of pottery? How many
pounds of clay will she have left over?
1
pounds left over
1_58 bags; 1__
10
5. A hairstylist schedules _14 hour to trim a
customer’s hair and _16 hour to style the
customer’s hair. The hairstylist plans to
work 3 _13 hours each day for 5 days each
week. How many appointments can the
hairstylist schedule each week if each
customer must be trimmed and styled?
10. Trina works after school and on weekends. She always works three days
each week. This week she worked 2 _34 hours on Monday, 3 _35 hours on
Friday, and 5 _12 hours on Saturday. Next week she plans to work the same
number of hours as this week, but will work for the same number of
hours each day. How many hours will she work on each day?
12. Justify Reasoning Indira and Jean begin their hike at 10 a.m. one
1
morning. They plan to hike from the 2 _25 -mile marker to the 8 __
-mile
10
marker along the trail. They plan to hike at an average speed of 3 miles
1
per hour. Will they reach the 8 __
-mile marker by noon? Explain your
10
reasoning.
1
7
Yes; they will hike a distance of 8 __
- 2 _25 = 5 __
miles.
10
10
At an average speed of 3 miles per hour, they will hike
9
7
÷ 3 = 1 __
hours. Noon is 2 hours
this distance in 5 __
10
10
1
from 10 a.m., so they will reach the 8 __
-mile marker
10
before noon.
13. Multiple Representations You are measuring walnuts for bananawalnut oatmeal and a spinach and walnut salad. You need _38 cup of
walnuts for the oatmeal and _34 cup of walnuts for the salad. You have a
_1 -cup scoop. Describe two different ways to find how many scoops of
4
walnuts you will need.
Sample answer: Add _38 + _34 = 1_18 to find the total amount
of walnuts needed. Then divide 1_18 ÷ _14 = 4_12 to find
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Name
how many scoops. Or, divide _38 ÷ _14 = 1_12 scoops for the
40 appointments
oatmeal and divide _34 ÷ _14 = 3 scoops for the salad. Then
add 1_12 + 3 = 4 _12.
Lesson 4.4
EXTEND THE MATH
PRE-AP
99
Activity available online
100
Unit 2
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Activity Kyle is hanging a new painting in his house. He knows that the painting is
twice as wide as it is tall. The painting is __14 as tall as the wall he is hanging it on. If the
wall is 8__45 feet high, what are the dimensions of the painting in inches?
52__45 inches wide by 26__25 inches tall
Solving Multistep Problems with Fractions and Mixed Numbers
100
MODULE QUIZ
Ready to Go On?
Ready
Assess Mastery
4.1 Applying GCF and LCM to Fraction Operations
Solve.
Use the assessment on this page to determine if students have
mastered the concepts and standards covered in this module.
3
1. _45 × _34
3. _38 + 2 _12
Response to
Intervention
2
1
Divide.
Enrichment
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Differentiated Instruction
Differentiated Instruction
• Reteach worksheets
• Challenge worksheets
ELL
• Success for English
Learners ELL
8. 3 _13 ÷ _23
10. 4 _14 ÷ 4 _12
Additional Resources
Assessment Resources includes:
• Leveled Module Quizzes
6. _13 ÷ _58
8
__
15
2
5
17
__
18
9. 1 _78 ÷ 2 _25
11. 8 _13 ÷ 4 _27
25
__
32
17
1__
18
4.4 Solving Multistep Problems with Fractions
and Mixed Numbers
PRE-AP
Extend the Math PRE-AP
Lesson Activities in TE
3
_
7
Divide.
Online and Print Resources
• Reading Strategies
23
__
30
4.3 Dividing Mixed Numbers
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Online Assessment
and Intervention
4. 1 _35 − _56
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9
__
14
7. Luci cuts a board that is _34 yard long into pieces that
are _38 yard long. How many pieces does she cut?
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instant scoring, feedback, and customized intervention
or enrichment.
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2_78
9
2. _57 × __
10
4.2 Dividing Fractions
5. _13 ÷ _79
Intervention
3
_
5
Personal
Math Trainer
Online Assessment
and Intervention
12. Jamal hiked on two trails. The first trail was 5 _13 miles
long, and the second trail was 1 _34 times as long as
the first trail. How many miles did Jamal hike?
14 _23 miles
ESSENTIAL QUESTION
13. Describe a real-world situation that is modeled by dividing two
fractions or mixed numbers.
Sample answer: You want to divide 3 _34 pounds of grapes
into bags that hold _34 pound each. Divide 3 _34 by _34 to find
that you can fill 5 bags.
Module 4
Common Core Standards
Lesson
Exercises
Common Core Standards
4.1
1–4
6.NS.4
4.2
5–7
6.NS.1
4.3
8–11
6.NS.1
4.4
12
6.NS.1
101
Unit 2 Module 4
101
MODULE 4 MIXED REVIEW
Assessment
Readiness
Assessment Readiness Tip Encourage students to check their
answers to multiplication problems by using division, and to division
problems by using multiplication, before filling in the answer bubble.
Item 3 Students should divide their answer by __97 to check it. If the
answer is correct, they should find the quotient to be the initial
value of 133 pennies.
Item 7 Students need to know that area is the product of the
length and width. Since students need to divide to find the width,
they should multiply their answer by 12 __12 and confirm that the area
is 103 __18 square feet.
1. Two sides of a rectangular fence are 5 _58 feet
long. The other two sides are 6 _1 feet long.
What is the perimeter?
Item 4 Students may apply the Commutative Property incorrectly
and find answer choice B to be another form of the expression.
Remind them that the Commutative Property applies only to
addition and multiplication, not subtraction or division.
Item 6 Students may take the reciprocal of just the fractional
part of the mixed number and find answer C to be the reciprocal.
Remind them that mixed numbers must be changed into
improper fractions before their reciprocals can be found.
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B 13 feet
5
D 35 __ feet
32
3
A GCF: 3; _
4
6
_
B GCF: 3;
8
3
C GCF: 6; _
4
6
_
D GCF: 6;
8
B $1.52
C
D $1.71
4. Which of these is the same as _35 ÷ _47 ?
3
7
A _÷ _
5
4
4 _
3
_
B
÷
7 5
3
4
C _× _
5
7
3 7
D _×_
5 4
102
4
A 4_
9
B 6
C
D 10
7
7. A rectangular patio has a length of 12 _12 feet
and an area of 103 _18 square feet. What is
the width of the patio?
1
A 4 _ feet
8
1
_
B 8 feet
4
1
C 16 _ feet
2
5
A -_
8
1
B -_
8
1
_
C
4
D 0.5
Mini-Task
9. Jodi is cutting out pieces of paper that
measure 8 _12 inches by 11 inches from a
larger sheet of paper that has an area of
1,000 square inches
a. What is the area of each piece of paper
that Jodi is cutting out?
5. Andy has 6 _23 quarts of juice. How many
_2 -cup servings can he pour?
3
Online
Assessment and
Intervention
3
B _
7
24
D __
7
D 33 feet
A $1.49
$1.68
7
A __
24
7
C _
3
8. Which number is greater than the absolute
value of -_38 ?
3. A jar contains 133 pennies. A bigger jar
contains 1 _27 times as many pennies.
What is the value of the pennies in the
bigger jar?
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4
18
2. Which shows the GCF of 18 and 24 with __
24
in simplest form?
Avoid Common Errors
Online
Assessment and
Intervention
6. What is the reciprocal of 3 _37 ?
Selected Response
7
A 11 _ feet
8
3
_
C 23 feet
4
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93 _12 square inches
b. What is the greatest possible number of
pieces of paper that Jodi can cut out of
the larger sheet?
10
Unit 2
Common Core Standards
Items
Grade 6 Standards
Mathematical Practices
1
6.NS.1
MP.4
2
6.NS.4
MP.7
3
6.NS.1
MP.4
4
6.NS.1
MP.2
5
6.NS.1
MP.4
6*
6.NS.1
MP.6, MP.7
7
6.NS.1
MP.4
8
6.NS.1
MP.2
9
6.NS.1
MP.4, MP.5
* Item integrates mixed review concepts from previous modules or a previous course.
Operations with Fractions
102