# Fraction Equivalence Using Multiplication and Division Mathematics Curriculum 4

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```New York State Common Core
4
Mathematics Curriculum
GRADE 4 • MODULE 5
Topic B
Fraction Equivalence Using
Multiplication and Division
4.NF.1, 4.NF.3b
Focus Standard:
4.NF.1
Instructional Days:
5
G3–M5
Fractions as Numbers on the Number Line
G5–M3
Addition and Subtraction of Fractions
G5–M4
Multiplication and Division of Fractions and Decimal Fractions
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual
fraction models, with attention to how the number and size of the parts differ even
though the two fractions themselves are the same size. Use this principle to recognize
and generate equivalent fractions.
In Topic B, students start to generalize their work with fraction equivalence. In Lessons 7 and 8, students
analyze their earlier work with tape diagrams and the area model in Lessons 3 through 5 to begin using
multiplication to create an equivalent fraction comprised of smaller units, e.g.,
Conversely,
students reason, in Lessons 9 and 10, that division can be used to create a fraction comprised of larger units
(or a single unit) that is equivalent to a given fraction, e.g.,
. The numerical work of Lessons 7
through 10 is introduced and supported using area models and tape diagrams.
In Lesson 11, students use tape diagrams to transition their knowledge of
fraction equivalence to the number line. They see that any unit fraction length
can be partitioned into n equal lengths. For example, each third in the interval
from 0 to 1 may be partitioned into 4 equal parts. Doing so multiplies both the
total number of fractional units (the denominator) and the number of selected
units (the numerator) by 4. On the other hand, students see that in some
cases fractional units may be grouped together to form some number of larger
fractional units. For example, when the interval from 0 to 1 is partitioned into
twelfths, one may group 4 twelfths at a time to make thirds. In doing so, both
the total number of fractional units and the number of selected units are
divided by 4.
Topic B:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Fraction Equivalence Using Multiplication and Division
1/7/14
5.B.1
Topic B 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
A Teaching Sequence Towards Mastery of Fraction Equivalence Using Multiplication and Division
Objective 1: Use the area model and multiplication to show the equivalence of two fractions.
(Lessons 7–8)
Objective 2: Use the area model and division to show the equivalence of two fractions.
(Lessons 9–10)
Objective 3: Explain fraction equivalence using a tape diagram and the number line, and relate that to
the use of multiplication and division.
(Lesson 11)
Topic B:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Fraction Equivalence using Multiplication and Division
1/7/14
5.B.2
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7
Objective: Use the area model and multiplication to show the equivalence
of two fractions.
Suggested Lesson Structure
Fluency Practice

Application Problem

Concept Development

Student Debrief

(12 minutes)
(4 minutes)
(34 minutes)
(10 minutes)
Total Time
(60 minutes)
Fluency Practice (12 minutes)
 Break Apart Fractions 4.NF.3
(4 minutes)
 Count by Equivalent Fractions 3.NF.3
(4 minutes)
 Draw Equivalent Fractions 4.NF.1
(4 minutes)
Break Apart Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lessons 1─3.
T:
S:
(Project a tape diagram of 3 fifths with the whole labeled.) Name the fraction of 1 whole that’s
.
T:
(Write = __.) Say the fraction.
S:
3 fifths.
T:
On your boards, write as a repeated addition sentence using unit fractions.
S:
(Write
T:
(Write = + + = __
S:
(Write = + + = 3
1
1
1
1
1
1
1
1
1
+ .)
1
1
.) On your boards, complete the number sentence.
)
Continue process for the following possible sequence:
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
1
1
1
1
1
= + + + + =5
1
and
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.3
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
1
1
1
1
1
= + + + + =5
1
.
Count by Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity prepares students for lessons throughout this module.
T:
S:
T:
Count from 0 to 10 by ones.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Count by 1 fourths to 10 fourths. Start at 0 fourths. (Write as students count.)
1
1
0
1
S:
1
1
2
1
1
, , , , , , , , , .
T:
4 fourths is the same as 1 of what unit?
S:
1 one.
T:
(Beneath 4 fourths, write 1.) 2 wholes is the same as how many fourths?
S:
8 fourths.
T:
(Beneath , write 2.) Let’s count to 1 fourths again but this time, say the whole numbers when
you come to a whole number. Start at 0.
S:
0, , , , 1, , , , 2, , .
1
1
Repeat process, counting by thirds to 10 thirds.
Draw Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 6.
T:
S:
(Write .) Say the fraction.
.
=
T:
On your boards, draw an area model to show .
S:
(Draw a model partitioned into 3 equal units. Shade 2 units.)
T:
(Write =
.) Draw a dotted horizontal line to find the equivalent fraction.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.4
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
(Draw a dotted horizontal line, breaking 3 units into 6 smaller units. Write = .)
Continue process for the following possible sequence:
=
, =
, =
1
, and = .
Application Problem (4 minutes)
Model an equivalent fraction for using an area model.
NOTES ON
MULTIPLE MEANS OF
REPRESENTATION:
Students working below grade level
and others may benefit from explicit
instruction as they decompose unit
fractions. When doubling the number
of units, instruct students to draw one
horizontal dotted line. When tripling,
draw two lines, and so on.
Note: This Application Problem reviews G4–M5–Lesson 6 and leads into today’s lesson as students find
equivalent fractions using multiplication.
Concept Development (34 minutes)
Materials: (S) Personal white boards
Problem 1: Determine that multiplying the numerator and denominator by n results in an equivalent
fraction.
MP.7
T:
Draw an area model representing 1 whole partitioned into thirds. Shade and
1
record below the area model. Draw 1 horizontal line across the area model.
S:
T:
S:
T:
S:
(Partition area model.)
What happened to the size of the fractional units?
The units got smaller.  The unit became half the size.
What happened to the number of units in the whole?
There were 3; now there are 6.  We doubled the total number of units.
T:
What happened to the number of selected units when we drew the dotted
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.5
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
line?
There was 1 unit selected, and now there are 2!  It doubled, too!
T:
That’s right. We can record the doubling of units with multiplication:
S:
Hey, I remember from third grade that is the same as .
T:
T:
S:
Yes, they are equivalent fractions.
Why didn’t doubling the number of selected units make the fraction larger?
We didn’t change the amount of the fraction just the size.  Yeah, so the size of the units became
half as big.
Draw an area model representing 1 whole partitioned with a vertical line into 2
halves.
T:
T:
S:
T:
MP.7
S:
T:
S:
T:
S:
T:
S:
1
1
.
1
1
1
Shade and record below the area model. If we want to rewrite using 4 times
as many units, what should we do?
Draw horizontal dotted lines, three of them.  Then, we can write a number
sentence using multiplication.  This time it’s 4 times as many, so we will
multiply the top number and the bottom number by 4.
Show me. (Allow time for students to partition the area model.) What
happened to the size of the fractional unit?
The size of the fractional unit got smaller.
What happened to the number of units in the whole?
There are 4 times as many.  They quadrupled.
What happened to the number of selected units?
There was 1, and now there are 4.  The number of selected units quadrupled!
Has the size of the selected units changed?
There are more smaller unit fractions instead of one bigger unit fraction, but the area is still the
same.
1
T: What can you conclude about and
S: They are equal!
T: Let’s show that using multiplication:
1
1
.
(
)
T:
When we quadrupled the number of units, the number of selected units quadrupled. When we
doubled the number of units, the number of selected units doubled. What do you predict would
happen to the shaded fraction if we tripled the units?
S: The number of units within the shaded fraction would triple, too.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.6
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Problem 2: Given an area model, determine an equivalent fraction for the area selected.
T:
1
(Display area model showing .) Work with your partner to determine an
1
equivalent fraction to .
S:
Let’s draw one horizontal line. That will double the number of units.  We
can draw two horizontal lines. That will triple the number of units and make
them smaller, too.  If we multiply the top and bottom numbers by 4, we
could quadruple the number of units. Each one will be a quarter the size,
too.
Circulate to listen for student understanding and to monitor their work. Reconvene to
examine one or more equivalent fractions.
T:
S:
T:
S:
Some groups drew one horizontal line. (Demonstrate.) Tell your partner what happened to the size
of the units.
The units got smaller.
Tell your partner what happened to the number of units.
There are twice as many units.
T:
Let’s record that:
T:
What is the relationship of the numerators, the top numbers, in the equivalent fractions?
S:
The numerator in is double the numerator in because we doubled the number of selected units.
 Since the size of the selected units are half as big, we doubled the numerator.
What is the relationship of the denominators, the bottom numbers, in the equivalent fractions?
T:
S:
1
1
.
1
1
The denominator in is double the denominator in because we doubled the number of units. 
Since the size of the units are half as big, we doubled the denominator.
Problem 3: Express an equivalent fraction using multiplication and verify by drawing an area model.
1
T:
Discuss with your partner how to find another way to name without drawing an area model first.
S:
Let’s triple the number of units in the whole.  So, we have to multiply the numerator and the
denominator by 3.  Or, we could double the top number and double the bottom number.
Now verify that the fraction you found is equivalent by drawing an area model.
(Work.)
T:
S:
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.7
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Problem Set (10 minutes)
Students should do their personal best to complete the
Problem Set within the allotted 10 minutes. For some
classes, it may be appropriate to modify the assignment by
specifying which problems they work on first. Some
problems do not specify a method for solving. Students
solve these problems using the RDW approach used for
Application Problems.
Student Debrief (10 minutes)
Lesson Objective: Use the area model and multiplication
to show the equivalence of two fractions.
The Student Debrief is intended to invite reflection and
active processing of the total lesson experience.
Invite students to review their solutions for the Problem
Set. They should check work by comparing answers with a
partner before going over answers as a class. Look for
misconceptions or misunderstandings that can be
addressed in the Debrief. Guide students in a conversation
to debrief the Problem Set and process the lesson.
You may choose to use any combination of the questions
below to lead the discussion.

What pattern did you notice for Problems 1(a–d)?

Discuss and compare with your partner your
answers to Problems 2(e) and 2(f).

In Problem 2, the unit fractions have different
denominators. Discuss with your partner how the
size of a unit fraction is related to the
denominator.

The numerator identifies the number of units
selected. Can the numerator be larger than the
denominator?
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.8
Lesson 7 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you
assess the students’ understanding of the concepts that were presented in the lesson today and plan more
effectively for future lessons. You may read the questions aloud to the students.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.9
Lesson 7 Problem Set 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents 1 whole.
1. The shaded unit fractions have been decomposed into smaller units. Express the equivalent fractions in a
number sentence using multiplication. The first one has been done for you.
a.
b.
1
1
c.
d.
2. Decompose the shaded fractions into smaller units using the area models. Express the equivalent
fractions in a number sentence using multiplication.
a.
b.
c.
d.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.10
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7 Problem Set 4•5
e. What happened to the size of the fractional units when you decomposed the fraction?
f.
What happened to the total number of units in the whole when you decomposed the fraction?
3. Draw three different area models to represent 1 third by shading.
Decompose the shaded fraction into (a) sixths, (b) ninths, and (c) twelfths.
Use multiplication to show how each fraction is equivalent to 1 third.
a.
b.
c.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.11
Lesson 7 Exit Ticket 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
1. Draw two different area models to represent 1 fourth by shading.
Decompose the shaded fraction into (a) eighths and (b) twelfths.
Use multiplication to show how each fraction is equivalent to 1 fourth.
a.
b.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.12
Lesson 7 Homework 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents 1 whole.
1. The shaded unit fractions have been decomposed into smaller units. Express the equivalent fractions in a
number sentence using multiplication. The first one has been done for you.
a.
b.
1
1
c.
d.
2. Decompose the shaded fractions into smaller units using the area models. Express the equivalent
fractions in a number sentence using multiplication.
a.
b.
c.
d.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
d.
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.13
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 7 Homework 4•5
3. Draw three different area models to represent 1 fourth by shading.
Decompose the shaded fraction into (a) eighths, (b) twelfths, and (c) sixteenths.
Use multiplication to show how each fraction is equivalent to 1 fourth.
a.
b.
c.
Lesson 7:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.14
Lesson 8 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 8
Objective: Use the area model and multiplication to show the equivalence
of two fractions.
Suggested Lesson Structure
Fluency Practice

Application Problem

Concept Development

Student Debrief

Total Time
(12 minutes)
(4 minutes)
(34 minutes)
(10 minutes)
(60 minutes)
Fluency Practice (12 minutes)
 Multiply Mentally 4.OA.4
(4 minutes)
 Count by Equivalent Fractions 3.NF.3
(4 minutes)
 Draw Equivalent Fractions 4.NF.1
(4 minutes)
Multiply Mentally (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–Module 3 content.
T:
S:
T:
S:
T:
S:
(Write 32 3 =
.) Say the multiplication sentence.
32 3 = 96.
(Write 32 3 = 96. Below it, write 32 20 =
.) Say the multiplication sentence.
32 20 = 640.
(Write 32 20 = 640. Below it, write 32 23 =
.) On your board, solve 32 23.
(Write 32 23 = 736.)
Repeat the process for the following possible sequence: 42 2, 42 20, 42 22 and 21 4, 21 40, 21 44.
Count by Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 4.
T:
Count by twos to 12.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.15
Lesson 8 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
T:
0, 2, 4, 6, 8, 10, 12.
Count by 2 thirds to 12 thirds. Start at 0 thirds. (Write as students count.)
S:
T:
S:
T:
S:
1 whole is the same as how many thirds?
3 thirds.
2 wholes is the same as how many thirds?
6 thirds.
T:
(Beneath , write 2.) 3 wholes is the same as
how many thirds?
S:
9 thirds.
T:
4 wholes is the same as how many thirds?
S:
12 thirds.
T:
(Beneath , write 4.) Count by 2 thirds again. This time, say the whole numbers when you arrive at
them.
S:
.
Repeat the process, counting by 2 sixths to 18 sixths.
Draw Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 7.
T:
S:
(Write .) Say the fraction.
.
T:
On your board, draw a model to show .
S:
(Draw a model partitioned into 2 equal units. Shade 1 unit.)
T:
(Write
.) Draw a dotted horizontal line to find the equivalent fraction. Then, complete
the number sentence.
S:
(Draw a dotted horizontal line, breaking 2 units into 4 smaller units. Write =
Continue the process for the following possible sequence:
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
=
, =
, =
, =
= .)
, and = .
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.16
Lesson 8 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Application Problem (4 minutes)
Saisha gives some of her chocolate bar, pictured below, to her younger brother Lucas. He says “Thanks for
of the bar ” Saisha responds “No I gave you of the bar ” Explain why both Lucas and Saisha are correct.
CHOCOLATE
Note: This Application Problem reviews content from G4–M5–Lesson 7. This bridges to today’s lesson where
students will determine equivalent fractions of non-unit fractions. Revisit this problem in the Debrief by
asking students to write the remaining portion as two equivalent fractions.
Concept Development (34 minutes)
Materials: (S) Personal white boards
Problem 1: Determine that multiplying both the numerator and denominator by n results in an equivalent
fraction.
T:
S:
T:
S:
T:
S:
MP.7
T:
S:
T:
Draw an area model to represent 2 thirds. Draw three
horizontal lines across the area model.
(Draw and partition the model.)
What happened to the size of the fractional units?
The units are 4 times as small because we divided each unit
into 4 smaller units.
What happened to the number of units?
There were 3, and now there are 12.  There are 4 times as
many units.
What happened to the number of selected units?
There were 2 units selected, now there are 8 units selected.
Discuss with your partner how to represent the equivalence of
and using multiplication.
S:
We can multiply the numerator and denominator by 4. We
can write
.
T:
How do you know the fraction is still representing the same
amount?
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.17
Lesson 8 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
T:
S:
T:
MP.7 T:
S:
I know it’s the same size because I didn’t change how much is selected.  There are more smaller
units instead of fewer bigger units, but the area of the selected fraction is still the same.  The
fractions are equivalent.
What was different about this problem than the ones we did yesterday?
The fraction that we are starting with doesn’t have as the numerator
We know any fraction can be decomposed into the sum of unit fractions. Yesterday, we saw that 1
third equals 4 twelfths. Today, we see that 2 thirds equals 4 twelfths plus 4 twelfths, or 8 twelfths.
Draw an area model to represent . Find an equivalent fraction with the
denominator of 12. Explain to a partner how this is done.
We partition each of the 6 units into 2 parts so that we have 12 units in the
whole.  We double the number of units to make twelfths.  There are
twice as many units in the whole and twice as many units selected, but the
parts are only half as big. 
T: What have we discovered about finding equivalent fractions?
S: The area of the fraction stays the same, but the number and size of the units change.  The number
of units increases. The size of the unit fraction decreases.
Problem 2: Determine that two fractions are equivalent using an area model and a
number sentence.
T:
S:
T:
S:
(Project
.) If the whole is the same, is this statement true
or false?
Three times 2 is 6, and 4 times 2 is 8. Yes, it’s true  If we
multiply both the numerator and denominator by 2, we get .
 Doubling the selected units and the number of units in the
whole has the same area as .
Represent the equivalence in a number sentence using
multiplication, and draw an area model to show the
equivalence.
(Do so, as pictured to the right.)
T:
(Project
.) If the wholes are the same, is this statement
true or false? How do you know? Discuss with your partner.
S: Three times 2 is 6, and times is
It’s false We didn’t
multiply by the same number.  This is false. I drew a model
for and then decomposed it into twelfths. There are 9 units
shaded, not 6.  The numerator is being multiplied by 2, and
the denominator is being multiplied by 3. They are not
equivalent fractions.
T:
With your partner, revise the right side of the equation to
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.18
Lesson 8 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
make a true number sentence.
S:
We could change to .  Or, we could change
denominator would be multiplied by 2.
to , because then both the numerator and
Problem 3: Write a number sentence using multiplication to show the equivalence of two fractions. Draw
the corresponding area model.
T:
Find an equivalent fraction without drawing an area
model first. Write on your personal board. How have
we found equivalent fractions?
S:
We’ve doubled tripled, or quadrupled the numerator
and the denominator.  We multiply the numerator
and denominator by the same number.
T:
Find an equivalent fraction to using multiplication.
S:
When I multiply the numerator and denominator by 2,
I get .
T:
S:
Use an area model to confirm your number sentence.
(Do so, correcting any errors as necessary. Answers
may vary.)
Problem Set (10 minutes)
Students should do their personal best to complete the Problem
Set within the allotted 10 minutes. For some classes, it may be
appropriate to modify the assignment by specifying which
problems they work on first. Some problems do not specify a
method for solving. Students solve these problems using the
RDW approach used for Application Problems.
NOTES ON
MULTIPLE MEANS OF
ENGAGEMENT
Invite students working above grade
level and others to test their
discoveries about multiplying fractions
by partitioning shapes other than
rectangles, such as circles and
hexagons. This work may best be
supported by means of concrete or
virtual manipulatives.
Student Debrief (10 minutes)
Lesson Objective: Use the area model and multiplication to show the equivalence of two fractions.
The Student Debrief is intended to invite reflection and active processing of the total lesson experience.
Invite students to review their solutions for the Problem Set. They should check work by comparing answers
with a partner before going over answers as a class. Look for misconceptions or misunderstandings that can
be addressed in the Debrief. Guide students in a conversation to debrief the Problem Set and process the
lesson.
You may choose to use any combination of the questions below to lead the discussion.

For Problems 3(a–d), how did you determine the number of horizontal lines to draw in each area
model?
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.19
Lesson 8 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM




For Problem 5(c), did you and your partner have
the same answer? Explain why you might have
Explain when someone might need to use
equivalent fractions in daily life.
How are we able to show equivalence without
having to draw an area model?
Think back to the Application Problem. What
fraction of the bar did Saisha receive?
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete
the Exit Ticket. A review of their work will help you assess
the students’ understanding of the concepts that were
presented in the lesson today and plan more effectively
for future lessons. You may read the questions aloud to
the students.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.20
Lesson 8 Problem Set 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents 1 whole.
1. The shaded fractions have been decomposed into smaller units. Express the equivalent fractions in a
number sentence using multiplication. The first one has been done for you.
a.
b.
c.
d.
2. Decompose the shaded fractions into smaller units, as given below. Express the equivalent fractions in a
number sentence using multiplication.
a. Decompose into tenths.
b. Decompose into fifteenths.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.21
Lesson 8 Problem Set 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
3. Draw area models to prove that the following number sentences are true.
a.
b.
c.
d.
4. Use multiplication to rename each fraction below.
a.
b.
c.
d.
5. Determine which of the following are true number sentences. Correct those that are false by changing
the right-hand side of the number sentence.
a.
b.
c.
d.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 8 Exit Ticket 4•5
Date
1. Use multiplication to create an equivalent fraction for the fraction below.
2. Determine if the following is a true number sentence. If needed, correct the statement by changing the
right-hand side of the number sentence.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.23
Lesson 8 Homework 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents 1 whole.
1. The shaded fractions have been decomposed into smaller units. Express the equivalent fractions in a
number sentence using multiplication. The first one has been done for you.
a.
b.
c.
d.
2. Decompose both shaded fractions into twelfths. Express the equivalent fractions in a number sentence
using multiplication.
a.
b.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.24
Lesson 8 Homework 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
3. Draw area models to prove that the following number sentences are true.
a.
b.
c.
d.
4. Use multiplication to create an equivalent fraction for each fraction below.
a.
b.
c.
d.
5. Determine which of the following are true number sentences. Correct those that are false by changing
the right-hand side of the number sentence.
a.
b.
c.
d.
Lesson 8:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and multiplication to show the equivalence of two
fractions.
1/7/14
5.B.25
Lesson 9 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9
Objective: Use the area model and division to show the equivalence of two
fractions.
Suggested Lesson Structure




Fluency Practice
Application Problem
Concept Development
Student Debrief
(12 minutes)
(6 minutes)
(32 minutes)
(10 minutes)
Total Time
(60 minutes)
Fluency Practice (12 minutes)
 Add and Subtract 4.NBT.4
(4 minutes)
 Find Equivalent Fractions 4.NF.1
(4 minutes)
 Draw Equivalent Fractions 4.NF.1
(4 minutes)
Add and Subtract (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews the year-long Grade 4 fluency standard for adding and subtracting using
the standard algorithm.
T:
S:
T:
S:
(Write 732 thousands 367 ones.) On your boards, write this number in standard form.
(Write 732,367.)
(Write 423 thousands 142 ones.) Add this number to 732,367 using the standard algorithm.
(Write 732,367 + 423,142 = 1,155,509 using the standard algorithm.)
Continue the process for 671,526 + 264,756.
T:
S:
T:
S:
(Write 916 thousands 450 ones.) On your boards, write this number in standard form.
(Write 916,450.)
(Write 615 thousands 137 ones.) Subtract this number from 916,450 using the standard algorithm.
(Write 916,450 – 615,137 = 301,313 using the standard algorithm.)
Continue the process for 762,162 – 335,616 and 500,000 – 358,219.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.26
Lesson 9 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Find Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 7.
1
2
×
×
1
2
(Write =
S:
1
.
2
T:
On your boards, complete the number sentence to make an equivalent fraction.
S:
(Write =
1
2
1×2
2×2
=
2
T:
. Point to .) Say the unit fraction.
2
4
= .)
1
2
4
8
Continue the process for the following possible suggestions: = ,
Draw Equivalent Fractions (4 minutes)
1
3
2
6
= ,
1
3
3
9
= ,
1
4
=
4
16
1
5
3
15
, and = .
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 8.
T:
2
3
(Write .) Say the fraction.
S:
2
.
3
T:
On your boards, draw a model to show .
T:
(Write =
= .) Draw a dotted horizontal line to find the equivalent fraction. Then, complete
6
3
×
the number sentence.
S:
2
3
2
×
2 2×2
3 3×2
4
= .
5 15
(Draw a dotted horizontal line, breaking 3 units into 6 smaller units. Write =
Continue the process for the following possible sequence:
2
3
=
9
3
4
3
5
, = 12 , = 10 , and
4
6
= .)
Application Problem (6 minutes)
What fraction of a foot is 1 inch? What
fraction of a foot is 3 inches? (Hint: 12 inches
= 1 foot.) Draw a tape diagram to model your
work.
Note: Students are asked to think about
fractions within a context, such as
measurement, that will be useful in upcoming
word problems. This measurement work will be
developed more in G4–Module 7.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.27
Lesson 9 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
NOTES ON
MULTIPLE MEANS FOR
EXPRESSION:
Concept Development (32 minutes)
Materials: (S) Personal white boards
6
Problem 1: Simplify by composing larger fractional units
12
using division.
6
T:
(Project area model showing .) What fraction does
12
the area model represent?
S:
6
.
12
As the conceptual foundation for
simplification is being set, the word
simplify is initially avoided with students
as they compose higher value units. The
process is rather referred to as
composition, the opposite of
decomposition, which relates directly to
their drawing, their work over the last
two lessons, and their work with whole
numbers. When working numerically,
the process is referred to at times as
renaming, again in an effort to relate to
whole number work.
T:
Discuss with a partner. Do you see any fractions
6
equivalent to 12?
S:
Half of the area model is shaded. The model shows .
T:
S:
T:
Which is the larger unit? Twelfths or halves?
Halves!
Circle the smaller units to make the larger units. Say the equivalent
fractions.
S:
6
1
= .
12 2
T:
S:
T:
S:
T:
T:
S:
1
2
6÷6
(Write
= and point to the denominator.) Twelve units were in
12 ÷ 6
the whole, and we made groups of 6 units. Say a division sentence to
record that.
12 ÷ 6 = 2.
(Record the 2 in the denominator, and point to the
numerator.) Six units were selected, and we made a
NOTES ON
group of 6 units. Say a division sentence to record
MULTIPLE MEANS FOR
that.
REPRESENTATION:
6 ÷ 6 = 1.
English language learners may confuse
(Record the 1 in the numerator.) We write
the terms decompose and compose.
6
6÷6 1
=
= , dividing both the numerator and the

Demonstrate that the prefix de12 12 ÷ 6 2
can be placed before some words
denominator by 6 to find an equivalent fraction.
to add an opposite meaning.
What happened to the size of the units and the total

Use gestures to clarify the
number of units?
meanings: Decompose is to take
The size of the units got bigger. There are fewer units
apart, and compose is to put
in the whole.  The units are 6 times as large, but the
together.
number of units is 6 times less.  The units got bigger.

Refresh students’ memory of
The number of units got smaller.
decomposition and composition in
the context of the operations with
whole numbers.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.28
Lesson 9 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
2
8
Problem 2: Simplify both and
3
1
as by composing larger fractional units.
12
4
2
T:
Draw an area model to represent . Group two
8
units to make larger units.
T:
Write =
= . How many groups of 2 are
8 8÷2
1! (Write 1 in the numerator.)
How many groups of 2 are in the whole?
4!
S:
T:
S:
2
2÷2
2
2÷2
1
T:
(Write =
= .) Talk to your partner about how we showed that 2 eighths is the same as 1 fourth.
8 8÷2 4
Discuss both the model and our use of division. (Allow students time to discuss.)
T:
Draw an area model to represent . Compose an equivalent fraction.
S:
We can make groups of 2.  No, that won’t work. Some of the groups could have shaded and
unshaded units.  Groups of 3 will work. That’s how many shaded units there are.
How many groups of 3 are shaded?
1.
How many groups of 3 in all?
4.
The new fraction is?
T:
S:
T:
S:
T:
3
12
S:
1
.
4
T:
Write the number sentence to show you
composed groups of 3.
S:
3
3÷3 1
=
=
12 12 ÷ 3 4
T:
Compare the area models for
S:
They both equal .
1
4
2
6
Problem 3: Simplify both and
T:
S:
T:
S:
3
12
2
8
and .
4
1
as by composing larger fractional units.
12
3
When we composed fractions in the last two
problems, what did you notice?
We divided to find equivalent fractions.  We
made equal groups to make large units.  We
composed a unit fraction from a non-unit
fraction.
2
4
Draw area models to show and . Rename
6
12
both fractions as the same unit fraction.
I can make groups of 2 in both area models. I could make groups of 3, but I won’t be making equal
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.29
Lesson 9 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
T:
S:
T:
S:
groups of shaded and unshaded units.  Four is a factor of both 4 and 12, so I can make groups of
4.  First, I made groups of 2 when I was working with 4 twelfths, but then I noticed I could make
groups of 2 again.  Hey, dividing by 2 twice is the same as dividing by 4.
Circle the groups, and express each composition in a number sentence using division.
2 2÷2 1 4
4÷4 1
=
= . =
= .
6 6 ÷ 2 3 12 12 ÷ 4 3
4
2
How are and related?
12
6
4
2
When I model and , I see that they both have the
12
6
1
1 4 2
same area as 3.  3 = 12 = 6.  The equivalent fraction
4
2
1
for 12 and 6 with the largest units is 3.  We composed
4
2
and into the same unit fraction.
12
6
Problem Set (10 minutes)
Students should do their personal best to complete the
Problem Set within the allotted 10 minutes. For some
classes, it may be appropriate to modify the assignment by
specifying which problems they work on first. Some
problems do not specify a method for solving. Students
solve these problems using the RDW approach used for
Application Problems.
NOTES ON
MULTIPLE MEANS OF
ENGAGEMENT:
Challenge students working above
grade level and others to couple the
expressions of fraction composition
with the related multiplication
expression of decomposition.
For example,
1
3
=
1×4
3×4
=
4
12
4
12
=
4÷4
12 ÷ 4
=
1
3
and
.
Student Debrief (10 minutes)
Lesson Objective: Use the area model and division to
show the equivalence of two fractions.
The Student Debrief is intended to invite reflection and
active processing of the total lesson experience.
Invite students to review their solutions for the Problem
Set. They should check work by comparing answers with a
partner before going over answers as a class. Look for
misconceptions or misunderstandings that can be
addressed in the Debrief. Guide students in a conversation
to debrief the Problem Set and process the lesson.
You may choose to use any combination of the questions
below to lead the discussion.
•
Look at Problems 1(a–d). Write some examples of
fractions where the denominator is a multiple of
the numerator. (Pause.) What do we know about
these fractions?
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.30
Lesson 9 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM






In Problems 3 and 4, does it matter how your
area models are shaded? Will you still result in a
Explain how two fractions can be composed into
the same larger unit fraction.
How can what you know about factors help
rename a fraction in larger units?
3
1
When we rename as , why is it helpful to
12
4
think about the factors of 3 and 12?
Contrast the following: renaming fractions when
you multiply versus when you divide and
decomposing versus composing fractions. For
each, discuss what happens to the size of the
units and the number of units.
3
Use what you learned today to determine if can
8
be renamed as a larger unit. Why or why not?
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete
the Exit Ticket. A review of their work will help you assess
the students’ understanding of the concepts that were
presented in the lesson today and plan more effectively for
future lessons. You may read the questions aloud to the
students.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.31
Lesson 9 Problem Set 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents 1 whole.
1. Compose the shaded fractions into larger fractional units. Express the equivalent fractions in a number
sentence using division. The first one has been done for you.
a.
b.
2 2÷2 1
=
=
4 4÷2 2
c.
d.
2. Compose the shaded fractions into larger fractional units. Express the equivalent fractions in a number
sentence using division.
a.
b.
c.
d.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.32
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9 Problem Set 4•5
e. What happened to the size of the fractional units when you composed the fraction?
f.
What happened to the total number of units in the whole when you composed the fraction?
3.
a. In the first area model, show 2 sixths. In the second area model, show 3 ninths. Show how both
fractions can be renamed as the same unit fraction.
b. Express the equivalent fractions in a number sentence using division.
4.
a. In the first area model below, show 2 eighths. In the second area model, show 3 twelfths. Show how
both fractions can be composed, or renamed, as the same unit fraction.
b. Express the equivalent fractions in a number sentence using division.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.33
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 9 Exit Ticket 4•5
Date
1.
a. In the first area model, show 2 sixths. In the second area model, show 4 twelfths. Show how both
fractions can be composed, or renamed, as the same unit fraction.
b. Express the equivalent fractions in a number sentence using division.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.34
Lesson 9 Homework 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents one whole.
1. Compose the shaded fractions into larger fractional units. Express the equivalent fractions in a number
sentence using division. The first one has been done for you.
a.
b.
2 2÷2 1
=
=
4 4÷2 2
c.
d.
2. Compose the shaded fractions into larger fractional units. Express the equivalent fractions in a number
sentence using division.
a.
b.
c.
d.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.35
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 9 Homework 4•5
e. What happened to the size of the fractional units when you renamed the fraction?
f.
What happened to the total number of units in the whole when you renamed the fraction?
3.
a. In the first area model, show 4 eighths. In the second area model, show 6 twelfths. Show how both
fractions can be composed, or renamed, as the same unit fraction.
b. Express the equivalent fractions in a number sentence using division.
4.
a. In the first area model below, show 4 eighths. In the second area model, show 8 sixteenths. Show
how both fractions can be composed, or renamed, as the same unit fraction.
b. Express the equivalent fractions in a number sentence using division.
Lesson 9:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.36
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 10
Objective: Use the area model and division to show the equivalence of two
fractions.
Suggested Lesson Structure




Fluency Practice
Application Problem
Concept Development
Student Debrief
(12 minutes)
(8 minutes)
(30 minutes)
(10 minutes)
Total Time
(60 minutes)
Fluency Practice (12 minutes)
 Count by Equivalent Fractions 3.NF.3
(4 minutes)
 Find Equivalent Fractions 4.NF.1
(4 minutes)
 Draw Equivalent Fractions 4.NF.1
(4 minutes)
NOTES ON
MULTIPLE MEANS OF
REPRESENTATION:
Count by Equivalent Fractions (4 minutes)
As you lead the Count by Equivalent
Fractions fluency activity, enunciate
the ending digraph /th/ of fraction
names to help English language
learners distinguish fractions from
whole numbers (e.g., fourths, not
fours).
Materials: (S) Personal white boards
Note: This fluency activity reinforces G4–Module 5 fraction concepts.
T:
S:
T:
Count by threes to 24.
0, 3, 6, 9, 12, 15, 18, 21, 24.
Count by 3 fourths to 24 fourths. (Write as students
count.)
S:
0 3 6 9
, , ,
4 4 4 4
T:
S:
T:
S:
T:
S:
,
12 15 18 21 24
, , , , .
4 4 4 4 4
0
1 whole is the same as how
4
many fourths?
4 fourths.
0
2 wholes is the same as how
many fourths?
8 fourths.
3 wholes is the same as how many fourths?
12 fourths.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Couple numbers on the board with
prepared visuals, if beneficial.
3
4
6
4
9
4
12
4
15
4
18
4
21
4
24
4
3
4
6
4
9
4
3
15
4
18
4
21
4
6
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.37
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
12
4
T:
(Beneath , write 3.) 4 wholes is the same as how many fourths?
S:
16 fourths.
T:
5 wholes is the same as how many fourths?
S:
20 fourths.
T:
6 wholes is the same as how many fourths?
S:
24 fourths.
T:
(Beneath , write 6.) Count by 3 fourths again. This time, say the whole numbers when you arrive
4
24
3
4
6
4
9
4
S: 0, , , , 3,
15 18 21
, , , 6.
4
4
4
Repeat the process, counting by 3 fifths to 30 fifths.
Find Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 8.
3
4
×
×
3
4
T:
(Write =
S:
3
.
4
T:
On your boards, complete the number sentence.
S:
(Write =
3
4
3×2
4×2
= . Point to .) Say the fraction.
8
6
8
= .)
Continue the process for the following possible suggestions:
3
4
=
9
12
,
2
3
4
6
= ,
2
5
=
4
10
,
4
5
=
8
10
, and
3
5
4
12
1
3
9
15
= .
Draw Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 9.
T:
(Project model with 2 out of 4 equal units shaded.) Draw the model, and write
the fraction that is shaded.
S:
(Draw model with 2 out of 4 equal units shaded. Write .)
T:
(Write =
= .) Compose the shaded units into 1 larger unit by circling.
4
÷
Then, complete the number sentence.
S:
2
4
2
÷
2
4
2÷2 1
= .)
4÷2 2
3 1 4 1
= , =
9 3 8 2
(Circle the shaded units into 1 larger unit. Write =
Continue the process for the following possible sequence:
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
,
2
8
1
4
= ,
5
10
1
2
= , and
= .
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.38
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Application Problem (8 minutes)
9
Nuri spent of his money on a book, and the rest of his money
12
on a pencil.
a. Express how much of his money he spent on the pencil
in fourths.
b. Nuri started with \$1. How much did he spend on the
pencil?
Note: This Application Problem connects G4–M5–Topic A and
G4–M5–Lesson 9 by finding the other fractional part of the
whole and expressing equivalent fractions. Using what students
know about money, ask why it is preferable to answer in fourths
rather than twelfths. Students connect fourths to quarters of a
dollar. Revisit this problem in the Debrief to express how much
money was spent on the book in fourths.
NOTES ON
MULTIPLE MEANS OF
REPRESENTATION:
Concept Development (30 minutes)
Materials: (S) Personal white boards
Problem 1: Simplify a fraction by drawing to find a common
factor, and relate to division.
10
12
There are multiple ways of showing a
given fraction using an area model. Area
models may, therefore, look different
from student to student. Allow students
to share how they have drawn different
area models and be accepting of those
that are mathematically correct.
T:
Draw an area model that represents .
T:
If we want to compose an equivalent fraction, what do
we do?
We make equal groups.  We divide the numerator
and the denominator by the same number.  We
should divide by 10. We divided by the same number
that was in the numerator yesterday.
Can I divide both the numerator and the denominator by 10?
No.
Discuss with your partner how to determine the largest possible unit.
We can try to make groups of 2, then 3, then 4, until we have the
largest number of units in a group with no remainder.  We can only
make equal groups of 2. The other numbers don’t divide evenly into
both the numerator and the denominator.
Show me. (Allow time for students to compose area model.) What
happened to the number of shaded units?
There were 10 units shaded, and now there are 5 groups of 2 units
S:
T:
S:
T:
S:
T:
S:
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.39
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
1
12
1
6
T:
Consider the unit fractions
and . What do you notice about their denominators?
S:
T:
S:
T:
S:
T:
6 is a factor of 12.
What about the numerators 10 and 5?
5 is a factor of 10!
List the factors of 10 and 12.
The factors of 12 are 1, 2, 3, 4, 6, and 12. The factors of 10 are 1, 2, 5, and 10.
1 and 2 are factors of both. We know then we can make equal groups of 2. Equal groups of 1 bring
us back to the original fraction.
Problem 2: Draw an area model of a number sentence that shows the simplification of a fraction.
6
6÷2 3
=
= .
10 10 ÷ 2 5
T:
Project
T:
S:
Draw an area model to show how we can we know this number sentence is true.
The numerator and denominator are both being divided by 2. I will circle groups of 2.  I know 2 is
a factor of 6 and 10, so I could make groups of 2.  There are 3 shaded groups of 2 and 5 total
3
groups of 2.  That’s !
5
Problem 3: Simplify a fraction by drawing to find different common factors, and relate to division.
T:
S:
8
8
With your partner, draw an area model to represent . Rename using larger fractional units. You
12
12
may talk as you work. (Circulate and listen.)
I can circle groups of 2 units.  2 is a factor of 8 and 12.  There are 6 groups of 2 units in the
4
whole.  Four groups are shaded. That’s .
6
T:
S:
What happens when I use 4 as a common factor instead of 2? Turn and talk.
Four is a factor of both 8 and 12. It works.  We can make
larger units with groups of 4.  Thirds are larger than sixths.
8 2
= .  We have fewer units, but they’re bigger.
12
T:
3
Express the equivalent fractions as a division number
sentence.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.40
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
8
8÷4
2
=
= .
12 12 ÷ 4 3

8
8÷2
4
=
= .
12 12 ÷ 2 6
2
3
4
6
T:
What can you conclude about and ?
S:
They are both equivalent to .
T:
What is true about dividing the numerator and the denominator in
S:
Two and 4 are both factors of 8 and 12.  The larger the factor used, the larger the fractional units
will be.
Interesting. Discuss what your classmate said. “The larger the factor, the larger the new fractional
units.”
When we divided by 2, we got sixths, and when we divided by 4, we got thirds. Thirds are larger.
Four is larger than 2. A bigger factor gave a bigger unit.  When the factor is larger, it means we
can make fewer units but bigger ones.
T:
S:
8
12
8
12
by 2 or by 4?
Problem 4: Simplify a fraction using the largest possible common factor.
T:
S:
T:
S:
T:
S:
T:
8
Discuss with your partner how to rename with the largest units possible without using an area
12
model.
Figure out the greatest number of units that can be placed in equal groups.  Divide the numerator
and denominator by the same number, just like we’ve been doing.  Find a factor of both 8 and 12,
and use it to divide the numerator and the denominator.
Express the equivalence using a division number sentence.
8
8÷2 4
=
= .
12 12 ÷ 2 6
8
8÷4
2
Four and 6 are still both even, so that wasn’t the largest factor.  =
= . The only
12 12 ÷ 4 3
common factor 2 and 3 have is 1, so 4 must be the largest factor they have in common.
How can we know we expressed an equivalent fraction with the largest units?
When we make equal groups, we need to see if we can make larger ones.  When we find the
factors of the numerator and denominator, we have to pick the largest factor. Four is larger than 2,
4
so dividing the numerator and denominator by 4 gets us the largest units.  When I found , I
6
realized 2 and 4 are both even, so I divided the numerator and denominator again by 2. Two and 3
only have a common factor of 1, so I knew I had made the largest unit possible.  Dividing by 2
twice is the same as dividing by 4. Just get it over with faster and divide by 4!
8
4
It’s not wrong to say that = . It is true. It’s just that at times it really is simpler to work with larger
12 6
units because it means the denominator is a smaller number.
Problem Set (10 minutes)
Students should do their personal best to complete the Problem Set within the allotted 10 minutes. For some
classes, it may be appropriate to modify the assignment by specifying which problems they work on first.
Some problems do not specify a method for solving. Students solve these problems using the RDW approach
used for Application Problems.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.41
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Student Debrief (10 minutes)
Lesson Objective: Use the area model and division to
show the equivalence of two fractions.
The Student Debrief is intended to invite reflection and
active processing of the total lesson experience.
Invite students to review their solutions for the Problem
Set. They should check work by comparing answers with a
partner before going over answers as a class. Look for
misconceptions or misunderstandings that can be
addressed in the Debrief. Guide students in a
conversation to debrief the Problem Set and process the
lesson.
You may choose to use any combination of the questions
below to lead the discussion.



In Problem 2(b), did you compose the same units
correct? Why?
In Problems 4(a–d), how is it helpful to know the
common factors for the numerators and
denominators?
In Problem 4, you were asked to use the largest
4 1
common factor to rename the fraction: = . By
4





8
2
doing so, you renamed using larger units. How
8
is renaming fractions useful?
Do fractions always need to be renamed to the
largest unit? Explain.
Why is it important to choose a common factor
to make larger units?
How can you tell that a fraction is composed of
the largest possible fractional units?
When you are drawing an area model and circling
equal groups, do all of the groups have to appear
the same in shape? How do you know that they
still show the same amount?
Explain how knowing the factors of the
numerator and the factors of the denominator
can be helpful in identifying equivalent fractions
of a larger unit size.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.42
Lesson 10 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete the Exit Ticket. A review of their work will help you
assess the students’ understanding of the concepts that were presented in the lesson today and plan more
effectively for future lessons. You may read the questions aloud to the students.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.43
Lesson 10 Problem Set 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents 1 whole.
1. Compose the shaded fraction into larger fractional units. Express the equivalent fractions in a number
sentence using division. The first one has been done for you.
a.
b.
4 4÷2 2
=
=
6 6÷2 3
c.
d.
2. Compose the shaded fractions into larger fractional units. Express the equivalent fractions in a number
sentence using division.
a.
b.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.44
Lesson 10 Problem Set 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
3. Draw an area model to represent each number sentence below.
a.
4
10
=
4÷2
10 ÷ 2
=
2
5
b.
6
9
=
6÷3 2
=
9÷3 3
4. Use division to rename each fraction given below. Draw a model if that helps you. See if you can use the
largest common factor.
a.
4
8
b.
8
12
c.
9
12
d.
10
15
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.45
Lesson 10 Exit Ticket 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
1. Draw an area model to show why the fractions are equivalent.
Show the equivalence in a number sentence using division.
4
10
=
2
5
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.46
Lesson 10 Homework 4 5
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Date
Each rectangle represents one whole.
1. Compose the shaded fraction into larger fractional units. Express the equivalent fractions in a number
sentence using division. The first one has been done for you.
a.
b.
4 4÷2 2
=
=
6 6÷2 3
c.
d.
2. Compose the shaded fractions into larger fractional units. Express the equivalent fractions in a number
sentence using division.
a.
b.
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.47
Lesson 10 Homework 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
3. Draw an area model to represent each number sentence below.
a.
12
16
=
12 ÷ 4 3
=
16 ÷ 4 4
b.
6
18
=
6÷3
18 ÷ 3
=
2
6
4. Use division to rename each fraction given below. Draw a model if that helps you. See if you can use the
largest common factor.
a.
6
9
b.
4
12
c.
10
15
d.
12
16
Lesson 10:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Use the area model and division to show the equivalence of two
fractions.
1/15/14
5.B.48
Lesson 11 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11
Objective: Explain fraction equivalence using a tape diagram and the
number line, and relate that to the use of multiplication and division.
Suggested Lesson Structure
Fluency Practice

Application Problem

Concept Development

Student Debrief

(12 minutes)
(5 minutes)
(33 minutes)
(10 minutes)
Total Time
(60 minutes)
Fluency Practice (12 minutes)
 Find the Quotient and Remainder 4.NBT.6
(4 minutes)
 Find Equivalent Fractions 4.NF.1
(4 minutes)
 Draw Equivalent Fractions 4.NF.1
(4 minutes)
Find the Quotient and Remainder (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M3–Lesson 28’s Concept Development.
T:
S:
(Write 6,765 ÷ 2.) On your boards, find the quotient and remainder.
(Solve for and write the quotient and remainder.)
Continue for the following possible sequence: 6,811 ÷ 5, 1,265 ÷ 4, and 1,736 ÷ 4.
Find Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 9.
T:
S:
2
2
( =
2
= . Point to .) Say the fraction.
.
T:
On your boards, fill in the missing number to find the equivalent fraction.
S:
(Write
2
=
2 2
2
= )
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.49
Lesson 11 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Continue process for the following possible sequence:
2
,
, , and 2.
Draw Equivalent Fractions (4 minutes)
Materials: (S) Personal white boards
Note: This fluency activity reviews G4–M5–Lesson 10.
T:
(Project model with 4 out of 10 equal units shaded.) Draw the model and write the fraction that is
S:
(Draw model with 4 out of 10 equal units shaded. Write .)
T:
(Write =
= ) Compose the shaded units into larger units by
circling. Then, complete the number sentence.
S:
(Circle the shaded units into 1 larger unit. Write
Continue the process for the following possible sequence:
=
, ,
2
2
= )
2
8
, and
.
Application Problem (5 minutes)
Kelly was baking bread but could only find her 8 cup measuring cup. She needs cup sugar, cup whole
wheat flour, and cup all purpose flour. How many cups will she need for each ingredient?
2
8
Solution 2
Solution 1
Note: This Application Problem places equivalent fractions into a context that
may be familiar to students. Multiple solution strategies are possible. The first
solution models the equivalency learned in G4–M5–Lessons 7 and 8. The
second solution uses number bonds to find unit fractions, reviewing G4–M5–
Topic A content.
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.50
Lesson 11 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
Concept Development (33 minutes)
Materials: (S) Personal white boards, ruler
Problem 1: Use a tape diagram and a number line to
find equivalent fractions for halves, fourths, and
eighths.
T:
S:
T:
S:
T:
S:
T:
S:
T:
S:
T:
S:
T:
S:
Draw a tape diagram to show 1 whole
partitioned into halves.
(Draw tape diagram.)
Shade Now, decompose halves to make
2
fourths. How many fourths are shaded?
2 fourths.
On your personal board, write what we did as a
multiplication number sentence.
2
2
2 2
2
Decompose fourths to make eighths. How many
4 eighths.
Write a multiplication number sentence to show that 2
fourths and 4 eighths are equal.
2
2 2
2
8

2
2
8
Label 2 on the number line. Decompose the
number line into fourths. What is equivalent
2
to on the number line?
2
2
. We showed that on the tape diagram.
Decompose the number line into eighths.
(Label the eighths.)
T:
What is 8 equal to on the number line?
T:
In order to preserve the pace of the
lesson, provide a tape diagram and
number line template for some
learners. Students may also choose to
transform the tape diagram into a
number line by erasing the top line,
labeling points, and extending the end
points.
Now, use a ruler to draw a number line
slightly longer than the tape diagram. Label
points 0 and 1 so they align with the ends of
the tape diagram.
(Draw number line).
T:
S:
S:
NOTES ON
MULTIPLE MEANS OF
REPRESENTATION:
2
8
. 
2
8
.  That also means 2
2
8
.
Explain what happened on the number line as you decomposed the half.
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.51
Lesson 11 4•5
NYS COMMON CORE MATHEMATICS CURRICULUM
S:
When we decomposed the half into fourths, it was like sharing a licorice strip with four people
instead of two.  We got 4 smaller parts instead of 2 bigger parts.  There are 4 smaller segments
in the whole instead of 2 larger segments.  We doubled the number of parts but made smaller
parts, just like with the area model.  It made 2 lengths that were the same length as 1 half.
Problem 2: Use a number line, multiplication, and division to decompose and compose fractions.
T:
T:
Partition a number line into thirds. Decompose 1
third into 4 equal parts.
Write a number sentence using multiplication to
show what fraction is equivalent to 1 third on this
number line.
S:
T:
S:
T:
S:
T:
S:
2
.
Explain to your partner why that is true.
It’s just like the area model. We made more smaller units but the lengths stayed the same, instead
of the area staying the same.  If we multiply a numerator and a denominator by the same
number, we find an equivalent fraction.  1 third was decomposed into fourths, so we multiplied
the number of units in the whole and the number of selected units by 4.
Write the equivalence as a number sentence using division.
2
2
.
Explain to your partner why that is true.
We can join four smaller segments to make one longer one that is the same as 1 third.  We can
group the twelfths together to make thirds.  Four copies of equals .  Just like the area
2
model, we are composing units to make a larger unit.
Problem 3: Decompose a non-unit fraction using a number line and division.
2
T:
Draw a number line. Partition it into fifths, label it, and locate .
S:
(Draw.)
T:
Decompose into 6 equal parts. First, discuss your
strategy with your partner.
I will make each fifth into 6 parts.  No, we have to
decompose 2 units, not 1 unit. Each unit will be
decomposed into 3 equal parts.  Two units are
becoming 6 units. We are multiplying the numerator
and the denominator by 3.
Write a number sentence to express the equivalent
fractions.
S:
T:
S:
2
2
2
.
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.52
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11 4•5
Problem Set (10 minutes)
Students should do their personal best to complete the Problem
Set within the allotted 10 minutes. For some classes, it may be
appropriate to modify the assignment by specifying which
problems they work on first. Some problems do not specify a
method for solving. Students solve these problems using the
RDW approach used for Application Problems.
Student Debrief (10 minutes)
Lesson Objective: Explain fraction equivalence using a tape
diagram and the number line, and relate that to the use of
multiplication and division.
NOTES ON
MULTIPLE MEANS OF
ENGAGEMENT:
Challenge students working above
grade level and others to discuss or
journal about the three models used
for finding equivalent fractions. Ask,
“How do the tape diagram and number
line relate to one another? When
might you choose to use a number line
rather than an area model? Why?”
The Student Debrief is intended to invite reflection and
active processing of the total lesson experience.
Invite students to review their solutions for the Problem
Set. They should check work by comparing answers with a
partner before going over answers as a class. Look for
misconceptions or misunderstandings that can be
addressed in the Debrief. Guide students in a conversation
to debrief the Problem Set and process the lesson.
You may choose to use any combination of the questions
below to lead the discussion.






In Problem 1, compare the distance from 0 to
each point on the number line you circled. What
do you notice?
In Problem 1, does the unshaded portion of the
tape diagram represent the same length from the
point to 1 on every number line? How do you
know?
Compare your number sentences in Problem 2.
Could they be rewritten using division?
In Problem 5, what new units were created when 2 fifths was decomposed into 4 equal parts?
How is modeling with a number line similar to modeling with an area model? How is it different?
In Grade 3, you found equivalent fractions by locating them on a number line. Do you now require a
number line to find equivalent fractions? What other ways can you determine equivalent fractions?
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.53
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11 4•5
Exit Ticket (3 minutes)
After the Student Debrief, instruct students to complete
the Exit Ticket. A review of their work will help you assess
the students’ understanding of the concepts that were
presented in the lesson today and plan more effectively
for future lessons. You may read the questions aloud to
the students.
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.54
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 11 Problem Set 4•5
Date
1. Label each number line with the fractions shown on the tape diagram. Circle the fraction that labels the
point on the number line that also names the selected part of the tape diagram.
1
a.
1
b.
c.
1
2. Write number sentences using multiplication to show
a. the fraction represented in 1(a) is equivalent to the fraction represented in 1(b).
b. the fraction represented in 1(a) is equivalent to the fraction represented in 1(c).
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.55
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11 Problem Set 4•5
3. Use each shaded tape diagram below as a ruler to draw a number line. Mark each number line with the
unit fractions shown on the tape diagram, and circle the fraction that labels the point on the number line
that also names the selected part of the tape diagram.
1
a.
1
b.
1
c.
4. Write number sentences using division to show
a. the fraction represented in 3(a) is equivalent to the fraction represented in 3(b).
b. the fraction represented in 3(a) is equivalent to the fraction represented in 3(c).
5. a. Partition a number line from 0 to 1 into fifths. Decompose into 4 equal lengths
b. Write a number sentence using multiplication to show what fraction represented on the number line
is equivalent to .
c. Write a number sentence using division to show what fraction represented on the number line is
equivalent to .
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.56
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 11 Exit Ticket 4•5
Date
1.
a. Partition a number line from 0 to 1 into sixths. Decompose into 4 equal lengths.
b. Write a number sentence using multiplication to show what fraction represented on the number line
is equivalent to .
c. Write a number sentence using division to show what fraction represented on the number line is
equivalent to .
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.57
NYS COMMON CORE MATHEMATICS CURRICULUM
Name
Lesson 11 Homework 4•5
Date
1. Label each number line with the fractions shown on the tape diagram. Circle the fraction that labels the
point on the number line that also names the selected part of the tape diagram.
a.
b.
c.
2. Write number sentences using multiplication to show
a. the fraction represented in 1(a) is equivalent to the fraction represented in 1(b).
b. the fraction represented in 1(a) is equivalent to the fraction represented in 1(c).
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14
5.B.58
NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 11 Homework 4•5
3. Use each shaded tape diagram below as a ruler to draw a number line. Mark each number line with the
unit fractions shown on the tape diagram, and circle the fraction that labels the point on the number line
that also names the selected part of the tape diagram.
1
a.
1
b.
1
c.
4. Write number sentences using division to show
a. the fraction represented in 3(a) is equivalent to the fraction represented in 3(b).
b. the fraction represented in 3(a) is equivalent to the fraction represented in 3(b).
5. a. Partition a number line from 0 to 1 into fourths. Decompose into 6 equal lengths.
b. Write a number sentence using multiplication to show what fraction represented on the number line
is equivalent to .
c. Write a number sentence using division to show what fraction represented on the number line is
equivalent to .
Lesson 11:
Date:
© 2014 Common Core, Inc. Some rights reserved. commoncore.org
Explain fraction equivalence using a tape diagram and the number line,
and relate that to the use of multiplication and division.
1/15/14