Download Operations with Fractions

Operations with
Fractions
?
MODULE
4
LESSON 4.1
ESSENTIAL QUESTION
Applying GCF and
LCM to Fraction
Operations
How can you use operations
with fractions to solve
real-world problems?
6.NS.4
LESSON 4.2
Dividing Fractions
6.NS.1
LESSON 4.3
Dividing Mixed
Numbers
6.NS.1
LESSON 4.4
Solving Multistep
Problems with
Fractions and Mixed
Numbers
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6.NS.1
Real-World Video
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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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Math On the Spot
Animated Math
Personal Math Trainer
Go digital with your
write-in student
edition, accessible on
any device.
Scan with your smart
phone to jump directly
to the online edition,
video tutor, and more.
Interactively explore
key concepts to see
how math works.
Get immediate
feedback and help as
you work through
practice sets.
75
Are YOU Ready?
Personal
Math Trainer
Complete these exercises to review skills you will need
for this module.
Write an Improper Fraction
as a Mixed Number
EXAMPLE
13 _
__
= 55 + _55 + _35
5
= 1 + 1 + _35
= 2 + _35
= 2_35
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Online Practice
and Help
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.
Write each improper fraction as a mixed number.
1. _94
2. _83
23
3. __
6
11
4. __
2
17
5. __
5
15
6. __
8
33
7. __
10
29
8. __
12
Multiplication Facts
EXAMPLE
7×6=
7 × 6 = 42
Use a related fact you know.
6 × 6 = 36
Think: 7 × 6 = (6 × 6) + 6
= 36 + 6
= 42
9. 6 × 5
10. 8 × 9
11. 10 × 11
12. 7 × 8
13. 9 × 7
14. 8 × 6
15. 9 × 11
16. 11 × 12
Division Facts
EXAMPLE
63 ÷ 7 =
Think:
63 ÷ 7 = 9
So, 63 ÷ 7 = 9.
7 times what number equals 63?
7 × 9 = 63
Divide.
76
Unit 2
17. 35 ÷ 7
18. 56 ÷ 8
19. 28 ÷ 7
20. 48 ÷ 8
21. 36 ÷ 4
22. 45 ÷ 9
23. 72 ÷ 8
24. 40 ÷ 5
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Multiply.
Reading Start-Up
Visualize Vocabulary
Use the ✔ words to complete the triangle. Write the review word
that fits the description in each section of the triangle.
part
of a whole
top number
of a fraction
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)
Understand Vocabulary
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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
3
_
and _53
5
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
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
77
GETTING READY FOR
Operations with Fractions
Understanding the standards and the vocabulary terms in the standards
will help you know exactly what you are expected to learn in this module.
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.
Key Vocabulary
What It Means to You
You will learn how to divide two fractions. You will also understand
the relationship between multiplication and division.
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 _
= __
·2
4 3
= 4_12
The recipe will make 4_21 servings.
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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Common Core
Standards
explained.
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78
Unit 2
What It Means to You
You can use greatest common factors and least common multiples
to simplify answers when you calculate with fractions.
EXAMPLE 6.NS.4
Add. Write the answer in simplest form.
1 _
Use the LCM of 3 and 6 as
_
+ 16 = _26 + _16
3
a common denominator.
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.
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6.NS.4
LESSON
4.1
?
Applying GCF and
LCM to Fraction
Operations
ESSENTIAL QUESTION
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.
How do you use the GCF and LCM when adding, subtracting,
and multiplying fractions?
Multiplying Fractions
To multiply two fractions you first multiply the numerators and then multiply
the denominators.
numerator × numerator
numerator
_____________________
= __________
denominator × denominator
denominator
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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.
EXAMPL 1
EXAMPLE
6.NS.4
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
Write the problem as a single fraction.
2
6× 2
= ______
7× 3
Simplify before multiplying using the GCF.
2× 2
= _____
7× 1
Multiply numerators. Multiply denominators.
1
= _47
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
YOUR TURN
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Multiply. Write each product in simplest form.
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and Help
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1. _16 × _35
2. _34 × _79
3. _37 × _23
4. _45 × _27
8
7
× __
5. __
10
21
6. _67 × _16
Multiplying Fractions and
Whole Numbers
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To multiply a fraction by a whole number, you can 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
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?
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?
5
18
_
× 18 = _59 × __
1
9
× 18
= 5______
9×1
2
1
5×2
= ____
1×1
10
= 10
= __
1
10 students have pets.
80
Unit 2
5
times 18
You can write __
9
three ways.
5
__
×18
9
5
__
· 18
9
5
__
(18)
9
Rewrite 18 as a fraction.
Simplify before multiplying using the GCF.
Multiply numerators. Multiply denominators.
Simplify by writing as a whole number.
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STEP 1
Reflect
7.
Analyze Relationships Is the product of a fraction less than 1 and a
whole number greater than or less than the whole number? Explain.
YOUR TURN
Multiply. Write each product in simplest form.
9.
3
_
× 20
5
10.
1
_
× 8
3
11.
1
_
× 14
4
12.
7
× 7
3 __
10
13.
3
× 10
2 __
10
8.
5
_
× 24
8
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Adding and Subtracting Fractions
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
6.NS.4
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8
1
_
Add __
15 + 6 . Write the sum in simplest form.
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STEP 1
Rewrite the fractions as equivalent fractions. Use the
LCM of the denominators.
8
8×2
16
__
→ _____
→ __
15
15 × 2
30
1
×
5
5
1
_ → ____ → __
6
6×5
30
STEP 2
The LCM of 15 and 6 is 30.
Mathematical Practices
Add the numerators of the equivalent fractions. Then simplify.
16 __
5
21
__
+ 30
= __
30
30
÷3
_____
= 21
30 ÷ 3
7
= __
10
Math Talk
Can you use another
common multiple of
the denominators in
Example 3 to find
the sum? Explain.
Simplify by dividing by the GCF.
The GCF of 21 and 30 is 3.
Reflect
14.
Can you also use the LCM of the denominators of the fractions to
8
_1
rewrite the difference __
15 - 6 ? What is the difference?
Lesson 4.1
81
YOUR TURN
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and Help
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Add or subtract. Write each sum or difference in simplest form.
5
15. __
+ _16
14
5
3
16. __
- __
12 20
5
17. __
- _38
12
3
18. 1 __
+ _14
10
19. _23 + 6 _15
20. 3 _16 - _17
Guided Practice
Multiply. Write each product in simplest form. (Example 1)
1. _12 × _58
2. _35 × _59
3. _38 × _25
4. 2 _38 × 16
5
5. 1 _45 × __
12
2
× 5
6. 1 __
10
Find each amount. (Example 2)
7. _14 of 12 bottles of water =
bottles
9. _35 of $40 restaurant bill = $
8. _23 of 24 bananas =
10. _56 of 18 pencils =
bananas
pencils
5
11. _38 + __
24
5
1
+ __
12. __
20
12
9
- _1
13. __
20 4
9
3
- __
14. __
10 14
5
15. 3 _38 + __
12
7
5
- __
16. 5 __
10 18
?
ESSENTIAL QUESTION CHECK-IN
17. How can using the GCF and LCM help you when you add, subtract, and
multiply fractions?
82
Unit 2
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Add or subtract. Write each sum or difference in simplest form.
Name
Class
Date
4.1 Independent Practice
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6.NS.4
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
4
is _23 full. How much does the bag of
peanuts weigh now? Show your work.
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Online Practice
and Help
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.
Fruit Salad
3_12 cups thinly sliced rhubarb
15 seedless grapes, halved
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?
_1 orange, sectioned
2
10 fresh strawberries, halved
_3 apple, cored and diced
5
_2 peach, sliced
3
1 plum, pitted and sliced
_1 cup fresh blueberries
4
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b. If Marianne’s father calls and says he
needs 13 pounds of potting soil, how
many additional bags should she buy?
a. What are the new amounts for the
oranges, apples, blueberries, and
peaches?
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?
b. One half of the woodwinds are
clarinets. What fraction of the band is
clarinets?
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.
c. One eighth of the brass instruments
are tubas. If there are 240 instruments
in the band, how many are tubas?
Lesson 4.1
83
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?
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?
b. Suppose the teacher begins recording on a digital camera with an
hour available. Show that there is 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.
c. How much time is left on the digital camera?
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 in which all bicycles cost _65 of the regular price. What is the
sale price of the bicycle?
26. Justify Reasoning To multiply a whole number by a fraction, you can
first write the whole number as a fraction with a denominator of 1.
Explain why following this step does not change the product.
84
Unit 2
Work Area
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25. Error Analysis To find the product _73 × _94 , Cameron found the GCF of
4
3 and 9. He then multiplied the fractions _17 and _49 to find the product __
63.
What was Cameron’s error?
LESSON
4.2 Dividing Fractions
?
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.
ESSENTIAL QUESTION
How do you divide fractions?
EXPLORE ACTIVITY 1
6.NS.1
Modeling Fraction Division
For some real-world problems, you may need to divide a fraction by
a fraction. For others, you may need to divide a fraction by a whole
number.
A You have _34 cup of salsa for making burritos. Each
burrito requires _18 cup of salsa. How many burritos
can you make?
3
4
To find the number of burritos that can
be made, you need to determine how
many _18 -cup servings are in _34 cup.
In the diagram, _34 of a whole is divided
into quarters and into eighths. How many
1
8
eighths are there in _34 ?
You have enough salsa to make
burritos.
© Houghton Mifflin Harcourt Publishing Company
B Five people share _12 pound of cheese
equally. How much cheese does each
person receive?
To find how much cheese each person
receives, you can divide _12 pound
into 5 equal parts. Use the diagram to
determine what fraction of a whole pound
each person receives.
Each person receives
pound.
Reflect
1. Write the division shown by the models in
A
and
B
.
Lesson 4.2
85
Reciprocals
Another way to divide fractions is to use reciprocals. Two numbers whose
product is 1 are reciprocals.
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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.
numerator · ___________
denominator = 1
____________
denominator
numerator
EXAMPLE 1
Prep for 6.NS.1
Find the reciprocal of each number.
A _29
9
_
2
Switch the numerator and denominator.
The reciprocal of _92 is _92.
Math Talk
Mathematical Practices
How can you check
that the reciprocal in
A is correct?
B _18
8
_
1
Switch the numerator and denominator.
The reciprocal of _81 is _81, or 8.
C 5
5 = _51
5
_
1
Rewrite as a fraction.
1
_
5
Switch the numerator and the denominator.
The reciprocal of 5 is _51.
Reflect
2. Is any number its own reciprocal? If so, what number(s)? Justify your answer.
4. The reciprocal of a whole number is a fraction with
numerator.
in the
YOUR TURN
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86
Unit 2
Find the reciprocal of each number.
7
_
5. 8
6. 9
1
__
7. 11
© Houghton Mifflin Harcourt Publishing Company
3. Communicate Mathematical Ideas Does every number have a
reciprocal? Explain.
EXPLORE ACTIVITY 2
6.NS.1
Using Reciprocals to Divide
A The diagram below models the division problem 5 ÷ _58.
5
5
8
5
8
5
8
5
8
5
8
5
8
5
8
5
8
Each unit is divided into eighths. How many sections representing _58 are
there in 5?
sections
5=
5 ÷ __
8
B Use the diagrams to complete the division problems.
5
4
5
8
5
8
5
8
5
8
5=
5 ÷ __
__
5 ÷ __
5=
__
4 8
8 8
C Complete the table. What do you notice about the
Quotient
quotient and product in each row? What do you notice
5=
about the divisor in column 1 and the multiplier in
5 ÷ __
8
column 2?
5 ÷ __
5=
__
4 8
5 ÷ __
5=
__
© Houghton Mifflin Harcourt Publishing Company
8
8
Product
8=
5 × __
5
5 × __
8=
__
4
5
5 × __
8=
__
8
5
Reflect
8. Make a Conjecture Use the pattern in the table to make a conjecture
about how you can use multiplication to divide a number by a fraction.
9. Sketch a model on a separate piece of paper that you could use to solve
the division problem _67 ÷ _27. How can you use the pattern you see in the
table in C to solve this division problem?
Lesson 4.2
87
Using Reciprocals to Divide Fractions
Dividing by a fraction is equivalent to
multiplying by its reciprocal.
1 _
_
÷ 14 = _45
5
4 _
1×_
_
= 45
1
5
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EXAMPLE 2
6.NS.1
Divide _59 ÷ _23 . Write the quotient in simplest form.
STEP 1
Animated
Math
Rewrite as multiplication, using the reciprocal of the divisor.
5 _
_
÷ 2 = _59 × _32
9 3
2 __
The reciprocal of __
is 32 .
3
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STEP 2
Multiply and simplify.
5 _
15
_
× 32 = __
9
18
= _56
5 _
_
÷ 23 = _56
9
Multiply the numerators. Multiply the denominators.
Write the answer in simplest form.
15 ÷ 3 __
______
=5
18 ÷ 3 6
YOUR TURN
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Divide.
9
11. __
÷ _35 =
10
9
10. __
÷ _25 =
10
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Guided Practice
1. _25
2. 9
10
3. __
3
Divide. (Explore 1, Explore 2, and Example 2)
4. _43 ÷ _53 =
?
3
5. __
÷ _45 =
10
ESSENTIAL QUESTION CHECK-IN
7. Explain how to divide a number by a fraction.
88
Unit 2
6. _12 ÷ _25 =
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Find the reciprocal of each fraction. (Example 1)
Name
Class
Date
4.2 Independent Practice
6.NS.1
8. 8 ÷ _23 =
9. 7 ÷ _34 =
4
÷ _25 =
10. __
15
11. _58 ÷ _78 =
7
÷ _38 =
12. __
16
7 =
13. _45 ÷ __
10
14. Alison has _12 cup of yogurt for making fruit
parfaits. Each parfait requires _18 cup of
yogurt. How many parfaits can she make?
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15. How many tenths are there in _45 ?
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18. A pitcher contains _23 quart of water. If an
equal amount of water is poured into each
of 6 glasses, how much water will each
glass contain?
19. 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?
20. You make a salad to share with friends.
Your sister eats _13 of it before they arrive.
a. You want to divide the leftover salad
evenly among six friends. What
expression describes the situation?
Explain.
16. Biology Suppose one honeybee makes
1
__
teaspoon of honey during its lifetime.
12
How many honeybees are needed to
make _12 teaspoon of honey?
b. What fraction of the original salad does
each friend receive?
17. A team of runners is needed to run a _14 -mile
1
relay race. If each runner must run __
mile,
16
how many runners will be needed?
21. Six people share _53 pound of peanuts
equally. What fraction of a pound of
peanuts does each person receive?
Lesson 4.2
89
9
22. Liam has __
gallon of paint for painting the birdhouses he sells at the craft
10
1
fair. Each birdhouse requires __
gallon of paint. How many birdhouses can
20
Liam paint? Show your work.
FOCUS ON HIGHER ORDER THINKING
Work Area
23. Interpret the Answer The length of a ribbon is _34 meter. Sun Yi needs
pieces measuring _13 meter for an art project. What is the greatest number
of pieces measuring _13 meter that can be cut from the ribbon? How much
ribbon will be left after Sun Yi cuts the ribbon? Explain your reasoning.
24. Represent Real-World Problems Write and solve a real-world problem
that can be solved using the expression _34 ÷ _16.
25. 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.
1
27. Make a Prediction Susan divides the fraction _58 by __
. Her friend Robyn
16
5
1
_
__
divides 8 by 32 . Predict which person will get the greater quotient. Explain
and check your prediction.
90
Unit 2
© Houghton Mifflin Harcourt Publishing Company
26. Communicate Mathematical Ideas The reciprocal of a fraction less than
1 is always a fraction greater than 1. Why is this?
Dividing Mixed
Numbers
LESSON
4.3
?
6.NS.1
Interpret and compute
quotients of fractions,
and solve word problems
involving division of fractions
by fractions.
ESSENTIAL QUESTION
How do you divide mixed numbers?
6.NS.1
EXPLORE ACTIVITY
Modeling Mixed Number Division
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?
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
1
4
1
4
1
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
2
1
4
B How many fourths are in 2_12?
Antoine has enough rice to make
sushi rolls.
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Reflect
1.
Communicate Mathematical Ideas Which mathematical operation
could you use to find the number of sushi rolls that Antoine can make?
Explain.
2.
Multiple Representations Write the division shown by the model.
3.
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.
Lesson 4.3
91
Using Reciprocals to Divide
Mixed Numbers
Math On the Spot
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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.
EXAMPLE 1
6.NS.1
One serving of Harold’s favorite cereal contains 1_25 ounces. How many
servings are in a 17_12 -ounce box?
My Notes
STEP 1
Write a division statement to represent the situation.
17_12 ÷ 1_25
STEP 2
You need to find how many
2
1
17__
groups of 1__
2.
5 are in
Rewrite the mixed numbers as fractions greater than 1.
35 _
÷ 75
17_12 ÷ 1_25 = __
2
STEP 3
Rewrite the problem as multiplication using the reciprocal of
the divisor.
35 _
35 _
__
÷ 75 = __
× 57
2
2
STEP 4
7
5
__
The reciprocal of __
5 is 7 .
Multiply.
5 35 5
35 _
__
× 57 = __
× _7
2
2
1
5×5
= _____
2× 1
25
, or 12_12
= __
2
Simplify first using the GCF.
Multiply numerators. Multiply denominators.
Write the result as a mixed number.
Reflect
92
Unit 2
4.
Analyze Relationships Explain how can you check the answer.
5.
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.
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There are 12_12 servings of cereal in the box.
YOUR TURN
6.
Sheila has 10 _21 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.
Personal
Math Trainer
Online Practice
and Help
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Solving Problems Involving Area
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.
Math On the Spot
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EXAMPL 2
EXAMPLE
6.NS.1
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
Write the situation as a division problem.
56 _23 ÷ 8 _12
STEP 2
Rewrite the mixed numbers as fractions greater than 1.
170 __
÷ 17
56 _23 ÷ 8 _12 = ___
3
2
STEP 3
Rewrite the problem as multiplication using the reciprocal of
the divisor.
Math Talk
Mathematical Practices
Explain how to find
the length of a rectangle
when you know the area
and the width.
170
170
17
2
___
÷ __
= ___
× __
17
3
2
3
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=
10 170 × 2
______
3 × 17
20
, or 6 _32
= __
3
1
Multiply numerators. Multiply denominators.
Simplify and write as a mixed number.
The width of the sandbox is 6 _23 feet.
Reflect
7.
Check for Reasonableness How can you determine if your answer
is reasonable?
Lesson 4.3
93
YOUR TURN
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Online Practice
and Help
8.
The area of a rectangular patio is 12_38 square meters.
The width of the patio is 2_34 meters. What is the length?
9.
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?
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Guided Practice
Divide. Write each answer in simplest form. (Explore Activity and Example 1)
1. 4_14 ÷ _34
2. 1_12 ÷ 2_14
3=
_____ ÷ __
_____ ÷ _____ =
_____ × _____ =
_____ × _____ =
4
4
4
2
4
2
3. 4 ÷ 1_18 =
4. 3_15 ÷ 1_17 =
5. 8_13 ÷ 2_12 =
6. 15_13 ÷ 3_56 =
Write each situation as a division problem. Then solve. (Example 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?
?
ESSENTIAL QUESTION CHECK-IN
9. How is dividing mixed numbers like dividing fractions?
94
Unit 2
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7. A sandbox has an area of 26 square feet, and
the length is 5_21 feet. What is the width of the
sandbox?
Name
Class
Date
4.3 Independent Practice
Personal
Math Trainer
6.NS.1
Online Practice
and Help
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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
1
4
1
4
1
4
1
4
1
1
4
1
4
1
4
1
4
1
4
1
4
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 that are _2 yard
3
3
long that she can cut. Use the model to find the number of such pieces
she can cut. How much ribbon will be left after she cuts these pieces?
1
1
3
1
3
1
1
3
1
3
1
3
1
1
3
1
3
1
3
1
1
3
1
3
1
3
1
3
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.
13. Multistep Zoey made 5 _21 cups of trail mix for a camping trip. She wants
to divide the trail mix into _34 -cup servings.
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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
b. What size would the servings need to be for everyone to have a
serving? Explain.
c. If Zoey decides to use the _34 -cup servings, how much more trail mix
will she need? Explain.
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.
Lesson 4.3
95
11
15. The area of a rectangular mirror is 11 __
16 square feet. The width of the
3
_
mirror is 2 4 feet. If there is a 5 foot tall space on the wall to hang the
mirror, will it fit? Explain.
16. Ramon has a rope that is 25 _21 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 _21 feet
long. The two boards had equal widths. The total area of the set
was 60 _83 square feet. What was the width?
FOCUS ON HIGHER ORDER THINKING
Work Area
2
18. Draw Conclusions Micah divided 11 _32 by 2 _56 and got 4 __
17 for an answer.
Does his answer seem reasonable? Explain your thinking. Then check
Micah’s answer.
20. Analyze Relationships Explain how you can find the missing number
= 2 _57 . Then find the missing number.
in 3 _45 ÷
96
Unit 2
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19. Explain the Error To divide 14 _23 ÷ 2 _34 , Erik multiplied 14 _23 × _43 . Explain
Erik’s error.
LESSON
4.4
?
Solving Multistep Problems
with Fractions and Mixed
Numbers
ESSENTIAL QUESTION
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.
How can you solve word problems involving more than one
fraction operation?
Solving Problems with
Rational Numbers
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.
EXAMPL 1
EXAMPLE
Problem
Solving
Math On the Spot
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6.NS.1
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?
Analyze Information
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Photodisc/Getty Images
Identify the important information.
• Jon needs _34 cup of dried lentils for soup and 1_12 cups for salad.
• Jon has a _18 -cup scoop.
• You need to find the total number of many scoops of lentils he needs.
Formulate a Plan
(
)
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.
First add to find the total amount of dried lentils Jon will need.
3
_
+ 1_12 = _34 + _32
4
= _34 + _64
= _94
= 2_14
John needs
1
2 __
cups of
4
lentils.
Lesson 4.4
97
Jon needs 2 _14 cups of dried lentils for both the soup and the salad.
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
×8
= 9____
4×1
Math Talk
2
Simplify before
multiplying using
the GCF.
1
Mathematical Practices
What if Jon had a _14-cup scoop
instead of a _18-cup scoop?
How many scoops would
he need? Explain.
18
= __
= 18
1
Jon will need 18 scoops of dried lentils to have enough for both the lentil
barley soup and the lentil salad.
Justify and Evaluate
You added _34 and 1_21 first to find the total number of cups of lentils. Then
you divided the sum by _81 to find the number of _18 -cup scoops.
YOUR TURN
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Math Trainer
Online Practice
and Help
1. Before making some soup, a chef mixes _21 cup of cream
with _34 cup of milk. If the scientist uses _81 cup of the
combined liquids to make one batch of soup, how
many batches of soup can be made?
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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 _61 yard long. How
many pieces will the student cut to use for the gift boxes? (Example 1)
?
ESSENTIAL QUESTION CHECK-IN
2. Describe a real-world problem that you could solve using more than one
fraction operation. Show how to solve your problem.
98
Unit 2
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Guided Practice
Name
Class
Date
4.4 Independent Practice
6.NS.1.1
3. Karl participated in a two-day charity walk.
His parents donated money for each mile
he walked. Karl walked 3 _14 miles on the first
day and 5 _12 miles on the second day. His
parents donated $175. How much money
did his parents pay for each mile?
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Images
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?
5. A hairstylist schedules _14 hour to trim a
customer’s hair and _61 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?
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and Help
1
6. A picture framer has a board 10 __
feet
12
3
_
long. The framer notices that 2 8 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?
7. Jim’s lawn is a rectangle that is 15 _56 yards
long and 10 _52 yards wide. Jim buys sod in
pieces that are 1 _31 yards long and 1 _13 yards
wide. How many pieces of sod will Jim need
to buy to cover his lawn with sod?
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?
9. Mark wants to paint a mural. He has
1 _13 gallons of yellow paint, 1 _14 gallons of
green paint, and _87 gallon of blue paint.
Mark plans to use _43 gallon of each color.
How many gallons of paint will he have left
after painting the mural?
Lesson 4.4
99
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?
FOCUS ON HIGHER ORDER THINKING
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.
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 _43 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.
100
Unit 2
© Houghton Mifflin Harcourt Publishing Company
12. Justify Reasoning Indira and Jean begin their hike at 10 a.m. one
1
morning. They plan to hike from the 2 _52 -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.
MODULE QUIZ
Ready
Personal
Math Trainer
4.1 Applying GCF and LCM to Fraction Operations
Solve.
Online Practice
and Help
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1. _45 × _34
9
2. _57 × __
10
3. _38 + 2 _12
4. 1 _35 − _56
4.2 Dividing Fractions
Divide.
5. _13 ÷ _79
6. _13 ÷ _58
7. Luci cuts a board that is _43 yard long into pieces that
are _38 yard long. How many pieces does she cut?
4.3 Dividing Mixed Numbers
Divide.
8. 3 _13 ÷ _23
10. 4 _14 ÷ 4 _12
9. 1 _78 ÷ 2 _25
11. 8 _13 ÷ 4 _27
4.4 Solving Multistep Problems with Fractions
and Mixed Numbers
© Houghton Mifflin Harcourt Publishing Company
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?
ESSENTIAL QUESTION
13. Describe a real-world situation that is modeled by dividing two
fractions or mixed numbers.
Module 4
101
MODULE 4
MIXED REVIEW
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Assessment Readiness
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Online Practice
and Help
1. Consider each equation. Is the equation true?
Select Yes or No for equations A–C.
5
A. _23 + _38 = __
24
7
B. 3 _34 - 2 _25 = 1 __
20
5
7
C. 8 _59 + 6 __
= 15 __
12
36
Yes
No
Yes
No
Yes
No
2. Party leftovers of 1 _34 pounds of cheese, 1 _23 pounds of crackers, and _43 gallon of
juice will be divided equally into containers.
Choose True or False for each statement.
A. If each juice container holds
_1 gallon of juice, then you need
8
6 juice containers.
B. If each cracker bag can contain _56 pound
of crackers, then you need 4 bags.
C. If there are 7 containers for cheese, then
each container has _14 pound of cheese.
True
False
True
False
True
False
4. Dan weighed his dogs Spot and Rover in March and in April, and recorded
the weight change for each dog. He recorded a change of 2 _12 pounds for
Spot and -2 _43 pounds for Rover. He says Spot’s weight changed by a greater
amount because 2 _12 > -2 _34. Do you agree? Explain.
102
Unit 2
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3. A rectangular stone in a piece of costume jewelry has a width of _45 inch and
7
an area of __
square inch. Is a silver wire with a length of 3 _12 inches long
10
enough to outline the stone? Explain how you solved this problem.