Develop Skills and Strategies
Lesson 17
Part 1: Introduction
CCSS
3.NF.A.3b
3.NF.A.3c
Find Equivalent Fractions
In Lesson 16, you learned that equivalent fractions tell about the same amount
of the whole. Take a look at this problem.
Izzy’s mom baked a cake. She put chocolate frosting on half of the cake and
vanilla frosting on half of the cake.
Then Izzy’s mom cut the cake into fourths. What is a different fraction that
names the part of the cake that has chocolate frosting?
Explore It
Use the math you already know to solve the problem.
Look at the picture above. What fraction of the cake has chocolate frosting?
How many equal parts should the cake be divided into to show fourths? On the picture above, draw lines to divide the cake into fourths.
How many fourths of the cake have chocolate frosting? Did the amount of cake with chocolate frosting change? Explain how you know
that ​ 1 ​of the cake is the same as ​ 2 ​ .
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4
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L17: Find Equivalent Fractions
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Part 1: Introduction
Lesson 17
Find Out More
In the last lesson, you looked at stacked models to understand equivalent fractions.
Both models were the same size, but divided into a different number of equal parts.
You looked for fractions that named the same amount of the whole.
You can also find equivalent fractions by dividing the same model in different ways.
The cake already showed ​ 1 ​ . You may have drawn lines in one of these ways to
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show fourths.
No matter how the fourths are made, the models show that ​ 2 ​of the cake has
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chocolate frosting.
You can also divide a number line in different ways to show that ​ 1 ​is equivalent to ​ 2 .​
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1
2
0
0
1
4
2
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1
Reflect
1 Izzy’s mom now wants to cut the cake into eighths. Explain how to figure out how
many eighths of the cake have chocolate frosting.
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Part 2: Modeled Instruction
Lesson 17
Read the problem below. Then explore different ways to think about
equivalent fractions.
Casen ate ​ 2 ​of an orange. Trey’s orange is the same size. He ate ​ 1 ​of it. Show
8
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4
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that the two boys ate the same amount.
Picture It
You can use models to help find equivalent fractions.
This model shows ​ 2 ​ .
This model shows ​ 1 ​ .
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4
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Look at the model of ​ 2 ​ . The solid lines show the division into fourths. The dotted
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lines show how each fourth was divided to make eighths.
Model It
You can also use a number line to help find equivalent fractions.
This number line shows both fourths and eighths.
1
4
0
2
4
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4
1
0 1 2 3 4 5 6 7 1
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8 8 8
8 8
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Part 2: Guided Instruction
Lesson 17
Connect It
Now you will solve the problem from the previous page using equations.
2 How do you know that ​ 2 ​of the first model are shaded? 8
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3 How do you know that ​ 1 ​of the second model is shaded? 4
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4 Explain how the models show that the fractions are equivalent. 5 How does the number line show that the fractions are equivalent? 6 Write sentences to show the fractions of the two oranges name the same amount.
use words: Two eighths is equal to .
use fractions: equals .
7 Explain how you can show two fractions are equivalent. Try It
Use what you just learned to solve these problems.
8 Draw a model to show ​ 2 ​ 5 ​  4 ​ .
3 ··
6
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9 Use the number line to find a fraction
equivalent to ​ 1 ​ .
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0
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Part 3: Modeled Instruction
Lesson 17
Read the problem below. Then explore different ways to write a whole number
as a fraction.
Kacey used 2 same-sized boards to build a birdhouse. He cut each board into
fourths. How can you write 2 as a fraction to show Kacey’s boards divided into
fourths?
Picture It
You can use models to help you write a whole number as a fraction.
The fraction strips below show 2 wholes, each divided into fourths.
1
4
1
4
1
4
1
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1
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4
Each part is ​ 1 ​of a whole. There are eight ​ 1 ​s in all.
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Model It
You can use a number line to help you write a whole number as a fraction.
This number line shows whole numbers on the top and fourths on the bottom.
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1
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Notice that each whole number has an equivalent fraction.
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L17: Find Equivalent Fractions
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Part 3: Guided Instruction
Lesson 17
Connect It
Now you will solve the problem from the previous page using equations.
10 Look at the models on the previous page. How many equal parts are in 1 whole?
Explain how you know. 11 How many parts are in 2 wholes? Explain how you know. 12 Write sentences to show the fraction that is equivalent to 2.
use words: Two wholes equals .
use a fraction: 2 equals .
13 Explain how to find a fraction equivalent to a whole number.
Try It
Use what you just learned to solve these problems.
14 Use the model to find a fraction
equivalent to 3.
15 Draw a model to show 3 5 ​  18  ​ .
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Part 4: Modeled Instruction
Lesson 17
Read the problem. Then explore different ways to write a whole number as a
fraction with a denominator of 1.
Justin picked 4 green peppers from his garden. He did not cut them into pieces.
How can you write 4 as a fraction to show Justin’s green peppers?
Picture It
You can use models to help you write a whole number as a fraction with a
denominator of 1.
Each circle stands for 1 green pepper.
They are not divided into pieces, so each whole is made of one part.
Model It
You can use a number line to help you write a whole number as a fraction with
a denominator of 1.
This number line shows whole numbers on the top and fractions on the bottom.
0
1
2
3
4
0
1
1
2
1
3
1
4
1
Notice that each whole number has an equivalent fraction. The spaces between
whole numbers are not divided into parts. Each whole number is made of one part,
so the denominator is 1.
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Part 4: Guided Instruction
Lesson 17
Connect It
Now you will solve the problem from the previous page.
16 Explain how you know each whole only has 1 part. 17 How many parts do the 4 green peppers make? 18 What does the numerator of a fraction show? 19 What does the denominator of a fraction show? 20 Write a fraction equivalent to 4.
number of parts
​
​  ​ 1​ Remember: ··························
number of equal parts in a whole 2
21 Explain how to write a whole number as a fraction with a denominator of 1.
Why does this work?
Try It
Use what you just learned to solve these problems.
22 Use the model to find a fraction equivalent to 6.
6 5 23 Draw a model to show 5 5 ​  5 ​ .
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Part 5: Guided Practice
Lesson 17
Study the model below. Then solve problem 24–26.
Student Model
The student used solid
lines to show fourths
and dotted lines to show
eighths.
Caleb and Hannah bought two melons that are the same size.
Caleb cut his into fourths. Hannah cut hers into eighths.
Hannah ate ​ 4 ​of her melon, and Caleb ate an equal amount of
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his melon. What fraction did Caleb eat?
Look at how you could show your work using a model.
Pair/Share
How could you solve
this problem using a
number line?
How many thirds are in 1
whole? How many
eighths are in 1 whole?
Solution: Caleb ate ​ 2 ​of his melon.
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3
8
24 Matt says ​   ​is equivalent to 1. Elisa says ​   ​is equivalent to 1.
3
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Who is correct?
Show your work.
Pair/Share
What is another fraction
that is equivalent to 1?
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Solution: L17: Find Equivalent Fractions
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Part 5: Guided Practice
25 Show two fractions that are equivalent to 5.
Show your work.
Lesson 17
There will be 5 wholes in
all. Think about how
many parts will be in
each whole.
Pair/Share
Solution: 26 Kaia ate ​ 3 ​of a banana. Zoie ate an equivalent amount. Which
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fraction shows how much Zoie ate? Circle the letter of the correct
answer.
A​  1 ​
How did you decide
what denominators to
use in your fractions?
Look at where ​ 3  ​is on a
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number line. What other
fraction could be shown
at the same point?
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B​  2 ​
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C​  5 ​
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D​  1 ​
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Landon chose A as the correct answer. How did he get that
answer?
Pair/Share
Does Landon’s answer
make sense?
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Part 6: Common Core Practice
Lesson 17
Solve the problems.
1 Which fraction shown is equivalent to }
​ 2 ​   ?
6
A
B CD
2 Which fraction is equivalent to 3?
3 ​
A​ }
1
1  ​
B​ }
3
4  ​
C​ }
1
6 ​
D​ }
3
3 Look at point P on the number line.
P
0
1
Look at number lines 3a–3c. Is the point on each number line equal to the number
shown by point P? Choose Yes or No.
a.
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No
Yes
No
Yes
No
1
c.
0
Yes
1
b.
0
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Part 6: Common Core Practice
Lesson 17
4 For numbers 4a–4d, choose Yes or No to indicate whether each number graphed on
the number line represents one whole.
a.
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b.
c.
d.
Yes
No
Yes
No
Yes
No
Yes
No
5 Use the number line below to find a fraction equivalent to 3.
Show your work.
0
1
2
3
Answer 3 is equivalent to 6 Draw a model to find a fraction equivalent to }
​ 1 ​   .
4
Show your work.
1  ​  is equivalent to Answer ​ }
4
Self Check Go back and see what you can check off on the Self Check on page 131.
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Develop Skills and Strategies
Lesson 17
Find Equivalent Fractions
Lesson Objectives
The Learning Progression
•Use fraction models and number lines to identify
and create equivalent fractions, including those that
are greater to or equal to one whole.
In previous lessons students learned that more than
one fraction can name the same amount or a point on a
number line. Students understand that because sixths
are smaller than halves, it takes more sixths to cover
the same area. In this lesson students extend these
ideas to include identifying and generating equivalent
fractions for numbers equal to or greater than one
whole using fraction models and number lines.
•Express whole numbers as fractions.
•Identify fractions that are equivalent to whole
numbers.
Prerequisite SkilLs
Students also explore and practice ways to write a
•Understand the meaning of fractions.
whole number as a fraction with a denominator of 1,
•Identify fractions represented by models.
such as ​ 3 ​ . Students reason about the size of fractions
•Understand that the size of a fractional part is
relative to the size of the whole.
to solve problems and explain why two fractions are or
•Understand how to use number lines to count and
identify fractional parts.
Vocabulary
1
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are not equivalent.
Students’ work with equivalent fractions in this lesson
provides a foundation for comparing fractions and for
solving problems involving fraction operations.
There is no new vocabulary.
Teacher Toolbox
Teacher-Toolbox.com
Prerequisite
Skills
3.NF.A.3b
3.NF.A.3c
✓✓
✓
✓
✓
Ready Lessons
Tools for Instruction
Interactive Tutorials
CCSS Focus
3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
b.Recognize and generate simple equivalent fractions, e.g., ​ 1 ​ 5 ​ 2 ​ , ​ 4 ​ 5 ​ 2 ​ . Explain why the fractions are equivalent, e.g., by
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4 ··
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3
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using a visual fraction model.
c.Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in
the form 3 2 ​  3 ​ ; recognize that ​ 6 ​ 5 6; locate ​ 4 ​and 1 at the same point of a number line diagram.
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STANDARDS FOR MATHEMATICAL PRACTICE: SMP 1, 2, 4, 6 (see page A9 for full text)
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Part 1: Introduction
Lesson 17
At a Glance
Develop skills and strategies
Students read a word problem and answer a series of
questions designed to apply what they know about
equivalent fractions.
Lesson 17
Part 1: introduction
ccss
3.nF.a.3b
3.nF.a.3c
Find equivalent Fractions
in Lesson 16, you learned that equivalent fractions tell about the same amount
of the whole. take a look at this problem.
Step By Step
Izzy’s mom baked a cake. She put chocolate frosting on half of the cake and
•Tell students that this page models how to think
about using more than one fraction to name the same
amount.
vanilla frosting on half of the cake.
Then Izzy’s mom cut the cake into fourths. What is a different fraction that
•Have students read the problem at the top of the
page.
names the part of the cake that has chocolate frosting?
explore it
•Work through Explore It as a class. Draw a model of
use the math you already know to solve the problem.
the cake on the board and shade ​ 1 ​to represent the
Look at the picture above. What fraction of the cake has chocolate frosting?
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1
2
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chocolate frosting. Ask a student to show one way to
How many equal parts should the cake be divided into to show fourths?
4
On the picture above, draw lines to divide the cake into fourths.
divide the cake into fourths on the board.
How many fourths of the cake have chocolate frosting?
•Ask student pairs or groups to discuss the last
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Did the amount of cake with chocolate frosting change? Explain how you know
that 1 of the cake is the same as 2 .
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Possible answer: no, the amount with chocolate frosting did not change. 1
question together and explain how they know. Ask
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is the same as 2 because the same amount of the cake is shaded. the only
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them to share why the answer makes sense to them.
thing that changed was how many pieces the cake was cut into.
Invite volunteers to share. Emphasize that the
amount of chocolate frosting did not change and
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L17: Find Equivalent Fractions
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that ​ 1 ​ and ​ 2 ​name the same amount of frosting.
2
4
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SMP Tip: Students rely on a picture or model of
the cake to help conceptualize and gain insight into
the problem’s solution and ask themselves if it
makes sense (SMP 1).
L17: Find Equivalent Fractions
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Copying is not permitted.
Mathematical Discourse
•Why does it make sense that ​ 2 ​ and ​ 1 ​can both name
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2
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the same amount of frosting?
Students should understand that the
denominators name different ways to divide the
same whole into equal parts and the
numerators name parts of the divided whole
that are shaded.
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Part 1: Introduction
Lesson 17
At a Glance
Students explore using the model for one fraction to
make an equivalent fraction by dividing each fractional
part into smaller, equal parts. They apply this idea by
showing equivalent fractions on a number line.
Part 1: introduction
Lesson 17
Find out More
In the last lesson, you looked at stacked models to understand equivalent fractions.
Both models were the same size, but divided into a different number of equal parts.
You looked for fractions that named the same amount of the whole.
You can also find equivalent fractions by dividing the same model in different ways.
Step By Step
The cake already showed 12 . You may have drawn lines in one of these ways to
··
show fourths.
•Read Find Out More as a class. Explain that
previously they showed two fraction models side-byNo matter how the fourths are made, the models show that 2 of the cake has
side to see that two fractions were equivalent. Draw a
chocolate frosting.
model of the cake showing two halves. Direct
You can also divide a number line in different ways to show that 1 is equivalent to 2.
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2
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students’ attention to the three ways to divide the
0
halves into fourths. Ask students to talk with a
0
partner to prove that all the ways to show ​ 2 ​ are
equivalent to ​ 1 ​ . Invite students to share.
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2
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•Emphasize that although each ​ 1 ​is a different shape,
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they cover the same amount and they both are the
1
2
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1
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1
reflect
1 Izzy’s mom now wants to cut the cake into eighths. Explain how to figure out how
many eighths of the cake have chocolate frosting.
Possible answer: Divide each fourth into 2 equal pieces to make eighths.
then count how many eighths are shaded. or divide each section on the
number line again and then count how many eighths are the same as 1 .
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same amount as ​ 1 ​ .
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•Draw the number line from the page on the board.
Point out that they can label the point ​ 1 ​above the
2
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line and label fourths below the line to show that
​ 2 ​ and ​ 1 ​name the same point.
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2
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•Ask pairs to discuss and answer the Reflect It
question. Have them draw the cake on a paper or
white board to show one way that ​ 1 ​is equivalent
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to ​ 4 ​ . For a quick assessment, have pairs hold up
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their paper or white board. Call on students to
explain how they divided the fourths into eighths.
Concept Extension
•Up to this point, students have relied on an
intuitive notion of congruence or have laid
equivalent fractions on top of each other to see
Real-World Connection
Use two pieces of fruit that are the same size to help
students reason that they are getting a fair share or
equivalent amount if they cut the same-size fruits
into halves two different ways. If two apples are
shown standing up, cut one apple in half using a
horizontal cut. Cut the other apple using a vertical
cut. Emphasize that it’s the same amount, only the
shapes are different. Students will need more
experiences to develop this understanding over time.
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that they cover the same amount or area. To help
students understand that each ​ 1 ​is the same
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amount, cut out 3 models of the cake and shade
the ways to show ​ 2 ​on each. Cut apart the
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second ​ 2 ​, rotate it, and place it on top of the ​ 2 ​ on
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4
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the first model. For the ​ 2 ​on the 3rd model, cut
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each ​ 1 ​apart, and cut it in half, so you can rotate
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the parts to cover the ​ 2 ​on the first model.
4
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Part 2: Modeled Instruction
Lesson 17
At a Glance
Part 2: Modeled instruction
Students use number lines and fraction models to find
equivalent fractions.
read the problem below. then explore different ways to think about
equivalent fractions.
Step By Step
•Explain to students that they will now explore more
ways to think about equivalent fractions.
•Read the problem at the top of the page as a class.
Explain that students can use models to prove that
the boys ate the same amount as long as the two
oranges are the same size.
•Read Picture It. Guide students to realize that they
can draw rectangle models or use circle models that
more closely resembles oranges. Focus students’
attention on the first model that shows ​ 2 ​ . Ask them
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if they can also see where the fourths are in the
model. [The solid lines show fourths.] Ask students
to explain how they can get eighths from fourths.
[Divide each fourth in half.] Ask students to talk
with a partner to explain if it makes sense that both
boys ate the same amount and why. Invite students to
share their ideas.
•Draw a number line from 0 to 1 that shows fourths.
Read Model It together and ask students how they
could create eighths on the same number line. Invite
a student to draw the eighths on the number line on
the board.
Lesson 17
Casen ate 2 of an orange. Trey’s orange is the same size. He ate 1 of it. Show
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4
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that the two boys ate the same amount.
Picture it
you can use models to help find equivalent fractions.
This model shows 2 .
This model shows 1 .
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4
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Look at the model of 2 . The solid lines show the division into fourths. The dotted
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lines show how each fourth was divided to make eighths.
Model it
you can also use a number line to help find equivalent fractions.
This number line shows both fourths and eighths.
1
4
0
2
4
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4
1
0 1 2 3 4 5 6 7 1
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8 8
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SMP Tip: Draw students’ attention to using this
simple circle model can make it easier to see why
the same amount of orange can be named by a
different fraction 1​  ​  2 ​ or ​ 1 ​   2​ , depending upon the
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number of pieces in each whole in the model (SMP 4).
Concept Extension
Mathematical Discourse
•Use the number line model on the board to extend
•Is it possible to use the fraction circle to find another
students’ thinking about other equivalent fractions
fraction equivalent to ​ 1 ​ ? How could this be done?
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A student may say that they could divide each
eighths into two equal parts or divide each
fourth into 3 equal parts. Ask students to
predict the unit fractions that the whole would
be divided into.
shown on the number line. Ask students to find
at least two other fractions that are equivalent
​3  ​ 3 ​ and ​ 6 ​   4​and explain why they are equivalent
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[same distance from 0 on the number line].
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Part 2: Guided Instruction
Lesson 17
At a Glance
Students revisit the problem on page 152 and solve the
problem using equations.
Lesson 17
connect it
now you will solve the problem from the previous page using equations.
Step By Step
2 How do you know that 2 of the first model are shaded?
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there are 8 equal parts and 2 are shaded.
•Read and discuss Connect It as a class. For number
6, write the equation ​ 2 ​ 5 ​ 1 ​on the board. Ask
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students if there is another way to write the
equation. ​3  ​  1 ​ 5 ​ 2 ​   4​Focus students’ attention on the
4
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two numerators. Point out that they divided each
1 part out of 4 into 2 parts. Point to the numerator
and explain that when they cut each ​ 1 ​into two equal
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parts, they created 8 parts in the whole or eighths.
Help students understand that the numbers match
what they did when they divided the ​ 1 ​ s into
2 smaller and equal parts.
Part 2: guided instruction
4
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•For problem 7, ask students to first explain their
thinking to a partner. Then invite students to share
how they would show that two fractions are
equivalent. Look for students to say that the two
fractions take up the same space or area on the
model or are at the same point on the number line
(and same distance from 0).
•Ask students to work on the Try It problems in pairs.
Draw a rectangle and number line showing thirds on
the board. Invite students to come to the board to
show how they filled in the models on the page.
Be ready to model another way to create sixths from
thirds. Draw vertical thirds and draw a horizontal
dotted line to cut the rectangle into top and bottom
regions, creating sixths.
3 How do you know that 1 of the second model is shaded?
4
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there are 4 equal parts and 1 is shaded.
4 Explain how the models show that the fractions are equivalent.
the amount that is shaded is the same.
5 How does the number line show that the fractions are equivalent?
1 and 2 are located at the same point on the number line.
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6 Write sentences to show the fractions of the two oranges name the same amount.
use words: Two eighths is equal to one fourth
use fractions:
2
equals
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1
4
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.
.
7 Explain how you can show two fractions are equivalent.
you can see if they show the same amount in the model, or you can see if
they are at the same point on a number line.
try it
use what you just learned to solve these problems.
8 Draw a model to show 2 5 4 .
3 ··
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Possible drawing:
9 Use the number line to find a fraction
equivalent to 1 .
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3
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Try It Solutions
8Solution: Students may draw vertical thirds using a
solid line and divide each third into 2 equal parts
using dotted vertical lines. Students may choose to
divide the model into vertical thirds and then draw
one horizontal line through the center of the model
to create 6 equal parts.
9Solution: Students use a hash mark to cut each third
into 2 equal parts on the number line. They label
SMP Tip: Students work with a partner to reason
about quantities and their relationships and then
create a meaningful representation of two
equivalent fractions (SMP 2).
Concept Extension
•Have students explore other ways that the model
could show thirds and then sixths. For example,
the students could draw three horizontal lines to
show thirds and then draw a vertical line down
the center of the rectangle.
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each ​ 1 ​ .
6
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Error Alert: Students who drew in too many
lines between each ​ 1 ​may be confused with drawing
3
··
two lines in each third, instead of creating 2 equal
intervals in each third. Explain that they can draw
in 3 equal intervals for each third, but that would
create ninths, not sixths.
L17: Find Equivalent Fractions
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Part 3: Modeled Instruction
Lesson 17
At a Glance
Part 3: Modeled instruction
Students explore different ways to write a whole
number as a fraction.
Lesson 17
read the problem below. then explore different ways to write a whole number
as a fraction.
Step By Step
Kacey used 2 same-sized boards to build a birdhouse. He cut each board into
fourths. How can you write 2 as a fraction to show Kacey’s boards divided into
fourths?
•Explain to students that they will now look at
fractional names for whole numbers.
Picture it
you can use models to help you write a whole number as a fraction.
•Read the problem at the top of the page as a class.
Ask students to picture the problem and the 2 samesized boards that are each cut into fourths.
The fraction strips below show 2 wholes, each divided into fourths.
1
4
1
4
1
4
1
4
1
4
1
4
1
4
1
4
Each part is 1 of a whole. There are eight 1s in all.
•Draw the models from Picture It on the board.
4
··
4
··
Model it
Remind students that, previously, they used similar
you can use a number line to help you write a whole number as a fraction.
models to count by ​ 1 ​ s. Count the fourths together
4
··
This number line shows whole numbers on the top and fourths on the bottom.
0
aloud and point to each section as students count
0
​1  ​  1 ​ , ​ 2 ​ , ​ 3 ​ , ​ 4 ​ , ​ 5 ​ , ​ 6 ​ , ​ 7 ​ , ​ 8 ​   2​ .
4 ··
4 ··
4 ··
4 ··
4 ··
4 ··
4 ··
4
··
•Emphasize that ​ 4 ​is also 1 whole board and ​ 8 ​is also
4
4
··
··
1
1
4
2
4
3
4
4
4
2
5
4
6
4
7
4
8
4
Notice that each whole number has an equivalent fraction.
2 whole boards.
•Draw the number line as it is shown in Model It on
the board. Ask what the numbers show above the
line [whole numbers] and what they represent below
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the line [fourths]. Point out that the whole number 1
and the fraction ​ 4 ​name the same point on the line.
4
··
Make the same point for 2 and ​ 8 ​on the number line.
4
··
•Ask students to use what they know to solve the
problem. 3​  ​ 8 ​   4​
4
··
L17: Find Equivalent Fractions
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Mathematical Discourse
•What is another way you could model the boards
problem to show the fraction for two boards?
Use fraction tiles or strips or draw pictures of
the boards.
175
Part 3: Guided Instruction
Lesson 17
At a Glance
Students revisit the problem on page 154. They write an
equation to show the fraction for two wholes if each
whole is divided into fourths. They use models to find
fractions that are equivalent to other whole numbers.
Part 3: guided instruction
Lesson 17
connect it
now you will solve the problem from the previous page using equations.
10 Look at the models on the previous page. How many equal parts are in 1 whole?
Explain how you know. Possible answer: there are 4 parts in 1 whole. each
fraction strip has 4 parts.
Step By Step
•Work through Connect It problems 10–12 together.
11 How many parts are in 2 wholes? Explain how you know. Possible answer:
there are 8 parts in 2 wholes. i counted the number of parts in both
•Write the fraction ​ 8 ​on the board. Ask students to
fraction strips.
4
··
talk to their partners and discuss what each number
12 Write sentences to show the fraction that is equivalent to 2.
use words: Two wholes equals
means and why the numerator is two times larger
than the denominator.
•Ask students questions such as: What do you know
when the numerator is the same number as the
denominator? [The fraction equals 1.] or What do you
know when the numerator is larger than the
denominator? [The fraction names a number larger
than one whole.]
•Organize students into small groups and instruct
them to work together on the Try It problems.
Circulate among the groups and choose groups to
share their ideas with the class.
•Draw the models from problems 14 and 15 on the
board. Bring up any questions that you heard
students have when they were working and invite
students to share ideas. Select groups to use the
models on the board to explain how they found the
equivalent fractions.
•During the discussion ask questions such as: How do
you know that the model shows ​ 6 ​ ? What does the
2
··
fraction ​ 18  ​tell you? [There are 18 parts and each
6
··
use a fraction: 2 equals
8
4
··
eight fourths
.
.
13 Explain how to find a fraction equivalent to a whole number.
Possible answer: you look at how many parts are in one whole, and that is
the denominator. then you count how many parts there are in all of the
wholes together, and that is the numerator.
try it
use what you just learned to solve these problems.
14 Use the model to find a fraction
equivalent to 3.
15 Draw a model to show 3 5 18 .
6
··
6
35
2
··
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Try It Solutions
14 Solution: ​  6 ​ ; Students may label each half and count
six ​ 1 ​ s.
2
··
2
··
15 Solution: Students may count each ​ 1 ​to get ​ 18  ​or use
6
··
6
··
multiplication (wholes that each have 6 equal parts.
That’s 18 sixths in all).
whole has 6 parts or there are 3 wholes divided into
Error Alert: Students who wrote ​ 6 ​may have
sixths. That’s 18 sixths in all.] Help students make
counted 6 equal parts in the 3 wholes. They do not
the connection to multiplication that the model
understand the meaning of ​ 6 ​ , which means each of
shows 3 wholes that each have 6 equal parts, which
the 3 wholes is divided into halves and that there are
is 18 parts in all.
6 halves in the 3 wholes.
6
··
2
··
SMP Tip: Discuss with students how important it
is to communicate clearly and precisely when
explaining how to find an equivalent fraction to a
whole. Encourage them to use precise math terms,
such as numerator, denominator, and equivalent when
sharing their thinking (SMP 6).
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Part 4: Modeled Instruction
Lesson 17
At a Glance
Part 4: Modeled instruction
Students explore different ways to write a whole
number as a fraction with a denominator of 1.
Lesson 17
read the problem. then explore different ways to write a whole number as a
fraction with a denominator of 1.
Step By Step
Justin picked 4 green peppers from his garden. He did not cut them into pieces.
How can you write 4 as a fraction to show Justin’s green peppers?
•Explain to students that they will now explore
another way to write a whole number as a fraction.
Picture it
you can use models to help you write a whole number as a fraction with a
denominator of 1.
•Read the problem at the top of the page as a class.
Ask students to picture the 4 whole green peppers in
their mind. Direct their attention to the models
shown in Picture It and explain that each circle
stands for one green pepper.
Each circle stands for 1 green pepper.
They are not divided into pieces, so each whole is made of one part.
Model it
•Ask student if each whole pepper is divided into
you can use a number line to help you write a whole number as a fraction with
a denominator of 1.
smaller parts. [No] Write the fraction ​ 4 ​on the board.
1
··
This number line shows whole numbers on the top and fractions on the bottom.
Explain that 4 is the numerator because there are
4 wholes with one part each. Point out that the
0
1
2
3
4
0
1
1
2
1
3
1
4
1
Notice that each whole number has an equivalent fraction. The spaces between
whole numbers are not divided into parts. Each whole number is made of one part,
so the denominator is 1.
denominator tells us the number of parts in each
whole, which is one part. Tell students to say “four
wholes” as they look at the fraction on the board.
•Draw the number line in Model It on the board.
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Explain that the numbers on the top of the line are
Copying is not permitted.
whole numbers. Ask students if the whole numbers
are shown with any smaller parts. [No] Write the
fractions ( ​ 1 ​ , ​ 2 ​ , etc.) under the whole numbers on
1 ··
1
··
the number line.
•Emphasize that the fraction ​ 1 ​is equivalent to
1
··
one whole, that ​ 2 ​is equivalent to 2, ​ 3 ​is equivalent
to 3, etc.
1
··
1
··
•For a quick assessment, ask students to write the
fraction for 3 wholes on their paper or white board
Mathematical Discourse
•How could you show the fraction for 2 green peppers
using the number 1 in the fraction?
​  2 ​
1
··
Concept Extension
•Some students may have difficulty understanding
and hold it up. Look for students to write ​ 3 ​ . Ask
1
··
the meaning of a fraction with 1 in the
them to explain to their partner what the numerator
denominator. Write the fraction ​ 2 ​on the board.
and denominator tells them in the fraction. Have
Remind them that ​ 2 ​is made up of 2 copies of ​ 1 ​ .
students share.
Write “​  2 ​” on the board. Explain that the 1 means
4
··
4
··
4
··
1
··
there is only one part in the whole or that it is just
one whole. Point out that the 2 in the numerator
tells us that there are 2 copies of 1 whole, or ​ 2 ​ .
1
··
L17: Find Equivalent Fractions
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177
Part 4: Guided Instruction
Lesson 17
At a Glance
Students revisit the problem on page 156 and discuss
the meaning of the 1 in fractions that are equivalent to
a whole number. They apply this thinking to find
fractions that are equivalent to other whole numbers.
Part 4: guided instruction
Lesson 17
connect it
now you will solve the problem from the previous page.
16 Explain how you know each whole only has 1 part.
the wholes are not divided into parts, so there is only 1 part.
Step By Step
•Work through the Connect It problems together. Ask
students to discuss problem 21 with a partner, write
a response and share ideas. Expect students to share
that the 1 tells us the whole is one whole part (not
divided into smaller parts) and the numerator tells
the number of wholes that are shaded or are
represented.
•Ask students to work with a partner to discuss and
respond to the Try It problems. Invite a student to
explain their reasoning behind each response.
4
17 How many parts do the 4 green peppers make?
18 What does the numerator of a fraction show? how many parts you have
19 What does the denominator of a fraction show?
how many parts in a whole
20 Write a fraction equivalent to 4.
4
number of parts
1 Remember: ··························
number of equal parts in a whole 2
1
··
21 Explain how to write a whole number as a fraction with a denominator of 1.
Why does this work?
Possible answer: Write the whole number as the numerator and use 1 for
the denominator. the whole number is the number of parts. the
denominator is the number of parts in the whole, which is 1.
try it
use what you just learned to solve these problems.
22 Use the model to find a fraction equivalent to 6.
6
1
··
65
23 Draw a model to show 5 5 5 .
1
··
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Try It Solutions
22 Solution: 6 5 ​  6 ​ .
1
··
Error Alert: Students who wrote ​ 1 ​are confused
6
··
about how to write a fraction for 6 wholes and do
not understand the meaning of ​ 6 ​ .
1
··
23 Solution: See possible student work above.
178
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Part 5: Guided Practice
Lesson 17
Part 5: guided Practice
Lesson 17
study the model below. then solve problem 24–26.
Student Model
The student used solid
lines to show fourths
and dotted lines to show
eighths.
Caleb and Hannah bought two melons that are the same size.
Part 5: guided Practice
Lesson 17
25 Show two fractions that are equivalent to 5.
There will be 5 wholes in
all. Think about how
many parts will be in
each whole.
Show your work.
Possible student work using models:
Caleb cut his into fourths. Hannah cut hers into eighths.
Hannah ate 4 of her melon, and Caleb ate an equal amount of
8
··
his melon. What fraction did Caleb eat?
Look at how you could show your work using a model.
Pair/share
Pair/share
How could you solve
this problem using a
number line?
How many thirds are in 1
whole? How many
eighths are in 1 whole?
Solution:
Solution:
caleb ate 2 of his melon.
2 ···
3
···
26 Kaia ate 3 of a banana. Zoie ate an equivalent amount. Which
6
··
24 Matt says
3 is equivalent to 1. Elisa says 8 is equivalent to 1.
8
··
3
··
Who is correct?
1
3
c
2
3
8 8
5
8
··
2
··
3
3
Landon chose a as the correct answer. How did he get that
answer?
Possible answer: he found a fraction that is equivalent
Pair/share
Does Landon’s answer
make sense?
to 2 , not 3 .
0 1 2 3 4 5 6 7 8
8 8 8
2
3
··
D 1
1
8
Look at where 3 is on a
6
··
number line. What other
fraction could be shown
at the same point?
3
··
0
Pair/share
answer.
b
1
0
fraction shows how much Zoie ate? Circle the letter of the correct
a 1
Show your work.
0
158
Possible answer: 10 ; 15
4
··
Possible student work using number lines:
What is another fraction
that is equivalent to 1?
How did you decide
what denominators to
use in your fractions?
6
··
8 8
6
··
Solution: they are both correct.
L17: Find Equivalent Fractions
159
L17: Find Equivalent Fractions
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At a Glance
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24 Solution: Both students are correct; The student may
Students use models and number lines to solve
problems involving fractions that are equivalent to
whole numbers.
draw 2 equal rectangles, dividing one into thirds
Step By Step
equivalent. (DOK 1)
•Ask students to solve the problems individually and
to label the fractional parts in their drawings.
•When students have completed each problem, have
them Pair/Share to discuss their solutions with a
partner or in a group.
8 shaded equal parts to show that both ​ 8 ​ and ​ 3 ​ are
8
··
3
··
25 Solution: Possible answers ​ 10  ​ and ​ 15  ​ or ​ 10  ​ and ​ 20  ​ ;
2
··
3
··
2
··
Students use two same-sized models to show
4
··
5 wholes that are each divided into two parts and
three parts or two parts and four parts. (DOK 1)
2
6 Solution: D; Identify the fraction in which both the
numerator and denominator can be multiplied by
Solutions
Ex Students knew that Caleb ate pieces cut into
fourths, so they looked at the number of fourths
that ​ 4 ​covered ​1 ​  2 ​  2​. In this problem, students must
8
··
and shading the whole. The other rectangle shows
4
··
look for larger-sized pieces that cover the same
amount in the model.
L17: Find Equivalent Fractions
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the same number to equal ​ 3 ​.
6
··
Explain to students why the other two answer
choices are not correct:
B and C are not correct because the student does
not understand that 3 out of 6 parts is the same
amount as 1 out of 2 parts. (DOK 3)
179
Part 6: Common Core Practice
Part 6: common core Practice
Lesson 17
Solve the problems.
1
2?
Which fraction shown is equivalent to }
6
B
C
B
C
D
0
1
1
2
1
3
1
4
1
0
1
1
2
1
3
1
4
1
0
1
4
2
4
3
4
1
0
1
4
2
4
3
4
4
4
b.
D
c.
3
}
1
1
}
3
4
}
1
6
}
3
d.
5
Lesson 17
For numbers 4a–4d, choose Yes or No to indicate whether each number graphed on
the number line represents one whole.
a.
Which fraction is equivalent to 3?
A
3
Part 6: common core Practice
4
A
2
Lesson 17
3 Yes
No
Yes
3 No
3 Yes
No
3 Yes
No
Use the number line below to find a fraction equivalent to 3.
Show your work.
Possible student work using a number line:
0
Look at point P on the number line.
P
0
0
1
a.
0
1
0
1
b.
c.
0
3 is equivalent to
Answer
Look at number lines 3a–3c. Is the point on each number line equal to the number
shown by point P? Choose Yes or No.
3 Yes
No
Yes
3 No
Yes
3 No
6
1
1
3
2
3
3
3
2
4
3
5
3
6
3
3
7
3
8
3
9
3
9
3
··
1.
Draw a model to find a fraction equivalent to }
4
Show your work.
2
1
is equivalent to
}
4
Answer
Possible student work using a model:
8
··
1
self check Go back and see what you can check off on the Self Check on page 131.
160
L17: Find Equivalent Fractions
L17: Find Equivalent Fractions
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At a Glance
4Solution: a. Yes; b. No; c. Yes; d. Yes (DOK 2)
Students answer questions about fractions that are
equivalent to whole numbers that might appear on a
mathematics test.
5Solution: ​  9 ​ ; See possible student work above.
(DOK 1)
3
··
6Solution: ​ 2 ​ ; See possible student work above.
(DOK 1)
Solutions
161
8
··
1Solution: D; Students see that if they divide each
third into two equal parts, they will get ​ 2 ​ . (DOK 1)
6
··
2Solution: A; Students understand that the numerator
shows that there are 3 wholes and the denominator
shows that each whole is one whole part. (DOK 1)
3Solution: a. Yes; b. No; c. No (DOK 2)
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Differentiated Instruction
Lesson 17
Assessment and Remediation
•Ask students to show a fraction that is equivalent to ​ 4 ​using a number line. Provide a number line from 0 to 4
1
··
showing halves (no fraction labels) and whole numbers labeled.
•For students who are struggling, use the chart below to guide remediation.
•After providing remediation, check students’ understanding. Ask students to use a number line to show a
fraction that is equivalent to ​ 3 ​ . Provide a number line showing thirds (no fraction labels) and whole numbers
1
··
labeled.
If the error is . . .
Students may . . .
To remediate . . .
students label the
not understand how to
Ask students to circle the whole numbers on the number line. Have
halves but do not
show that two fractions
them count by ​ 1 ​ s and label each half ( ​ 1 ​ , ​ 2 ​ , ​ 3 ​ , ​ 4 ​ , etc.) Review with
clearly show that ​ 4 ​
on a number line are
students the meaning of ​ 4 ​ . Be sure students understand that the
and ​ 8 ​are on the
equivalent or are
denominator names the size of the parts, which are one whole in size,
same point on the
confused as to what the
and that the numerator tells how many there are of those wholes.
number line and,
fraction ​ 4 ​ means.
1
··
Asks students to write the fraction for the whole numbers on the top
1
··
2
··
therefore, are
2
··
2 ··
2 ··
2 ··
2
··
1
··
part of the number line ( ​ 1 ​ , ​ 2 ​ , ​ 3 ​ , ​ 4 ​ ). Direct students to put a point at ​ 4 ​
1 ··
1 ··
1 ··
1
··
1
··
and explain in words that ​ 4 ​ and ​ 8 ​name that same point.
equivalent.
1
··
2
··
Hands-On Activity
Challenge Activity
Explore equivalent fractions with differently
shaped parts.
Explore equivalent fractions with differently
shaped parts.
Materials: graph paper with 4 3 4 squares outlined
Materials: graph paper with 4 3 4 squares
Have students work in pairs to explore different ways
outlined This activity is an extension of the
to divide the 4 3 4 square into two equal areas to
Hands-On activity. Instead of having students find
show the fraction ​ 2 ​ . Students will discover that each
2
··
half may be a different shape, but they cover the
same area or amount of the square.
When a student finds a new way to divide the 4 3 4
square into two equal areas, have them share with
the class. Ask students how they can prove that each
ways to show ​ 1 ​ , have students explore ways to
2
··
divide their large square into equal parts to show ​ 4 ​ .
4
··
Again, students will discover that each ​ 1 ​may be a
4
··
different shape, but each ​ 1 ​covers the same area
4
··
or amount of the square. They will also see that
each ​ 1 ​is equivalent to ​ 4  ​  .
4
··
16
··
half is really ​ 1 ​of the square. Students should
2
··
recognize that each ​ 1 ​should have 8 of the 16 smaller
squares.
2
··
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181