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Transcript
```Number and Operations – Fractions
→ Extend understanding of fraction equivalence and ordering.
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
Lesson
47
48
49
50
51
52
53
54
CC.4.NF.1
CC.4.NF.1
CC.4.NF.1
CC.4.NF.1
CC.4.NF.1
CC.4.NF.2
CC.4.NF.2
CC.4.NF.2
Equivalent Fractions . . . . . . . . . . . . .
Generate Equivalent Fractions . . . . . . . .
Simplest Form . . . . . . . . . . . . . . . .
Common Denominators . . . . . . . . . . .
Problem Solving • Find Equivalent Fractions
Compare Fractions Using Benchmarks . . . .
Compare Fractions . . . . . . . . . . . . . .
Compare and Order Fractions . . . . . . . .
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. 93
. 95
. 97
. 99
.101
.103
.105
.107
Add and Subtract Parts of a Whole . . . . . .
Write Fractions as Sums . . . . . . . . . . . .
Rename Fractions and Mixed Numbers . . . . .
Add and Subtract Mixed Numbers . . . . . . .
Subtraction with Renaming . . . . . . . . . .
Algebra • Fractions and Properties of Addition
Add Fractions Using Models . . . . . . . . . .
Subtract Fractions Using Models . . . . . . . .
Add and Subtract Fractions . . . . . . . . . .
Problem Solving • Multistep
Fraction Problems . . . . . . . . . . . . . . .
Multiples of Unit Fractions . . . . . . . . . . .
Multiples of Fractions . . . . . . . . . . . . .
Multiply a Fraction by a Whole Number
Using Models . . . . . . . . . . . . . . . . .
Multiply a Fraction or Mixed Number by
a Whole Number . . . . . . . . . . . . . . .
Problem Solving • Comparison Problems
with Fractions . . . . . . . . . . . . . . . . .
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.109
.111
.113
.115
.117
.119
.121
.123
.125
→ Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
© Houghton Mifflin Harcourt Publishing Company
Lesson
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Lesson
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Lesson
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55
56
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59
60
61
62
63
64
Lesson 65
Lesson 66
Lesson 67
Lesson 68
Lesson 69
CC.4.NF.3a
CC.4.NF.3b
CC.4.NF.3b
CC.4.NF.3c
CC.4.NF.3c
CC.4.NF.3c
CC.4.NF.3d
CC.4.NF.3d
CC.4.NF.3d
CC.4.NF.3d
CC.4.NF.4a
CC.4.NF.4b
CC.4.NF.4b
CC.4.NF.4c
CC.4.NF.4c
. . .127
. . .129
. . .131
. . .133
. . .135
. . .137
→ Understand decimal notation for fractions, and compare decimal fractions.
Lesson
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Lesson
Lesson
Lesson
Lesson
70
71
72
73
74
75
CC.4.NF.5
CC.4.NF.5
CC.4.NF.6
CC.4.NF.6
CC.4.NF.6
CC.4.NF.7
Equivalent Fractions and Decimals . . .
Add Fractional Parts of 10 and 100 . .
Relate Tenths and Decimals . . . . . .
Relate Hundredths and Decimals . . . .
Relate Fractions, Decimals, and Money .
Compare Decimals . . . . . . . . . . .
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.145
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v
Name
LESSON
47
1
Equivalent Fractions
CC.4.NF.1
OBJECTIVE Use models to show equivalent fractions.
__.
Write two fractions that are equivalent to 2
6
2.
Step 1 Make a model to represent __
6
The rectangle is divided into 6 equal parts, with 2 parts shaded.
Step 2 Divide the rectangle from Step 1 in half.
The rectangle is now divided into 12 equal parts, with 4 parts shaded.
4 . So, __
2 and ___
4 are equivalent.
The model shows the fraction ___
12
6
12
© Houghton Mifflin Harcourt Publishing Company
Step 3 Draw the same rectangle as in Step 1, but with only
3 equal parts. Keep the same amount of the rectangle shaded.
The rectangle is now divided into 3 equal parts, with 1 part shaded.
1. So, __
2 and __
1 are equivalent.
The model shows the fraction __
3
6
3
Use models to write two equivalent fractions.
__
1. 2
4
__
2. 4
6
1
4
__, __
2 8
Number and Operations–Fractions
Check students’ models.
2
8
__, ___
3 12
93
Name
1
Equivalent Fractions
CC.4.NF.1
Use the model to write an equivalent fraction. Possible answers are given.
1.
4
__
6
2
__
3
=
2.
3
__
4
6
__
8
=
8 = __
4
3. ___
10
5
7
__ ≠ ___
4. 1
2
12
8
__ ≠ ___
5. 3
4
12
4
__ = __
6. 2
3
6
4
__ ≠ ___
7. 5
8
10
4
__ = ___
8. 2
6
12
1
20 = __
9. ____
100
5
9
__ ≠ ___
10. 5
8
10
Problem Solving
11. Jamal finished __56 of his homework.
Margaret finished __34 of her
10
homework, and Steve finished __
12
of his homework. Which two
students finished the same amount
of homework?
Jamal and Steve
94
12. Sophia’s vegetable garden is divided
into 12 equal sections. She plants
carrots in 8 of the sections. Write
two fractions that are equivalent to
the part of Sophia’s garden that is
planted with carrots.
2, __
4
3 6
Lesson 47
© Houghton Mifflin Harcourt Publishing Company
Tell whether the fractions are equivalent. Write = or ≠.
Name
LESSON
48
1
Generate Equivalent Fractions
CC.4.NF.1
OBJECTIVE Use multiplication to generate equivalent fractions.
4.
Write an equivalent fraction for __
5
Step 1 Choose a whole number, like 2.
Step 2 Create a fraction using 2 as the numerator and denominator: __22
This fraction is equal to 1. You can multiply a number by 1
without changing the value of the number.
2: 4
× 2 = ___
8.
__ by __
_____
Step 3 Multiply 4
5
2 5 × 2 10
8 are equivalent.
4 and ___
So, __
5
10
4.
Write another equivalent fraction for __
5
Step 1 Choose a different whole number, like 20.
20.
Step 2 Create a fraction using 20 as the numerator and denominator: ___
20
20: 4
× 20 = ____
80 .
__ by ___
_______
Step 3 Multiply 4
5
20 5 × 20 100
© Houghton Mifflin Harcourt Publishing Company
80 are equivalent.
4 and ____
So, __
5
100
Write two equivalent fractions.
2
1. __
6
4
2. ___
10
8 , ___
12
___
6 , ___
8
___
18 24
__
3. 3
8
20 30
__
4. 3
5
18
9 , ___
___
24 48
Number and Operations–Fractions
21
12
___, ___
20 35
95
Name
1
Generate Equivalent Fractions
Write two equivalent fractions for each.
1
1. __
3
2
2. __
3
1
×2=2
_____
__
3×2 6
CC.4.NF.1
1
3. __
2
4, ___
8
__
4
4. __
5
80
8 , ____
___
2
4
__, __
4 8
6 12
10 100
1
× 4 = ___
4
_____
3 × 4 12
Tell whether the fractions are equivalent.
Write = or ≠.
3
1 = ___
5. __
4
12
5
__ ≠ ___
6. 4
5
10
2
__ ≠ __
7. 3
8
6
6
__ = __
8. 3
4
8
10
__ = ___
9. 5
6
12
5
6 ≠ __
10. ___
12
8
4
__ = ___
11. 2
5
10
3
__ ≠ ___
12. 2
4
12
13. Jan has a 12-ounce milkshake.
Four ounces in the milkshake are
vanilla, and the rest is chocolate.
What are two equivalent fractions
that represent the fraction of the
milkshake that is vanilla?
1
2
__ and __
3
6
96
4
14. Kareem lives __
of a mile from the
10
mall. Write two equivalent fractions
that show what fraction of a mile
Kareem lives from the mall.
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
2
8
__ and ___
5
20
Lesson 48
Name
LESSON
49
1
Simplest Form
CC.4.NF.1
OBJECTIVE Write and identify equivalent fractions in simplest form.
A fraction is in simplest form when 1 is the only factor that the
numerator and denominator have in common.
__ is in simplest form.
Tell whether the fraction 7
8
Look for common factors in the numerator and the denominator.
Step 1 The numerator of 7_8 is 7. List all
the factors of 7.
1× 7= 7
Step 2 The denominator of 7_8 is 8. List
all the factors of 8.
1× 8= 8
The factors of 7 are 1 and 7.
2× 4= 8
The factors of 8 are 1, 2, 4, and 8.
Step 3 Check if the numerator and
The only common factor of 7 and 8
7
_
denominator of 8 have any
is 1.
common factors greater than 1.
So, 7_8 is in simplest form.
© Houghton Mifflin Harcourt Publishing Company
Tell whether the fraction is in simplest form. Write yes or no.
4
1. ___
10
2
2. __
8
no
3
3. __
5
no
yes
Write the fraction in simplest form.
4
4. ___
12
6
5. ___
10
1
__
3
Number and Operations–Fractions
3
6. __
6
3
__
5
1
__
2
97
Name
1
Simplest Form
CC.4.NF.1
Write the fraction in simplest form.
6
1. ___
10
__
2. 6
8
6 = _______
6 ÷ 2 = __
3
___
10
10 ÷ 2
__
3. 5
5
3
__
4
5
100
5. ____
100
2
6. __
6
1 or 1
__
2
__
3
1 or 1
__
1
2
7. __
8
1
__
3
1
8
4. ___
12
4
8. ___
10
1
__
4
2
__
5
6 ≠ ___
1
9. ___
12
12
60
6 = ____
13. ___
10
100
5
__ ≠ __
10. 3
4
6
3
6 = __
11. ___
10
5
1
3 ≠ __
12. ___
12
3
9
___ ≠ ___
14. 11
12
10
8
2 = ___
15. __
5
20
1
__ = __
16. 4
8
2
Problem Solving
17. At Memorial Hospital, 9 of the
12 babies born on Tuesday were
boys. In simplest form, what
fraction of the babies born on
Tuesday were boys?
3
__
4
98
18. Cristina uses a ruler to measure the
length of her math textbook. She
4 meter long.
says that the book is __
10
Is her measurement in simplest
form? If not, what is the length
of the book in simplest form?
2 meter
No; __
5
Lesson 49
© Houghton Mifflin Harcourt Publishing Company
Tell whether the fractions are equivalent.
Write = or ≠.
Name
LESSON
50
1
Common Denominators
OBJECTIVE Use equivalent fractions to represent a pair of fractions with a common denominator.
CC.4.NF.1
A common denominator is a common multiple of the
denominators of two or more fractions.
2 and __
3 as a pair of fractions with common denominators.
Write __
3
4
Step 1 Identify the denominators
of 2_3 and 3_4 .
Step 2 List multiples of 3 and 4.
Circle common multiples.
Step 3 Rewrite 2_3 as a fraction with a
denominator of 12.
Step 4 Rewrite 3_4 as a fraction with a
denominator of 12.
2 and __
3
__
3
4
The denominators are 3 and 4.
3: 3, 6, 9, 12, 15, 18
4: 4, 8, 12, 16, 20
12 is a common multiple of 3 and 4.
2
× 4 = ___
8
__ = 2
______
3
3 × 4 12
3
× 3 = ___
9
__ = 3
______
4
4 × 3 12
2 and __
3 as ___
8 and ___
9.
So, you can rewrite __
3
4
12
12
© Houghton Mifflin Harcourt Publishing Company
Write the pair of fractions as a pair of fractions with
a common denominator. Possible answers are given.
1 and __
1
1. __
2
3
2 and __
5
2. __
4
8
4
5
__, __
8 8
3
2
__, __
6 6
1 and __
3
3. __
2
5
1 and __
5
4. __
4
6
10
3 , ___
___
5 , ___
6
___
12 12
10 10
2 and __
2
5. __
5
3
4 and ___
7
6. __
5
10
10
6 , ___
___
15 15
Number and Operations–Fractions
8 , ___
7
___
10 10
99
Name
1
Common Denominators
CC.4.NF.1
Write the pair of fractions as a pair of fractions with a
common denominator. Possible answers are given.
3
__ and __
1. 2
3
4
2
__ and __
2. 1
4
3
3 and __
1
3. ___
10
2
Think: Find a common multiple.
3: 3, 6, 9, 12, 15
4: 4, 8, 12, 16, 20
8 , ___
9
___
12 12
3 and __
3
4. __
5
4
3 , ___
8
___
12 12
2 and __
7
5. __
4
8
7
4, __
__
8 8
15
12, ___
___
20 20
2 and ___
5
6. __
3
12
5
8 , ___
___
12 12
5
3 , ___
___
10 10
1 and __
1
7. __
4
6
2
3 , ___
___
12 12
2
__ ≠ __
8. 1
2
5
3
__ = __
9. 1
2
6
3
__ = __
12. 6
8
4
2
__ ≠ __
13. 3
4
3
5
3 ≠ __
10. __
4
6
3
6 = __
11. ___
10
5
4
2 ≠ __
14. ___
10
5
3
__ = ___
15. 1
4
12
Problem Solving
16. Adam drew two same size
rectangles and divided them into
the same number of equal parts.
He shaded 1_3 of one rectangle
and 1_4 of other rectangle. What
is the least number of parts into
which both rectangles could be
divided?
12 parts
100
17. Mera painted equal sections of her
bedroom wall to make a pattern.
She painted 2_5 of the wall white
and 1_2 of the wall lavender. Write
an equivalent fraction for each
using a common denominator.
4 and ___
5
10
10
Lesson 50
© Houghton Mifflin Harcourt Publishing Company
Tell whether the fractions are equivalent. Write = or ≠.
Name
LESSON
51
Problem Solving • Find Equivalent Fractions
OBJECTIVE Use the strategy make a table to solve problems using equivalent fractions.
1
CC.4.NF.1
Kyle’s mom bought bunches of balloons for a family party.
Each bunch has 4 balloons, and __14 of the balloons are blue.
If Kyle’s mom bought 5 bunches of balloons, how many
balloons did she buy? How many of the balloons are blue?
What do I need to find?
I need to find how
many balloons Kyle’s
mom bought and how
many of the balloons
are blue.
What information do I
need to use?
Each bunch has 1 out
of 4 balloons that are
blue, and there are
5 bunches.
How will I use the
information?
I will make a table to
find the total number
balloons Kyle’s mom
bought and the
fraction of balloons
that are blue.
Solve the Problem
© Houghton Mifflin Harcourt Publishing Company
I can make a table.
Number of Bunches
1
2
3
4
5
Total Number of Blue Balloons
Total Number of Balloons
1
__
4
2
__
8
3
___
4
___
5
___
12
16
20
Kyle’s mom bought 20 balloons. 5 of the balloons are blue.
Make a table to solve.
Check students’ tables.
1. Jackie is making a beaded bracelet. 2. Ben works in his dad’s bakery
The bracelet will have no more than
packing bagels. Each package can
1
__
have no more than 16 bagels. __34 of
bracelet will be green. What other
the bagels in each package are plain.
fractions could represent the part
What other fractions could represent
of the beads on the bracelet that
the part of the bagels in each
will be green?
package that will be plain?
2, __
3, ___
4
__
6 9 12
Number and Operations–Fractions
9 , ___
12
6, ___
__
8 12 16
101
Name
Problem Solving • Find Equivalent
Fractions
Solve each problem.
1
CC.4.NF.1
1. Miranda is braiding her hair. Then she will
attach beads to the braid. She wants __13 of the
beads to be red. If the greatest number of
beads that will fit on the braid is 12, what other
fractions could represent the part of the beads
that are red?
2, __
3, ___
4
__
6 9 12
2. Ms. Groves has trays of paints for students in
her art class. Each tray has 5 colors. One of the
colors is purple. What fraction of the colors in
20 trays is purple?
20 or __
1
____
3. Miguel is making an obstacle course for field
day. At the end of every sixth of the course,
there is a tire. At the end of every third of the
course, there is a cone. At the end of every
half of the course, there is a hurdle. At which
locations of the course will people need to go
through more than one obstacle?
5
1, __
2 and
__, __
At the 1
3 2 3
final locations
4. Preston works in a bakery where he puts
muffins in boxes. He makes the following table
to remind himself how many blueberry muffins
should go in each box.
Number of Blueberry Mufﬁns
2
4
8
Total Number of Mufﬁns
6
12
24
36
How many blueberry muffins should Preston
put in a box with 36 muffins?
12 blueberry muffins
102
Lesson 51
© Houghton Mifflin Harcourt Publishing Company
100
Name
1
Compare Fractions Using Benchmarks
LESSON
52
CC.4.NF.2
OBJECTIVE Compare fractions using benchmarks.
A benchmark is a known size or amount that helps you understand
a different size or amount. You can use __12 as a benchmark.
Sara reads for __36 hour every day after school. Connor reads for
2
__
hour. Who reads for a longer amount of time?
3
Compare the fractions.
3
__
6
2
__
3
Step 1 Divide one circle into 6 equal parts.
Divide another circle into 3 equal parts.
Step 2 Shade __36 of the first circle. How many
parts will you shade? 3 parts
JXiX
:feefi
Step 3 Shade __23 of the second circle.
How many parts will you shade? 2 parts
Step 4 Compare the shaded parts of each circle.
Half of Sara’s circle is shaded. More than half
3
__ is less than 2
__.
6
3
2
3 < __
__
6
3
© Houghton Mifflin Harcourt Publishing Company
So, Connor reads for a longer amount of time.
2 and __
3. Write < or >.
1. Compare __
8
4
'
(
)
1
8
1
8
1
8
1
8
1
8
1
8
1
8
(
'
1
8
(
)
1
4
2 <
__
8
1
4
(
1
4
1
4
3
__
4
Compare. Write < or >.
8
1 < ___
2. __
4
10
__ > 1
__
3. 7
8
3
5 < 1
__
4. ___
12
2
8
2 < ___
5. __
8
12
__ > 4
__
6. 4
6
8
7 > 2
__
7. ___
12
4
Number and Operations–Fractions
103
Name
1
Compare Fractions Using Benchmarks
CC.4.NF.2
Compare. Write < or >.
1 < ___
6
1. __
8
10
4 < 4
__
2. ___
12
6
__ < 1
__
3. 2
8
2
3 < 3
__
4. __
5
3
5
__ > ___
5. 7
8
10
9 > 1
__
6. ___
12
3
4 < 7
__
7. __
6
8
__ < 2
__
8. 2
4
3
__ > 1
__
9. 3
5
4
6 > 2
__
10. ___
10
5
2
__ < ___
11. 1
8
10
__
12. 2
3
4 < 5
__
13. __
5
6
__ < 5
__
14. 3
5
8
__ > 3
__
15. 8
8
4
1. ___
6 is
__ is less than __
Think: 1
8
2 10
__.
more than 1
2
Problem Solving
16. Erika ran __38 mile. Maria ran __34 mile.
Who ran farther?
Maria
104
17. Carlos finished __13 of his art project
on Monday. Tyler finished __12 of
his art project on Monday. Who
finished more of his art project
on Monday?
Tyler
Lesson 52
© Houghton Mifflin Harcourt Publishing Company
5
> ___
12
Name
1
Compare Fractions
LESSON
53
CC.4.NF.2
OBJECTIVE Compare fractions by first writing them as fractions with a common numerator
or a common denominator.
Theo filled a beaker __24 full with water. Angelica filled a
beaker __38 full with water. Whose beaker has more water?
2 and __
3.
Compare __
4
8
Step 1 Divide one beaker into 4 equal parts.
Divide another beaker into 8 equal parts.
Theo
2 of the first beaker.
4
__ of the second beaker.
8
1
4
Half of Theo’s beaker is shaded. Less than half of
2
3
__
__
4 is greater than 8 .
1
4
2 > __
3
__
1
4
4
8
So, Theo’s beaker has more water.
© Houghton Mifflin Harcourt Publishing Company
'
2 and __
3.
2. Compare __
3
6
(
)
(
1
2
1
4
'
(
)
1
3
1
2
1
4
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
4
Step 4 Compare the shaded parts of each beaker.
1 and __
1.
1. Compare __
2
4
Angelica
1
4
Which is greater?
1
4
1
__
2
1
6
(
1
3
1
6
1
6
Which is less?
1
3
1
6
1
6
1
6
3
__
6
Compare. Write <, > , or = .
1 < 3
__
3. __
2
4
Number and Operations–Fractions
6 < 5
__
4. ___
12
8
__ = 4
__
5. 2
3
6
__ > 1
__
6. 3
8
4
105
Name
1
Compare Fractions
CC.4.NF.2
Compare. Write <, > , or = .
3 < __
5
1. __
4
6
2 > __
2
3. __
4
5
2
1 = ___
2. __
10
5
Think: 12 is a common
denominator.
7
__ < ___
4. 3
5
10
1
4 > __
5. ___
12
6
1
__ < __
8. 2
5
2
__ = 2
__
9. 4
4
8
1
__ = __
6. 2
6
3
2
__ < __
7. 1
3
4
7 > __
2
10. ___
12
4
3
__ < __
11. 1
8
4
Problem Solving
12. A recipe uses __23 cup of flour and
5
__
cup of blueberries. Is there more
8
flour or more blueberries in the
recipe?
flour
106
13. Peggy completed __56 of the math
homework and Al completed __45 of
the math homework. Did Peggy
or Al complete more of the math
homework?
Peggy
Lesson 53
© Houghton Mifflin Harcourt Publishing Company
3 = ______
3 × 3 = ___
9
__
4 4 × 3 12
5
5 × 2 = 10
__ = ______
___
6 6 × 2 12
9 < 10
___
___
12 12
Name
1
Compare and Order Fractions
LESSON
54
CC.4.NF.2
OBJECTIVE Compare and order fractions.
3, __
1, and __
1 in order from least to greatest.
Write __
8 4
2
Step 1 Identify a common denominator.
Multiples of 8: 8, 16, 24
Multiples of 4: 4, 8, 12, 16
Multiples of 2: 2, 4, 6, 8
Use 8 as a common denominator.
Step 2 Use the common denominator
to write equivalent fractions.
3
__
8
1
× 2 = __
2
__ = 1
______
4
4× 2 8
1= 1
× 4 = __
4
__
______
2
2× 4 8
Step 3 Compare the numerators.
2< 3< 4
Step 4 Order the fractions from least
to greatest, using < or > symbols.
2 < __
3 < __
4
__
8
8
8
1 < __
3 < __
1.
So, __
4 8 2
© Houghton Mifflin Harcourt Publishing Company
Write the fraction with the greatest value.
2, __
1, __
1
1. __
3 4 6
2
__
3
3 , __
1, __
2
2. ___
10 2 5
1
__
2
1, ___
5 , ___
9
3. __
8 12 10
9
___
10
Write the fractions in order from least to greatest.
9 , __
1, __
4
4. ___
10 2 5
1
4 < ___
9
__ < __
2 5 10
Number and Operations–Fractions
3, __
7, __
1
5. __
4 8 2
1
3 < __
7
__ < __
2 4 8
2, __
3, __
5
6. __
3 4 6
2
3 < __
5
__ < __
3 4 6
107
Name
1
Compare and Order Fractions
CC.4.NF.2
Write the fractions in order from least to greatest.
2 , ___
8
__, ___
1. 5
8 12 10
2, __
5
__, __
2. 1
5 3 8
Use benchmarks and a number line.
5 is close to __
1. ___
2 is close to 0. ___
8 is close to 1.
Think: __
'
2 12
)
()
10
(
)
,
/
(
/
('
2 < __
5 < ___
8
___
12
8
1
5 < __
2
__ < __
5 8 3
10
1, __
2, ___
6
3. __
2 5 10
7 , ___
4, ___
5
4. __
6 12 10
6
2
1 < ___
__ < __
5 2 10
1, __
3, __
1
5. __
4 6 8
5 < ___
7 < __
4
___
10
3, ___
7
__, __
6. 1
8 6 12
8 ,
7. ____
100
3 < ___
7
1
__ < __
8 6 12
12
1
1 < __
3
__ < __
8 4 6
6
7, __
1
__, __
8. 3
4 8 5
3, ___
7
__
5 10
8 < __
3 < ___
7
____
100
5
1 < __
3 < __
7
__
10
5
4
8
Problem Solving
9. Amy’s math notebook weighs
1
__
pound, her science notebook
2
weighs __78 pound, and her history
notebook weighs __34 pound. What
are the weights in order from
lightest to heaviest?
3 pound, __
7 pound
1 pound, __
__
2
108
4
8
10. Carl has three picture frames. The
thicknesses of the frames are
3
4
__
inch, __
inch, and __56 inch. What are
5
12
the thicknesses in order from least
to greatest?
3 inch, __
4 inch, __
5 inch
___
12
5
6
Lesson 54
© Houghton Mifflin Harcourt Publishing Company
8
Name
1
Add and Subtract Parts of a Whole
LESSON
55
CC.4.NF.3a
OBJECTIVE Understand that to add or subtract fractions, they must refer to parts of the same wholes.
Justin has 3_8 pound of cheddar cheese and 2_8 pound of brick cheese.
How much cheese does he have in all?
1
Step 1 Use fraction strips to model the problem.
Use three 1_8 -strips to represent 3_8 pound
of cheddar cheese.
1
8
Step 2 Join two more 1_8 -strips to represent the
amount of brick cheese.
5
_
8
1
8
1
1
8
Step 3 Count the number of 1_8 -strips. There are
five 1_ -strips. Write the amount as a
8
fraction. Justin has
1
8
1
8
1
8
1
8
pound of cheese.
1
8
2=5
3 + __
__
__
8 8 8
Step 4 Use the model to write an equation.
Suppose Justin eats 1_8 pound of cheese. How much cheese is left?
Step 1 Use five 1_8 -strips to represent the 5_8 pound of
cheese.
Step 2 Remove one 1_8 -strip to show the amount eaten.
1
1
8
1
8
1
8
Step 3 Count the number of 1_8 -strips left. There are
4
four 1_ fraction strips. There is _8 pound left.
8
1
8
1=4
5
__ - __
__
8 8 8
Step 4 Write an equation for the model.
© Houghton Mifflin Harcourt Publishing Company
1
8
Use the model to write an equation.
1.
1
1
1
5
1
1
5
1
5
5
1
5
2.
1
1
5
1
5
1
5
1
5
1
1
3
1
3
1 + __
3 = __
4
__
5 5 5
3.
1
1
4
1
4
1
4
1
1
4
Number and Operations–Fractions
3
5
1
4
1 = __
4
3 + __
__
4
2 - __
1 = __
1
__
4
4.
1
1
4
1
4
1
4
1
4
3
3
1
1 1 1 1 1
6 6 6 6 6
2 = __
3
5
__ - __
6 6 6
109
Name
1
Add and Subtract Parts of a Whole
CC.4.NF.3a
Use the model to write an equation.
1.
1
Think:
3
__
5
2
__
8
+
8
2=5
3 + __
__
__
8 8 8
=
2.
5
__
8
3.
1
4 - __
3 = __
1
__
5 5 5
5
1 + __
2 = __
3
__
4
4
4
Use the model to solve the equation.
5.
4.
2 + __
3=
__
6 6
5
5
__
6
3 - __
2=
__
5 5
1
__
5
Problem Solving
6. Jake ate 4_8 of a pizza. Millie ate 3_8
of the same pizza. How much of
the pizza was eaten by Jake and
Millie?
7
__ of the pizza
8
7. Kate ate 1_4 of her orange. Ben
ate 2_4 of his banana. Did Kate and
Ben eat 1_4 + 2_4 = 3_4 of their fruit?
Explain.
No; one whole refers to an orange
and the other whole to a banana.
110
Lesson 55
© Houghton Mifflin Harcourt Publishing Company
1
Name
LESSON
56
1
Write Fractions as Sums
CC.4.NF.3b
OBJECTIVE Decompose a fraction by writing it as a sum of fractions with the same denominators.
A unit fraction tells the part of the whole that 1 piece represents.
A unit fraction always has a numerator of 1.
4
Bryan has __
10 pound of clay for making clay figures. He wants
1
to use __
10 pound of clay for each figure. How many clay figures
can he make?
4
Use fraction strips to write __
10 as a sum of unit fractions.
4
Step 1 Represent __
10 with fraction strips.
1
1
__
Step 2 Each __
10 is a unit fraction. Write a 10 addend
1
4
__
for each __
10 -strip you used to show 10 .
1 1 __
1
1 1 __
1 1 __
__
Step 3 Count the number of addends. The number
of addends represents the number of clay
figures Bryan can make.
So, Bryan can make 4 clay figures.
10
10
10
10
Write the fraction as the sum of unit fractions.
© Houghton Mifflin Harcourt Publishing Company
1.
2.
3=
__
6
1
__
6
+
1
__
6 +
1
__
6
2
__ =
4
3.
1
__
4
1
__
+
4
+
1
__
5 +
4.
4=
__
8
1
__
8
1
__
+
8 +
Number and Operations–Fractions
1
__
8 +
1
__
8
5=
__
5
1
__
5
1
__
5 +
1
__
5 +
1
__
5
111
Name
1
Write Fractions as Sums
CC.4.NF.3b
Write the fraction as a sum of unit fractions.
1+1
1+1
__
__ + __
__
5 5 5 5
4=
1. __
5
3=
2. __
8
1 +__
1
1 + __
__
8
8
8
__ four times.
5
6 =
3. ___
12
1 + ___
1
1 + ___
___
12 12 12
1 + ___
1 + ___
1
+ ___
12 12 12
4=
4. __
4
1 + __
1 + __
1
1
__ + __
4 4 4 4
Write the fraction as a sum of fractions three different ways.
7
5. ___
10
__
6. 6
6
2 + ___
3 + ___
2
7 = ___
___
10 10 10 10
7 = ___
4 + ___
2 + ___
1
___
10 10 10 10
7 = ___
5 + ___
1 + ___
1
___
10 10 10 10
6
4 + __
1 + __
1
__ = __
6 6 6 6
2 + __
2 + __
2
6
__ = __
6 6 6 6
6
3 + __
2 + __
1
__ = __
6 6 6 6
7. Miguel’s teacher asks him to
color 4_8 of his grid. He must use
3 colors: red, blue, and green.
There must be more green sections
than red sections. How can Miguel
color the sections of his grid to
8. Petra is asked to color 6_6 of her
grid. She must use 3 colors: blue,
red, and pink. There must be more
blue sections than red sections
or pink sections. What are the
different ways Petra can color the
sections of her grid and follow all
the rules?
1
1 blue, and __
2 green
__ red, __
8
8
8
2 red, __
1 pink;
3 blue, __
__
112
6
6
6
1 red, __
1 pink;
4 blue, __
__
6
6
6
3 blue, __
1 red, __
2 pink
__
6
6
6
Lesson 56
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Name
LESSON
57
1
Rename Fractions and Mixed Numbers
CC.4.NF.3b
OBJECTIVE Write fractions greater than 1 as mixed numbers and write mixed numbers
as fractions greater than 1.
A mixed number is made up of a whole number and
a fraction. You can use multiplication and addition to
rename a mixed number as a fraction greater than 1.
5 as a fraction.
Rename 2__
6
First, multiply the denominator, or the number
of parts in the whole, by the whole number.
2
6 × 2 = 12
total number
17 of parts
5
=
6 number of
6
parts in the whole
12 + 5 = 17
5 = ___
17.
So, 2__
6
6
You can use division to write a fraction greater than 1
as a mixed number.
16 as a mixed number.
Rename ___
3
__ as a mixed number, divide the numerator by
To rename 16
3
the denominator.
© Houghton Mifflin Harcourt Publishing Company
Use the quotient and remainder to write a mixed number.
1.
___ = 5__
So, 16
3
3
,
*q ( (,
(
Write the mixed number as a fraction.
2=
1. 3 __
3
11
___
3
__ =
2. 43
5
23
___
5
Write the fraction as a mixed number.
2
1
6__
6__
32
19
5
3
___
___
5.
6.
=
=
5
3
Number and Operations–Fractions
__ =
3. 43
8
15 =
7. ___
4
35
___
8
__
33
4
__ =
4. 21
6
51 =
8. ___
10
13
___
6
1
5___
10
113
Name
1
Rename Fractions and Mixed Numbers
CC.4.NF.3b
Write the mixed number as a fraction.
__
1. 23
5
Think:
__ + 5
__ + 3
__.
Find 5
5 5 5
13
___
5
__
2. 41
3
__
5. 41
8
7
6. 1___
10
__
3. 12
5
13
___
3
7
__
11
___
5
__
7. 51
2
17
___
10
33
___
8
__
4. 32
3
3
__
8. 23
8
11
___
2
19
___
8
Write the fraction as a mixed number.
___
10. 20
10
__
51
6
___
11. 15
8
__
17
8
2
23
13. ___
10
19
14. ___
5
11
15. ___
3
__
34
5
3
2___
10
___
12. 13
6
__
21
6
9
16. __
2
__
32
3
__
41
2
Problem Solving
17. A recipe calls for 2 2_4 cups of raisins,
but Julie only has a 1_4 -cup measuring
cup. How many 1_4 cups does Julie
need to measure out 2 2_4 cups of
raisins?
1 cups
ten __
4
114
18. If Nancy needs 3 1_4 cups of oatmeal,
how many 1_4 cups of oatmeal will
she use?
1 cups
thirteen __
4
Lesson 57
© Houghton Mifflin Harcourt Publishing Company
___
9. 31
6
Name
LESSON
58
1
CC.4.NF.3c
OBJECTIVE Add and subtract mixed numbers.
1
__ + 2__
Find the sum. 31
4
4
Add the whole number and fraction parts.
• Add the whole numbers: 3 + 2 = 5
1 1 2
• Add the fractions: __ + __ = __
4 4 4
Write the sum as a mixed number, so the
1 + 2__
1 = 5__
2
fractional part is less than 1. 3__
4
4
4
1
__ - 3__
Find the difference. 45
8
8
Subtract the fraction and the
whole number parts.
5 1 4
• Subtract the fractions: __ - __ = __
8 8 8
• Subtract the whole numbers:
4- 3= 1
5 - 3__
1 = 1__
4
4__
8
8
8
© Houghton Mifflin Harcourt Publishing Company
Find the sum or difference.
1.
4
3 __
5
__
+ 43
5
__
82
5
2.
5.
3
2__
8
__
+ 81
8
__
104
8
9
6. 11___
10
7
- 3___
10
2
8___
10
Number and Operations–Fractions
2
7__
3
__
- 31
3
1
4__
3
3.
7
4___
12
5
+ 6___
12
11
__
4. 123
4
1
- 6__
4
2
6__
4
7.
__
73
5
3
+ 4__
5
__
121
5
8.
3
8__
6
1
- 3__
6
__
52
6
115
Name
1
CC.4.NF.3c
Find the sum. Write the sum as a mixed number,
so the fractional part is less than 1.
1.
__
64
5
3
+ 3 __
5
__ = 10 2
__
97
5
5
2.
__
41
2
1
+ 2 __
2
7
3.
__
22
3
2
+ 3 __
3
__
61
3
4.
__
64
5
4
+ 7 __
5
__
14 3
5
5.
__
93
6
2
+ 2 __
6
__
11 5
6
6.
4
8 ___
12
6
+ 3 ___
12
10
11 ___
12
7.
__
43
8
5
+ 1 __
8
6
8.
5
9 ___
10
3
+ 6 ___
10
8
15 ___
10
11.
__
64
5
3
- 3__
5
__
31
5
12.
Write the fraction as a mixed number.
__
67
8
3
- 4 __
8
__
24
8
10.
__
42
3
__
- 31
3
__
11
3
__
73
4
1
- 2 __
4
__
52
4
Problem Solving
13. James wants to send two gifts
by mail. One package weighs 2 3_4
pounds. The other package weighs
1 3_4 pounds. What is the total weight
of the packages?
__ pounds
42
4
116
14. Tierra bought 4 3_8 yards blue ribbon
and 2 1_8 yards yellow ribbon for a
craft project. How much more blue
ribbon than yellow ribbon did
__ yards
22
8
Lesson 58
© Houghton Mifflin Harcourt Publishing Company
9.
Name
1
Subtraction with Renaming
LESSON
59
CC.4.NF.3c
OBJECTIVE Rename mixed numbers to subtract.
a mixed number from a whole number.
2
__ - 2__
Find the difference. 31
3
3
Step 1 Model the number you are subtracting
from, 3 1_3 .
Step 2 Because you cannot subtract 2_3 from 1_3
without renaming, change one of the
1 strips to three 1_3 strips. Then subtract
by crossing out two wholes and
two 1_3 strips.
1So, 3__
3
2 = __
2.
2__
3 3
__
Find the difference. 2 - 11
4
Step 1 Model the number you are subtracting
from, 2.
© Houghton Mifflin Harcourt Publishing Company
Step 2 Because you cannot subtract 1_4 from
1 without renaming, change one
of the 1 strips to four 1_4 strips. Then
subtract by crossing out one whole
and one 1_4 strip.
1 = __
3.
So, 2 - 1__
4
4
Find the difference.
3
__
2
5
__
1. 3 - 2 =
5
1 - 1__
3=
2. 2__
4
4
3.
2
__
4
__
33
5
4
- 2__
5
4
__
5
Number and Operations–Fractions
4.
1
3___
12
___
-211
12
2
___
12
5.
__
45
8
7
__
-2
8
6
1__
8
117
Name
1
Subtraction with Renaming
CC.4.NF.3c
Find the difference.
__
51
3
2
-3 __
3
_
→ 44__3
→ 3_2__3
2.
9.
13.
__
71
6
5
- 2__
6
_
2
4 __
6
1
3 __
4
3
-1 __
4
_
2
1__
4
3.
__
-3 2
5
__
23
5
__
12
3
3
5. 12 ___
10
7
-7 ___
10
__
6
4___
10
6
__
51
4
3
-2 __
4
_
2
2__
4
__
73
5
4
-4 __
5
_
__
24
5
4.
3
9 __
8
7
-8 __
8
_
4
__
8
8.
1
10 __
2
1
-8 __
2
__
2
6.
1
8 __
6
__
-3 5
6
_
__
42
6
7.
10.
3
9 ___
12
7
-4 ___
12
__
8
4___
12
11.
1
9 ___
10
7
-8 ___
10
__
4
___
10
12.
14.
5
4 __
8
7
-1 __
8
_
__
26
8
15.
1
5 ___
12
8
-3 ___
12
__
5
1___
12
16.
__
91
3
2
__
3
_
2
8__
3
7
3
-1 __
5
2
5__
5
Problem Solving
17. Alicia buys a 5-pound bag of rocks
for a fish tank. She uses 1 1_8 pounds
for a small fish bowl. How much
is left?
__ pounds
37
8
118
18. Xavier made 25 pounds of
roasted almonds for a fair. He has
3 1_2 pounds left at the end of the
fair. How many pounds of roasted
almonds did he sell at the fair?
__ pounds
211
2
Lesson 59
© Houghton Mifflin Harcourt Publishing Company
1.
Name
LESSON
60
1
Algebra • Fractions and Properties
CC.4.NF.3c
so you can use mental math to find sums.
The Commutative Property of Addition states that
when the order of two addends is changed,
the sum is the same.
6+ 3= 3+ 6
The Associative Property of Addition states that
when the grouping of addends is changed, the sum
is the same.
(3 + 6) + 4 = 3 + (6 + 4)
3 + 4__
7 + 6__
5.
Use the properties and mental math to add 10__
8
8
8
3 + 4__
7 + 6__
5
Step 1 Look for fractions that combine to make 1.
10__
8
8
8
© Houghton Mifflin Harcourt Publishing Company
Step 2 Use the Commutative Property to order the
3 + 4__
5 + 4__
7
7 + 6__
5 = 10__
3 + 6__
with a sum of 1 are together.
10__
8
8
8
8
8
8
7
5) + 4__
__ + 6__
Step 3 Use the Associative Property to group
= (103
8
8
8
the other mixed number.
7
= (17) + 4__
8
Step 5 Write the sum.
__
= 217
8
Use the properties and mental math to find the sum.
(
)
1 + 1 __
4
2 + 4 __
1. 3 __
5
5
5
(
)
7 + 1 ___
3
4 + 6 ___
2. 5 ___
10
10
10
__
92
5
(
)
5 + 3 ___
7
11 + 1 ___
4. 2 ___
12
12
12
11
7 ___
12
Number and Operations–Fractions
(
4
13 ___
10
(
16
)
7 + 63
1
__ + __
5. 4 __
8
8 8
3
11 __
8
)
1
__ + 5 + 3 __
3. 7 3
4
4
(
)
2 + 41
4
__ + 7 __
6. 9 __
6
6
6
__
21 1
6
119
Name
1
CC.4.NF.3c
Use the properties and mental math to find the sum.
(
)
1
__ + 22
__ + 1__
1. 51
3
3
3
__ + (4)
51
3
(
(
)
1
__ + 4 2
__ + 5__
4. 6 3
4
4
4
(
2
17__
5
)
(
2
4 + 9 __
__ + 10 __
5. 6 3
6
6
6
__
162
4
)
(
__
112
5
)
1 + 20___
2 + 15 ___
7
8. 14___
10
10
10
1
12__
8
)
1
4 + 3 __
__ + 1__
6. 62
5
5
5
__
263
6
7 + 31
1
__ + 1__
7. 7__
8
8
8
)
4
__ + 3 2
__ + 5 __
3. 8 1
5
5
5
__
165
8
1
9__
3
(
(
)
5 + 27
__ + 3 __
__
2. 101
8
8
8
(
)
2 + 8___
5
7 + 9___
9. 13___
12
12
12
2
31___
12
50
10. Nate’s classroom has three tables
of different lengths. One has a
length of 4 1_2 feet, another has a
length of 4 feet, and a third has
a length of 2 1_2 feet. What is the
length of all three tables when
pushed end to end?
11 feet
120
11. Mr. Warren uses 2 1_4 bags of mulch
for his garden and another
4 1_4 bags for his front yard. He also
uses 3_4 bag around a fountain.
How many total bags of mulch
does Mr. Warren use?
1 bags
7__
4
Lesson 60
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Name
1
LESSON
61
CC.4.NF.3d
OBJECTIVE Use models to represent and find sums involving fractions.
Fractions with like denominators have the same denominator. You
can add fractions with like denominators using a number line.
4 + __
1.
Model __
6
6
Step 1 Draw a number line labeled with
sixths. Model the fraction 4_6 by
starting at 0 and shading 4 sixths.
'
1 more sixth.
'
(
'
-
(
-
*
-
+
-
,
-
-
(
'
-
Step 3 How many sixths are there in all?
5 sixths
)
-
(
-
)
-
*
-
+
-
,
-
-
Write the number of sixths as a fraction.
5
5 sixths = __
6
4 + __
1 = __
5
__
6
6
1 + __
4.
1. Model __
5 5
5
__
5
1 + __
4=
__
© Houghton Mifflin Harcourt Publishing Company
5
5
6
'
'
,
(
(
,
)
,
*
,
+
,
,
,
Find the sum. Use a model to help.
2 + ___
4
2. ___
10 10
'
6
___
10
1
__ + __
3. 1
4 4
(
' ( ) * + , - . / 0 ('
(' (' (' (' (' (' (' (' (' (' ('
'
2
__
4
Number and Operations–Fractions
'
+
(
(
+
)
+
*
+
+
+
121
Name
1
CC.4.NF.3d
Find the sum. Use fraction strips to help.
2 + __
1=
1. __
6 6
)
-
3
__
6
4 + ___
5 =
2. ___
10 10
3
__
3
2 + __
1=
4. __
4 4
3
__
4
9
___
10
(
-
1 + __
2=
3. __
3 3
2 + ___
4 =
5. ___
12 12
6
___
12
1 + __
2=
6. __
6 6
3
__
6
3 + ___
9 =
7. ___
12 12
12
___
12
4=
__ + __
8. 3
8 8
7
__
1 + __
2=
10. __
5 5
3
__
3 + __
1=
9. __
4 4
4
__
4
8
5
4
11. Lola walks __
10 mile to her friend’s
5
house. Then she walks __
10 mile to
the store. How far does she walk
in all?
9 mile
___
10
13. Jacqueline buys 2_4 yard of green
ribbon and 1_4 yard of pink ribbon.
How many yards of ribbon does
3
__ yard
4
122
12. Evan eats 1_8 of a pan of lasagna
and his brother eats 2_8 of it. What
fraction of the pan of lasagna do
they eat in all?
3
__ of the pan
8
14. Shu mixes 32_ pound of peanuts
with 1_3 pound of almonds. How
many pounds of nuts does Shu mix
in all?
3 pound
__
3
Lesson 61
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Name
1
Subtract Fractions Using Models
LESSON
62
CC.4.NF.3d
OBJECTIVE Use models to represent and find differences involving fractions.
You can subtract fractions with like denominators using fraction strips.
5 - __
2.
Model __
8
8
2.
Step 2 Subtract __
8
Think: How many eighths are taken away?
Cross out 2 of the shaded eighths.
Step 3 Count the shaded eighths that remain.
There are 3 eighths remaining.
Step 4 Write the number of eighths that remain as
a fraction.
3
3 eighths = __
8
5 - __
2 = __
3
__
8
8
8
3 - __
2.
1. Model __
3 3
© Houghton Mifflin Harcourt Publishing Company
3 - __
2=
__
3
3
1
__
3
Subtract. Use fraction strips to help.
5 - __
1
2. __
6 6
5
1=
__ - __
6 6
6 - ___
3
3. ___
10 10
4
__
6
Number and Operations–Fractions
6 - ___
3 =
___
10
10
3
___
10
123
Name
1
Subtract Fractions Using Models
Subtract. Use fraction strips to help.
__
1= 3
__ - __
1. 4
5
5 5
CC.4.NF.3d
2
__
4
3 - __
1=
2. __
4 4
)
+
1=
__ - __
3. 5
6 6
4
__
6
1=
__ - __
4. 7
8 8
2=
5. 1 - __
3
1
__
3
8 - ___
2 =
6. ___
10 10
3 - __
1=
7. __
4 4
2
__
4
7 - __
5=
8. __
6 6
6
__
8
6
___
10
2
__
6
Problem Solving
9. Ena is making trail mix. She buys
the items shown in the table. How
many more pounds of pretzels
Item
Pretzels
Peanuts
Raisins
5 pound
__
8
10. How many more pounds of granola
than banana chips does she buy?
Banana Chips
Granola
Weight
(in pounds)
7
_
© Houghton Mifflin Harcourt Publishing Company
Use the table for 9 and 10.
8
4_
8
2_
8
3_
8
5_
8
2
__ pound
8
124
Lesson 62
Name
LESSON
63
1
CC.4.NF.3d
OBJECTIVE Solve word problems involving addition and subtraction with fractions.
You can find and record the sums and the differences of fractions.
2 + __
4
6
6
Step 1 Model it.
Step 2 Think: How many
sixths are there in all?
There are 6 sixths.
6
6 sixths = __
6
Step 3 Record it.
Write the sum as an
2 + __
4 = __
6
__
6 6 6
6 - ___
2
Subtract. ___
10
10
Step 1 Model it.
Step 2 Think: There are
6 tenths. I take away
2 tenths. How many
tenths are left?
There are 4 tenths left.
4
4 tenths = ___
10
Step 3 Record it.
Write the difference as
a subtraction equation.
6 - ___
2 = ___
4
___
10
10
10
© Houghton Mifflin Harcourt Publishing Company
Find the sum or difference.
1. 7 eighth-size parts - 4 eighth-size parts =
7 - __
4=
__
8 8
3
__
8
7
___
11 - ___
4 =
2. ___
12 12
2 + __
2=
5. __
4 4
3 eighth-size parts
12
4
__
4
Number and Operations–Fractions
4
___
2 + ___
2 =
3. ___
10 10
4 - __
3=
6. __
5 5
10
1
__
5
6 - __
4=
4. __
8 8
2
__
1 + __
2=
7. __
3 3
3
__
8
3
125
Name
1
CC.4.NF.3d
Find the sum or difference.
___
4 + ___
8 = 12
1=
__ - __
1. ___
2. 3
12
12 12
6 6
+
()
/
()
2
__
6
4 - __
3=
3. __
5 5
1
__
5
3
__
)
-
6 + ___
3 =
4. ___
10 10
9
___
3=
5. 1 - __
8
5
__
8
2=
__ + __
6. 1
4 4
9 - ___
5 =
7. ___
12 12
4
___
5 - __
2=
8. __
6 6
3
__
6
2 + __
1 =
9. __
3 3
10
12
4
3
__
3
Problem Solving
10. Guy finds how far his house is from
several locations and makes the
table shown. How much farther
away from Guy’s house is the
library than the cafe?
Distance from Guy’s House
Location
Library
School
Store
5 mile
___
10
Cafe
Yogurt Shop
11. If Guy walks from his house to
school and back, how far does
he walk?
Distance
(in miles)
9
__
10
5
__
10
7
__
10
4
__
10
6
__
10
© Houghton Mifflin Harcourt Publishing Company
Use the table for 10 and 11.
10 mile
___
10
126
Lesson 63
Name
1
Problem Solving • Multistep
Fraction Problems
LESSON
64
CC.4.NF.3d
OBJECTIVE Use the strategy act it out to solve multistep fraction problems.
3
_
5
Jeff runs mile each day. He wants to know how many days
he has to run before he has run a whole number of miles.
What do I need to find?
I need to find how many days
Jeff needs to run 3_5 mile
until
he has run a whole number
of miles.
What information do I need
to use?
3
__
Jeff runs 5 mile a day. He
wants the distance run to be
a whole number .
How will I use the
information?
© Houghton Mifflin Harcourt Publishing Company
I can use a number line and
act out
patterns
to
the problem.
Solve the Problem
Describe how to act it out.
Use a number line.
' ,( ), ,*
__ mile
Day 1: 3
5
__ mile
Day 2: 6
5
Number and Operations–Fractions
(* (+ * (( -, ., /, 0, ) ((, ()
, , ,
,
3
__
5 +
3
__
5 =
6
__
5
__ mile more
1 whole mile and 1
5
3
9
3
3
__
__
__
__
9
__
5
5
5
+
+
= 5
Day 3: mile
5
__ mile more
1 whole mile and 4
5
3
3
3
3
12
__
__
__
__
___
12
___
5
5
5
5
+
+
+
= 5
Day 4:
mile
5
__ mile more
2 whole miles and 2
5
3
3
3
3
15
3
__
__
__
__
__
___
15
___
5
5
5
5
5
+
+
+
+
= 5
Day 5:
mile
5
3 whole miles
So, Jeff will run a total of 3 miles in 5 days.
1. Lena runs 2_3 mile each day. She
wants to know how many days
she has to run before she has
run a whole number of miles.
3 days
+
,
2. Mack is repackaging 6_8 -pound bags
of birdseed into 1-pound bags of
birdseed. What is the least number
of 6_8 -pound bags of birdseed he
needs in order to fill 1-pound bags
without leftovers?
6 -pound bags
four __
8
127
Name
1
Problem Solving • Multistep Fraction Problems
CC.4.NF.3d
1. Each child in the Smith family was given an orange cut
into 8 equal sections. Each child ate 5_8 of the orange. After
combining the leftover sections, Mrs. Smith noted that
there were exactly 3 full oranges left. How many children
are in the Smith family?
1
8
1
8
3
8
1
8
1
8
+
1
8
3
8
1
8
1
8
1
8
+
1
8
3
8
1
8
+
1
8
3
8
1
8
1
8
+
1
8
3
8
1
8
1
8
+
1
8
3
8
1
8
1
8
+
1
8
3
8
1
8
1
8
+
1
8
3
8
1
8
=
3
There are 8 addends, so there are 8 children in the Smith family.
8 children
2. Val walks 2 3_5 miles each day. Bill runs 10 miles once every
4 days. In 4 days, who covers the greater distance?
Val
© Houghton Mifflin Harcourt Publishing Company
them into bags that hold 65_ pound of peanuts. How many
fill the 5_6 -pound bags without having any peanuts left over?
five 2-pound bags
4. A carpenter has several boards of equal length. He cuts
3
_ of each board. After cutting the boards, the carpenter
5
notices that he has enough pieces left over to make up
the same length as 4 of the original boards. How many
10 boards
128
Lesson 64
Name
LESSON
65
1
Multiples of Unit Fractions
CC.4.NF.4a
OBJECTIVE Write a fraction as a product of a whole number and a unit fraction.
A unit fraction is a fraction with a numerator of 1. You can write
a fraction as the product of a whole number and a unit fraction.
7
Write __
10 as the product of a whole number and a
unit fraction.
7
Write __
10 as the sum of unit fractions.
1
1
1
1
___
___
___
7 = ___
___
+ 10 + 10 + 10
10
10
1
___
+
+
10
1
___
1
___
+
10
10
Use multiplication to show repeated addition.
7 =
1
___
× ___
7
10
10
1
___
7 =
× 10
So, ___
7
10
.
The product of a number and a counting number is a multiple of the
number. You can find multiples of unit fractions.
1.
List the next 4 multiples of __
8
Make a table and use repeated addition.
1
1 × __
8
1
2 × __
8
__
3× 1
8
1
__
1 + __
1
__
1 + __
1 + __
1
__
1
__
8
2
__
© Houghton Mifflin Harcourt Publishing Company
8
8
8
8
8
1 are
The next 4 multiples of __
8
8
3
__
8
2
__
8
8
,
1
4 × __
8
1
5 × __
8
1 + __
1 + __
1 + __
1
__
8 8 8 8
4
__
8
1 + __
1 + __
1 + __
1 + __
1
__
3
__
8
,
4
__
8
8
8
5
__
8
, and
8
5
__
8
8
8
.
Write the fraction as the product of a whole number and a unit fraction.
1
1
1
5 =
2 × __
__ =
__ =
5 × ___
7 × __
1. 2
2. ___
3. 7
5
12
2
5
12
2
List the next four multiples of the unit fraction.
3
4
5
2
__
__
__
__
__,
__,
4. 1
5. 1
4
4
4
4
4
6
,
,
,
Number and Operations–Fractions
2
__
6
,
3
__
6
,
4
__
6
,
5
__
6
129
Name
1
Multiples of Unit Fractions
CC.4.NF.4a
Write the fraction as a product of a whole number and a unit fraction.
1
1
__
7 = 7 × __
5× 1
__ =
__ =
5 × __
1. 5
2. __
3. 5
6
8
3
6
8
3
4=
7. __
6
1
9 × ___
10
1
4 × __
6
3=
5. __
4
8 =
8. ___
20
1
3 × __
4
11 =
6. ___
12
1
8 × ___
20
List the next four multiples of the unit fraction.
2
3
4
5
__
__
__
__
1,
__,
10. 1
11. __
5
5
5
5
8
,
,
, 5
13 =
9. ____
100
2
__
8
,
3
__
8
,
1
11 × ___
12
1
13 × ____
100
4
__
8
,
5
__
8
Problem Solving
12. So far, Monica has read 5_6 of a
book. She has read the same
number of pages each day for
5 days. What fraction of the book
1 of the book
__
6
130
13. Nicholas buys 3_8 pound of cheese.
He puts the same amount of
cheese on 3 sandwiches. How
much cheese does Nicholas put on
each sandwich?
1 pound
__
8
Lesson 65
© Houghton Mifflin Harcourt Publishing Company
9 =
4. ___
10
Name
LESSON
66
1
Multiples of Fractions
CC.4.NF.4b
OBJECTIVE Write a product of a whole number and a fraction as a product of a whole number
and a unit fraction.
You have learned to write multiples of unit fractions. You can also
write multiples of other fractions.
2.
Write the next 4 multiples of __
5
Make a table.
2
1 × __
5
2
2 × __
5
__
3× 2
5
2
4 × __
5
2
5 × __
5
2
__
2
2
__ + __
5
5
4
__
5
2 + __
2
2 + __
__
2
2
2+ 2
__ + __
__ + __
5
5
5
5
8
__
5
2 + __
2 + __
2 + __
2
2 + __
__
5
2
__
5
5
5
6
__
5
5
4
__
2 are
So, the next 4 multiples of __
6
__
5
5
8
__
5
5
10
___
5
5
10
___
5 .
,
,
, and
5
2 as the product of a whole number and a unit fraction.
Write 3 × __
5
2.
Use a number line. Make three jumps of __
5
6
25_
33_
5 5
© Houghton Mifflin Harcourt Publishing Company
'
(
,
)
,
1
23_
5
5
5
*
,
+
,
,
,
1
43_
5
5
,
.
,
1
63_
5
/
,
0
,
1
83_
5
1
2 = __
6, or 6 × __
So, 3 × __
5.
5 5
List the next four multiples of the fraction.
6
12
15
9
___
__
___
3, __
__,
1. __
2. 5
4
4
4
4
6
,
,
, 4
10
___
6 ,
15
___
6
20
___
6 ,
,
25
___
6
Write as the product of a whole number and a unit fraction.
3.
4.
'
( ) * + , - . / 0 ('
/ / / / / / / / / /
3=
3 × __
8
1
9 × __
8
Number and Operations–Fractions
'
(
*
2=
4 × __
3
)
*
*
*
+
*
,
*
- . / 0 ('
* * * * *
1
8 × __
3
131
Name
1
Multiples of Fractions
CC.4.NF.4b
List the next four multiples of the fraction.
6
9
12
15
__
___
___
__, __
__,
1. 3
2. 2
5
5
5
5
5
6
,
,
,
4,
3. __
8
8
__
8
,
12
___
8
16
___
8
,
20
___
,
8
4
__
6
5,
4. ___
10
6
__
6
,
8
__
6
,
15
___
10 ,
10
___
10 ,
10
___
,
20
___
10 ,
6
25
___
10
Write the product as the product of a whole number and a unit fraction.
5.
6.
'
(
,
)
,
4=
2 × __
5
*
,
+
,
,
,
- . / 0 ('
, , , , ,
1
8 × __
5
'
(
*
2=
5 × __
3
)
*
*
*
+
*
,
*
- . / 0 ('
* * * * *
1
10 × __
3
7. Jessica is making 2 loaves of
banana bread. She needs 3_4 cup of
sugar for each loaf. Her measuring
cup can only hold 1_4 cup of sugar.
How many times will Jessica need
to fill the measuring cup in order
to get enough sugar for both
6
132
8. A group of four students is
performing an experiment with
salt. Each student must add 3_8
teaspoon of salt to a solution.
The group only has a 1_8 -teaspoon
measuring spoon. How many
times will the group need to fill
the measuring spoon in order to
perform the experiment?
12
Lesson 66
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Name
LESSON
67
1
Multiply a Fraction by a
Whole Number Using Models
CC.4.NF.4b
OBJECTIVE Use a model to multiply a fraction by a whole number.
You can use a model to multiply a fraction by a whole number.
3.
Find the product of 4 × __
5
__ each.
Use fraction strips. Show 4 groups of 3
5
1 1 1 1 1
3
__ = __
1 group of 3
5 5 5 5 5
5
5
1
5
1
5
1
5
1
5
1
5
6
__ = __
2 groups of 3
5
5
1
5
1
5
1
5
1
5
1
5
9
__ = __
3 groups of 3
5
5
1
5
1
5
1
5
1
5
1
5
12
__ = ___
4 groups of 3
5
5
3 = ___
12.
So, 4 × __
5
5
Multiply.
2.
© Houghton Mifflin Harcourt Publishing Company
1.
5=
2 × __
6
10
___
6
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
1
8
7=
3 × __
8
2=
3. 6 × __
3
12
___
3
9 =
4. 2 × ___
10
5=
6. 4 × __
8
20
___
8
2=
7. 7 × __
5
Number and Operations–Fractions
18
___
10
14
___
5
21
___
8
3=
5. 5 × __
4
15
___
4
4=
8. 8 × __
6
32
___
6
133
Name
1
Multiply a Fraction by a Whole
Number Using Models
CC.4.NF.4b
Multiply.
5=
1. 2 × __
6
10
___
6
2=
2. 3 × __
5
6
__
3 =
3. 7 × ___
10
21
___
5 =
4. 3 × ___
12
15
___
3=
5. 6 × __
4
18
___
2=
6. 4 × __
8
8
__
8
7=
8. 2 × __
8
14
___
4=
9. 6 × __
5
24
___
5
10
___
3
4
8
10
Problem Solving
10. Matthew walks 5_8 mile to the bus
stop each morning. How far will
he walk in 5 days?
25
___ miles
8
134
11. Emily uses 2_3 cup of milk to make
one batch of muffins. How many
cups of milk will Emily use if she
makes 3 batches of muffins?
6 cups
__
3
Lesson 67
© Houghton Mifflin Harcourt Publishing Company
2=
7. 5 × __
3
12
5
Name
LESSON
68
1
Multiply a Fraction or Mixed
Number by a Whole Number
CC.4.NF.4c
OBJECTIVE Multiply a fraction by a whole number to solve a problem.
To multiply a fraction by a whole number, multiply the numerators.
Then multiply the denominators.
A recipe for one loaf of bread calls for 2 1_4 cups of flour.
How many cups of flour will you need for 2 loaves of bread?
Step 1 Write and solve an equation.
1 as a fraction.
__. Write 2__
Write 2 as 2
1
4
2 × __
9
2 × 2 1_4 = __
1
4
× 9
= 2______
1× 4
Multiply the numerators.
Then multiply the denominators.
___
= 18
4
Simplify.
}
}
}
}
Step 2 Write the product as a mixed number.
18
1 + __
1 + __
1+1
1 + __
1 + __
1 + __
1 + __
1 + __
1 + __
1 + __
1 + __
1 + __
1 + __
1 + __
1
1 + __
1 + __
___ = __
__ + __
4
4
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
4
1
=
© Houghton Mifflin Harcourt Publishing Company
=
4
+
__
42
4 , or
+
1
__
4
+
__
41
2
__
41
2
So, you will need
1
__
4
1
+
1
6
2___
10
Number and Operations–Fractions
1
1
1 + __
+ __
4 4
Combine the wholes. Then combine the
remaining parts.
Add. Write the sum as a mixed number.
cups of flour.
Multiply. Write the product as a mixed number.
__
__
2=
3=
11
14
1. 3 × __
2. 4 × __
5
8
5
8
3 =
4. 2 × 1___
10
+
2=
5. 4 × 1__
3
__
62
3
1=
3. 5 × __
3
1=
6. 7 × 1__
6
2
1__
3
1
8__
6
135
Name
1
Multiply a Fraction or Mixed
Number by a Whole Number
CC.4.NF.4c
3=
3. 5 × __
4
__
33
4
1=
4. 4 × 1__
5
4
4__
5
1=
5. 2 × 2__
3
__
42
3
1=
6. 5 × 1__
6
__
55
6
7=
7. 2 × 2__
8
__
56
8
3=
8. 7 × 1__
4
__
121
4
3=
9. 8 × 1__
5
__
124
5
Problem Solving
10. Brielle exercises for 3_4 hour each
day for 6 days in a row. Altogether,
how many hours does she exercise
during the 6 days?
2 hours
4__
4
136
11. A recipe for quinoa calls for
2 2_3 cups of milk. Conner wants to
make 4 batches of quinoa. How
much milk does he need?
2 cups
10__
3
Lesson 68
© Houghton Mifflin Harcourt Publishing Company
Multiply. Write the product as a mixed number.
5
4
3 = 1___
3 = 1__
1. 5 × ___
2. 3 × __
10
5
10
5
Name
LESSON
69
1
Problem Solving • Comparison
Problems with Fractions
CC.4.NF.4c
OBJECTIVE Use the strategy draw a diagram to solve comparison problems with fractions.
The Great Salt Lake in Utah is about 4_5 mile above sea level. Lake
Titicaca in South America is about 3 times as high above sea level as
the Great Salt Lake. About how high above sea level is Lake Titicaca?
Solve the Problem
What do I need to find?
I need to find about how high above
sea level Lake Titicaca is.
Great Salt Lake
sea level.
times as high above
How will I use the information?
© Houghton Mifflin Harcourt Publishing Company
I can draw a diagram to compare
the heights.
2
2__
So, Lake Titicaca is about 5
4
__
5
4
__
5
4
__
5
t
Write an equation and solve.
t is the height above sea level of
Lake Titicaca, in miles.
4
__
5
3
t=
×
Write an equation.
12
___
5
t=
Multiply.
2
2__
5
t=
Write the fraction
as a mixed number.
miles above sea level.
1. Amelia is training for a triathlon.
She swims 3_5 mile. Then she runs
about 6 times farther than she
Amelia run?
__ miles
33
5
Number and Operations–Fractions
Lake Titicaca
4
__
5
}
What information do I need to use?
4
__
The Great Salt Lake is about 5
mile above sea level. Lake Titicaca
3
Draw a comparison model. Compare
the heights above sea level of the Great
Salt Lake and Lake Titicaca, in miles.
2. Last week, Meg bought 1 3_4 pounds
of fruit at the market. This week, she
buys 4 times as many pounds of fruit
as last week. In pounds, how much
fruit does Meg buy this week?
7 pounds
137
Name
1
Problem Solving • Comparison
Problems with Fractions
CC.4.NF.4c
1. A shrub is 1 2_3 feet tall. A small tree is 3 times
as tall as the shrub. How tall is the tree?
t is the height of the tree, in feet.
2
t = 3 × 1__
3
__
t=3×5
3
15
___
t=
3
t=5
shrub
tree
So, the tree is 5 feet tall.
__
12
3
__
12
3
__
12
3
__
12
3
5 feet
2. You run 1 3_4 miles each day. Your friend runs
4 times as far as you do. How far does your
friend run each day?
7 miles
© Houghton Mifflin Harcourt Publishing Company
3. At the grocery store, Ayla buys 1 1_3 pounds of
ground turkey. Tasha buys 2 times as much
ground turkey as Ayla. How much ground
2 pounds
2__
3
4. When Nathan’s mother drives him to school, it
takes 51_ hour. When Nathan walks to school, it
takes him 4 times as long to get to school. How
long does it take Nathan to walk to school?
4
__ hour
5
138
Lesson 69
Name
LESSON
70
1
Equivalent Fractions and Decimals
CC.4.NF.5
OBJECTIVE Record tenths and hundredths as fractions and decimals.
20
Lori ran ___
mile. How many tenths of a mile did she run?
100
20
Write ___
as an equivalent fraction with a denominator of 10.
100
Step 1 Think: 10 is a common factor of the numerator and the denominator.
Step 2 Divide the numerator and denominator by 10.
20 = __________
20 ÷ 10 = ___
2
____
100
100 ÷ 10
10
2 mile.
So, Lori ran __
10
Use a place-value chart.
20
Step 1 Write ___
as an equivalent decimal.
100
Ones
0
· Tenths
·
2
Hundredths
0
Step 2 Think: 20 hundredths is
Ones
0
2
tenths
0
hundredths
· Tenths
·
2
So, Lori ran 0.2 mile.
© Houghton Mifflin Harcourt Publishing Company
Write the number as hundredths in fraction form and decimal form.
9
1. ___
10
4
3. ___
10
2. 0.6
60 ; 0.60
____
90 ; 0.90
____
100
40 ; 0.40
____
100
100
Write the number as tenths in fraction form and decimal form.
70
4. ____
100
80
5. ____
100
7 ; 0.7
___
10
Number and Operations–Fractions
6. 0.50
8 0.8
___
10;
5 ; 0.5
___
10
139
Name
1
Equivalent Fractions and Decimals
CC.4.NF.5
Write the number as hundredths in fraction form
and decimal form.
5
1. ___
10
5 = _______
50
5 × 10 = ____
___
10
10 × 10
100
Think: 5 tenths is the same as 5 tenths and 0 hundredths.
Write 0.50.
50 ; 0.50
____
100
9
2. ___
10
3. 0.2
90 ; 0.90
____
4. 0.8
20 ; 0.20
____
100
80 ; 0.80
____
100
100
Write the number as tenths in fraction form
and decimal form.
10
6. ____
100
4 ; 0.4
___
10
7. 0.60
1 ; 0.1
___
6 ; 0.6
___
10
10
Problem Solving
6
8. Billy walks __
mile to school
10
6
each day. Write __
as hundredths in
10
fraction form and in decimal form.
60 ; 0.60
____
100
140
9. Four states have names that begin
with the letter A. This represents
0.08 of all the states. Write 0.08 as
a fraction.
8
____
100
Lesson 70
© Houghton Mifflin Harcourt Publishing Company
40
5. ____
100
Name
Add Fractional Parts of 10 and 100
LESSON
71
OBJECTIVE Add fractions when the denominators are 10 or 100.
1
CC.4.NF.5
35
Sam uses 100 glass beads for a project. Of the beads, ___
are gold
100
4
__
and 10 are silver. What fraction of the glass beads are gold or silver?
35 and ___
4.
100
10
Step 1 Decide on a common denominator. Use 100 .
4 as an equivalent fraction with a denominator of 100.
Step 2 Write __
10
4 = ________
4 × 10 = ____
40
___
10
10 × 10
100
35
40
___
100 and 100 .
35 + ____
40 = ____
75
____
100
100
100
Use 100 as the denominator.
75
____
So, 100 of the glass beads are gold or silver.
Step 1 Write each amount as a fraction of a dollar.
© Houghton Mifflin Harcourt Publishing Company
26
\$0.26 = ___
of a dollar
100
26
59
and ___
.
100
100
26 + ____
59 = ____
85
____
100
100
100
59
\$0.59 = ___
of a dollar
100
100 is the common denominator.
Step 3 Write the sum as a decimal.
85 = 0.85
____
100
So, \$0.26 + \$0.59 = \$0.85 .
Find the sum.
95
75 + ___
2 = ____
1. ____
100 10
100
Number and Operations–Fractions
2. \$0.73 + \$0.25 = \$ 0.98 .
98
73 + ____
25 = ____
____
100 100
100
141
Name
1
Add Fractional Parts of 10 and 100
CC.4.NF.5
Find the sum.
2 + ____
43
1. ___
10 100
2
Think: Write __
as a fraction with
10
a denominator of 100:
2 × 10 = ____
20
_______
20
43
63
___
+ ___
= ___
100
100
100
10 × 10
100
63
____
100
17 + ___
6
2. ____
100 10
9 + ___
4
3. ____
100 10
7 + ____
23
4. ___
10 100
49
____
77
____
100
93
____
100
100
5. \$0.48 + \$0.30
6. \$0.25 + \$0.34
7. \$0.66 + \$0.06
\$0.78
\$0.59
\$0.72
38
8. Ned’s frog jumped ___
meter. Then
100
4
__
his frog jumped 10 meter. How far
did Ned’s frog jump in all?
78 meter
____
100
142
5
9. Keiko walks __
kilometer from
10
school to the park. Then she
19
kilometer from the park
walks ___
100
to her home. How far does Keiko
walk in all?
69 kilometer
____
100
Lesson 71
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Name
1
Relate Tenths and Decimals
LESSON
72
CC.4.NF.6
OBJECTIVE Record tenths as fractions and as decimals.
Write the fraction and the decimal that are shown by the point
on the number line.
0
10
5
10
10
10
0.0
0.5
1.0
Step 1 Count the number of equal parts of the whole shown
on the number line. There are ten equal parts.
This tells you that the number line shows tenths.
Step 2 Label the number line with the missing fractions.
What fraction is shown by the point on the number line?
0
10
1
10
2
10
3
10
4
10
0.0
5
10
6
10
7
10
8
10
9
10
0.5
10
10
1.0
The fraction shown by the point on
8
the number line is __
.
10
Step 3 Label the number line with the missing decimals.
What decimal is shown by the point on the number line?
0
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
© Houghton Mifflin Harcourt Publishing Company
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
The decimal shown by the point on
the number line is 0.8.
So, the fraction and decimal shown by the point on the number line
8
and 0.8.
are __
10
Write the fraction or mixed number and the
decimal shown by the model.
1.
0
10
5
10
10
10
0.0
0.5
1.0
2 ; 0.2
___
10
Number and Operations–Fractions
2.
0
2 10
5
2 10
2 10
10
2.0
2.5
3.0
6 ; 2.6
2 ___
10
143
Name
1
Relate Tenths and Decimals
CC.4.NF.6
Write the fraction or mixed number and the decimal
shown by the model.
1.
2.
Think: The model is divided
into 10 equal parts.
Each part represents
one tenth.
6 ; 0.6
___
2 ; 1.2
1___
10
10
3. 2 0
10
5
2 10
2.0
2.5
4.
0
4 10
4.0
3 ; 2.3
2___
10
5
4 10
4 10
10
4.5
5.0
8 ; 4.8
4___
10
Write the fraction or mixed number as a decimal.
1
6. 3___
10
0.4
7
7. ___
10
3.1
0.7
5
8. 6___
10
9
9. ___
10
6.5
0.9
Problem Solving
10. There are 10 sports balls in the
equipment closet. Three are
kickballs. Write the portion of
the balls that are kickballs as a
fraction, as a decimal, and in word
form.
3 ; 0.3; three tenths
___
10
144
11. Peyton has 2 pizzas. Each pizza is
cut into 10 equal slices. She and
her friends eat 14 slices. What part
of the pizzas did they eat? Write
1.4 pizzas
Lesson 72
© Houghton Mifflin Harcourt Publishing Company
4
5. ___
10
Name
LESSON
73
1
Relate Hundredths and Decimals
CC.4.NF.6
OBJECTIVE Record hundredths as fractions and as decimals.
Write the fraction or mixed number and the
decimal shown by the model.
Step 1 Count the number of
and the total number of
squares in the whole model.
Step 2 Write a fraction to
represent the part of the
53
Number
______________________
Total number of squares
100
© Houghton Mifflin Harcourt Publishing Company
Step 3 Write the fraction in
decimal form.
Total number of squares: 100
53 .
The fraction shown by the model is ____
100
53 .
Think: The fraction shown by the model is ____
100
53 .
0.53 names the same amount as ____
100
The decimal shown by the model is 0.53.
53 and 0.53.
The fraction and decimal shown by the model are ____
100
Write the fraction or mixed number and the
decimal shown by the model.
1.
2.
24 ; 0.24
____
100
Number and Operations–Fractions
96 ; 0.96
____
100
145
Name
1
Relate Hundredths and Decimals
CC.4.NF.6
Write the fraction or mixed number and the
decimal shown by the model.
1.
Think: The whole is
divided into
one hundred
equal parts, so
each part is one
hundredth.
77 ;
____
100
2.
0
100
10
100
20
100
30
100
40
100
50
100
60
100
0
0.10
0.20 0.30 0.40 0.50 0.60
29 ; 0.29
____
0.77
100
3.
4.
20 4 30 4 40 4 50 4 60 4 70 4 80
4 100
100
100
100
100
100
100
4.20
4.30
4.40
4.50
4.60
4.70
4.80
62 ; 4.62
4____
100
54 ; 1.54
1____
100
Write the fraction or mixed number as a decimal.
11
6. 8____
100
0.37
98
7. ____
100
8.11
0.98
50
8. 25____
100
6
9. ____
100
25.50
0.06
Problem Solving
10. There are 100 pennies in a dollar.
What fraction of a dollar is 61
pennies? Write it as a fraction, as
a decimal, and in word form.
61 ; 0.61; sixty-one hundredths
____
100
146
11. Kylee has collected 100 souvenir
thimbles from different places
she has visited with her family.
Twenty of the thimbles are carved
from wood. Write the fraction of
thimbles that are wooden as a
decimal.
0.20
Lesson 73
© Houghton Mifflin Harcourt Publishing Company
37
5. ____
100
Name
LESSON
74
Relate Fractions, Decimals, and Money
OBJECTIVE Translate among representations of fractions, decimals, and money.
1
CC.4.NF.6
Write the total money amount. Then write the amount as a
fraction and as a decimal in terms of a dollar.
Step 1 Count the value of coins from greatest to least.
Write the total money amount.
\$ 0.25
\$ 0.35
\$ 0.40
\$ 0.45
\$ 0.50
Step 2 Write the total money amount as a fraction of a dollar.
The total money amount is \$0.50, which is the same as 50 cents.
Think: There are 100 cents in a dollar.
50 cents = ____
50
So, the total amount written as a fraction of a dollar is: __________
100 cents 100
Step 3 Write the total money amount as a decimal.
© Houghton Mifflin Harcourt Publishing Company
Think: I can write \$0.50 as 0.50.
50 written as a fraction of a dollar, and
The total money amount is ____
100
0.50 written as a decimal.
Write the total money amount. Then write the amount as a fraction
or a mixed number and as a decimal in terms of a dollar.
1.
2.
80 ; 0.80
\$0.80; ____
100
Number and Operations–Fractions
45 ; 1.45
\$1.45; 1____
100
147
Name
1
Relate Fractions, Decimals, and Money
CC.4.NF.6
Write the total money amount. Then write the amount as a
fraction or a mixed number and as a decimal in terms of dollars.
1.
2.
56 ; 0.56
\$0.56; ____
100
18 ; 0.18
\$0.18, ____
100
Write as a money amount and as a decimal in terms of dollars.
79
4. ____
100
25
3. ____
100
\$0.25; 0.25
31
5. ____
100
\$0.79; 0.79
8
6. ____
100
\$0.31; 0.31
42
7. ____
100
\$0.08; 0.08
\$0.42; 0.42
Write the money amount as a fraction in terms of dollars.
9. \$0.03
10. \$0.66
11. \$0.95
87
____
3
____
66
____
95
____
100
100
100
12. \$1.00
100
1
____, or __
100
1
100
Write the total money amount. Then write the amount as a
fraction and as a decimal in terms of dollars.
13. 2 quarters 2 dimes
14. 3 dimes 4 pennies
70 ; 0.70
\$0.70; ____
100
34 ; 0.34
\$0.34; ____
100
15. 8 nickels 12 pennies
52 ; 0.52
\$0.52; ____
100
Problem Solving
16. Kate has 1 dime, 4 nickels, and 8
pennies. Write Kate’s total amount
as a fraction in terms of a dollar.
38
____
100
148
75
17. Nolan says he has ___
of a dollar.
100
If he only has 3 coins, what are the
coins?
three quarters
Lesson 74
© Houghton Mifflin Harcourt Publishing Company
8. \$0.87
Name
LESSON
75
1
Compare Decimals
CC.4.NF.7
OBJECTIVE Compare decimals to hundredths by reasoning about their size.
Alfie found 0.2 of a dollar and Gemma found 0.23
of a dollar. Which friend found more money?
To compare decimals, you can use a number line.
Step 1 Locate each decimal on a number line.
0.0
0.10
0.20
0.30
Step 2 The number farther to the right is greater.
0.23 > 0.2, so Gemma found more money.
To compare decimals, you can compare equal-size parts.
Step 1 Write 0.2 as a decimal in hundredths.
0.2 is 2 tenths, which is equivalent to 20 hundredths.
0.2 = 0.20
Step 2 Compare.
23 hundredths
is greater than
20 hundredths.
© Houghton Mifflin Harcourt Publishing Company
so 0.23 > 0.2.
So, Gemma found more money.
Compare. Write <, >, or =.
1. 0.17 > 0.13
2. 0.8 > 0.08
3. 0.36 < 0.63
4. 0.4 = 0.40
5. 0.75 > 0.69
6. 0.3 < 0.7
7. 0.45 > 0.37
8. 0.96 > 0.78
Number and Operations–Fractions
149
Name
1
Compare Decimals
CC.4.NF.7
Compare. Write <, >, or =.
1. 0.35 < 0.53
2. 0.6 = 0.60
3. 0.24 < 0.31
Think: 3 tenths is less
than 5 tenths.
So, 0.35 < 0.53
4. 0.94 > 0.9
5. 0.3 < 0.32
6. 0.45 > 0.28
7. 0.39 < 0.93
Use the number line to compare. Write true or false.
0
0.1
8. 0.8 > 0.78
0.2
0.3
0.4
9. 0.4 > 0.84
true
false
0.5
0.6
0.7
0.8
10. 0.7 < 0.70
false
0.9
1.0
11. 0.4 > 0.04
true
Compare. Write true or false.
false
13. 0.24 = 0.42
false
14. 0.17 < 0.32
true
15. 0.85 > 0.82
true
Problem Solving
16. Kelly walks 0.7 mile to school.
Mary walks 0.49 mile to school.
Write an inequality using <, >,
or = to compare the distances
they walk to school.