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Common Core Math Grade 3 Comparing Fractions

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Published by Stella Seremetaki TRN 1788154, 2019-10-30 10:54:48

Common Core Math Grade 3 Comparing Fractions

Common Core Math Grade 3 Comparing Fractions

2014

Common Core

Mathematics  Teacher Resource Book 3

Table of Contents

Ready® Common Core Program Overview A6

Supporting the Implementation of the Common Core A7

Answering the Demands of the Common Core with Ready A8

The Standards for Mathematical Practice A9

Depth of Knowledge Level 3 Items in Ready Common Core A10

Cognitive Rigor Matrix A11

Using Ready Common Core A12

Teaching with Ready Common Core Instruction A14

Content Emphasis in the Common Core Standards A16

Connecting with the Ready® Teacher Toolbox A18

Using i-Ready® Diagnostic with Ready Common Core A20

Features of Ready Common Core Instruction A22

Supporting Research A38

Correlation Charts A42
Common Core State Standards Coverage by Ready Instruction A46
Interim Assessment Correlations

Lesson Plans (with Answers)

CCSS Emphasis

Unit 1: Operations and Algebraic Thinking, Part 1 1

Lesson 1 Understand the Meaning of Multiplication 3 M
11 M
CCSS Focus - 3.OA.A.1  Embedded SMPs - 2–4 23 M
33 M
Lesson 2 Use Order and Grouping to Multiply 41 M
49 M
CCSS Focus - 3.OA.B.5  Embedded SMPs - 2–4, 7 59 M
67
Lesson 3 Split Numbers to Multiply

CCSS Focus - 3.OA.B.5  Embedded SMPs - 1, 2, 4, 7, 8

Lesson 4 Understand the Meaning of Division

CCSS Focus - 3.OA.A.2  Embedded SMPs - 1–4, 6

Lesson 5 Understand How Multiplication and Division Are Connected

CCSS Focus - 3.OA.B.6  Embedded SMPs - 2, 6

Lesson 6 Multiplication and Division Facts

CCSS Focus - 3.OA.A.4, 3.OA.C.7  Embedded SMPs - 1, 2, 6–8

Lesson 7 Understand Patterns

CCSS Focus - 3.OA.D.9  Embedded SMPs - 2, 4, 6 ,7

Unit 1 Interim Assessment

M = Lessons that have a major emphasis in the Common Core Standards
S/A = Lessons that have supporting/additional emphasis in the Common Core Standards

Unit 2: Number and Operations in Base Ten CCSS Emphasis
70

Lesson 8 Use Place Value to Round Numbers 72 S/A
82 S/A
CCSS Focus - 3.NBT.A.1  Embedded SMPs - 1, 5–8 94 S/A
103
Lesson 9 Use Place Value to Add and Subtract

CCSS Focus - 3.NBT.A.2  Embedded SMPs - 1, 2, 7, 8

Lesson 10 Use Place Value to Multiply

CCSS Focus - 3.NBT.A.3  Embedded SMPs - 2, 7, 8

Unit 2 Interim Assessment

Unit 3: Operations and Algebraic Thinking, Part 2 106

Lesson 11 Solve One-Step Word Problems Using Multiplication and Division 109 M
121 M
CCSS Focus - 3.OA.A.3  Embedded SMPs - 1–4, 7 131 M
141
Lesson 12 Model Two-Step Word Problems Using the Four Operations

CCSS Focus - 3.OA.D.8  Embedded SMPs - 1, 2, 4, 5

Lesson 13 Solve Two-Step Word Problems Using the Four Operations

CCSS Focus - 3.OA.D.8  Embedded SMPs - 1–5

Unit 3 Interim Assessment

Unit 4: Number and Operations—Fractions 144

Lesson 14 Understand What a Fraction Is 146 M
154 M
CCSS Focus - 3NF.A.1  Embedded SMPs - 2–4, 6 162 M
170 M
Lesson 15 Understand Fractions on a Number Line 182 M
190 M
CCSS Focus - 3.NF.A.2a, 2b  Embedded SMPs - 2, 3, 7 199

Lesson 16 Understand Equivalent Fractions

CCSS Focus - 3.NF.A.3a  Embedded SMPs - 2–4

Lesson 17 Find Equivalent Fractions

CCSS Focus - 3.NF.A.3b, 3c  Embedded SMPs - 1, 2, 4, 6

Lesson 18 Understand Comparing Fractions

CCSS Focus - 3.NF.A.3d  Embedded SMPs - 2, 3, 7

Lesson 19 Use Symbols to Compare Fractions

CCSS Focus - 3.NF.A.3d  Embedded SMPs - 2–4, 7

Unit 4 Interim Assessment

M = Lessons that have a major emphasis in the Common Core Standards
S/A = Lessons that have supporting/additional emphasis in the Common Core Standards

Unit 5: Measurement and Data CCSS Emphasis
202
Lesson 20 Tell and Write Time 207 M
215 M
CCSS Focus - 3.MD.A.1  Embedded SMPs - 1, 3, 4, 6 225 M
235 M
Lesson 21 Solve Problems About Time 245 S/A
255 S/A
CCSS Focus - 3.MD.A.1  Embedded SMPs - 1, 3–6 265 S/A
275 M
Lesson 22 Liquid Volume 283 M
293 M
CCSS Focus - 3.MD.A.2  Embedded SMPs - 2, 4, 6 303 S/A
315
Lesson 23 Mass 318
320 S/A
CCSS Focus - 3.MD.A.2  Embedded SMPs - 1, 2, 4–6 328 S/A
338 S/A
Lesson 24 Solve Problems Using Scaled Graphs 346

CCSS Focus - 3.MD.B.3  Embedded SMPs - 1, 2, 4, 6, 7

Lesson 25 Draw Scaled Graphs

CCSS Focus - 3.MD.B.3  Embedded SMPs - 1, 2, 4, 6, 7

Lesson 26 Measure Length and Plot Data on Line Plots

CCSS Focus - 3.MD.B.4  Embedded SMPs - 1, 4–6

Lesson 27 Understand Area

CCSS Focus - 3.MD.C.5a, 5b, 6  Embedded SMPs - 2, 3, 5

Lesson 28 Multiply to Find Area

CCSS Focus - 3.MD.C.7a, 7b  Embedded SMPs - 4, 6–8

Lesson 29 Add Areas

CCSS Focus - 3.MD.C.7c, 7d  Embedded SMPs - 3, 5, 7

Lesson 30 Connect Area and Perimeter

CCSS Focus - 3.MD.D.8  Embedded SMPs - 1–7

Unit 5 Interim Assessment

Unit 6: Geometry

Lesson 31 Understand Properties of Shapes

CCSS Focus - 3.G.A.1  Embedded SMPs - 5, 6, 7

Lesson 32 Classify Quadrilaterals

CCSS Focus - 3.G.A.1  Embedded SMPs - 3, 5, 7

Lesson 33 Divide Shapes Into Parts With Equal Areas

CCSS Focus - 3.G.A.2  Embedded SMPs - 2, 4, 5

Unit 6 Interim Assessment

M = Lessons that have a major emphasis in the Common Core Standards
S/A = Lessons that have supporting/additional emphasis in the Common Core Standards

Focus on Math Concepts

Lesson 18

Understand Comparing Fractions

LESSON OBJECTIVES THE LEARNING PROGRESSION

• Understand that in order to compare two fractions, In this lesson, students apply their understanding of
students must reason about the size of unit fractions fractions to compare two fractions. They use fraction
shown by the denominators and number of parts models and number lines to help them reason about
shown in the numerator in each fraction. the size of unit fractions.

• Analyze the numerators and denominators in Students learn to look carefully at the numerators and
fractions to be compared to determine if the denominators of both fractions they are comparing to
fractions have the same numerators or denominators. determine if the numerators or the denominators are
the same. If the denominators are the same, the
• Explain why one fraction is smaller or larger when students know the fractions are built from the same
comparing two fractions using models or number size unit fraction and the fraction with the most parts
lines. in the numerator is larger. If the numerators are same,
then students reason about the size of the unit fractions
PREREQUISITE SKILLS used to make each of the two fractions.

• Understand the meaning of fractions. Students apply this understanding to solve problems
that involve comparing fractions and explain why one
• Identify fractions represented by models and number fraction is larger or smaller.
lines.
Teacher Toolbox Teacher-Toolbox.com
• Understand that the size of a fractional part is
relative to the size of the whole. Ready Lessons Prerequisite 3.NF.A.3d
Tools for Instruction Skills
• Identify equivalent fractions and explain why they ✓
are equivalent. ✓✓
✓✓
VOCABULARY Interactive Tutorials

There is no new vocabulary. Review the following key
terms.

unit fraction: a fraction with a numerator of 1; other
fractions are built from unit fractions

numerator: the number above the fraction bar in a
fraction that tells the number of equal parts out of the
whole

denominator: the number below the fraction bar in a
fraction that tells the number of equal parts in the whole

CCSS Focus

3.NF.A.3 E xplain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
d. C ompare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that

comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the
symbols ., 5, or ,, and justify the conclusions, e.g., by using a visual fraction model.

STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 3, 7 (see page A9 for full text)

182 L18: Understand Comparing Fractions

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Part 1: Introduction Lesson 18

AT A GLANCE Lesson 18 Part 1: Introduction Focus on Math Concepts

Students explore what it means to compare fractions. Understand Comparing Fractions CCSS
They compare fractions with the same denominators
using models and number lines. 3.NF.A.3d

STEP BY STEP How do we compare fractions?

• Remind students that they’ve compared whole When you compare fractions, you figure out which is smaller, which is larger, or if they
numbers such as 345 and 509 to see which is smaller, are the same size.
larger, or if the two numbers are the same.
You can use models or number lines to help you compare two fractions.

2
14
4

• Introduce the Question at the top of the page. Ask 0 1 2 3 1
4 4 4

students to identify the two fractions shown by the Both of these show that 1 is less than 2 .
·4· ·4·

The size of the wholes must be

two square fraction models, tell which is larger, and the same to compare fractions.

explain If not, it might look like 1 is
·4·
] [than ​·14 · ​. 
[explain
how they know.  ·​24 · ​ covers more of the square greater than 2 .
Asks students to use the number line to ·4·
why ​·42 · ​ is larger than ​·14 · ​.   ·​24 · ​ has more fourths
Think Sometimes when you compare fractions, the denominators are the same.

The models below show two wholes that are the same size Circle the model of
divided into sixths. the fraction that
is less.
]and is farther away from 0. Direct students’ attention
Think about how many unit fractions it takes to make each
fraction you are comparing.

to the two different size wholes that show ​·41 · ​ and ​·42 · ​.  11 11111
66 66666

Emphasize that when you compare two fractions, the It takes two 1 s to make ·26·. It takes five ·16·s to make ·65·.
·6·
size of the wholes must be the same.
2 is made of fewer unit fractions than ·65·. So, 2 is less than ·65·.
·6· ·6·

• Have a student to read the Think statement aloud. 162 L18: Understand Comparing Fractions ©Curriculum Associates, LLC Copying is not permitted.

Ask students what two fractions are shown by the

models. 3​  ·​62 · ​ and ​·65 · ​ 4 ​ Write the two fractions on the

board. Remind students what a unit fraction is and

explain that each of the fractions was made up of the Hands-On Activity
unit fraction ​·61 · ​.  Explain that you can use unit
fractions and fraction models to show if a fraction Use fraction bars to compare two fractions.

is larger or smaller. The fraction model shows that

it takes more copies of ​·61 · ​ to make ​·65 · ​ than to make Materials: fraction bars or circles, blank paper
only ​·62 · ​. 
Have students work in pairs and use a blank paper
Mathematical Discourse
as their work space. Ask each pair to put the one
• What is the same about the two fractions?
The denominators are the same size. whole fraction bar at the top of their paper. Instruct
one student in the pair to build the fractions ​·25 · ​ and ​·45 · ​
• What is different about the two fractions? and have the other point to the larger fraction and
They are built with a different number of unit
fractions or they have a different numerator. explain why it is larger. Ask a student to share why.

Do the same to compare the fractions ​·83 · ​ and ​·58 · ​,  ​·1 30·  ​ 
and ​·1 60·  ​ . 

L18: Understand Comparing Fractions 183
©Curriculum Associates, LLC  Copying is not permitted.

Part 1: Introduction Lesson 18

AT A GLANCE Part 1: Introduction Lesson 18

Students explore ways to compare fractions that have Think You can compare fractions with like numerators and different denominators.
the same numerators, but different denominators.
Think about two different unit fractions from the same It’s like cutting up a piece
STEP BY STEP of paper. The more
whole, such as 1 and 1 . pieces you cut the paper
• Read the Think statement together. On the board, ·3· ·8· into, the smaller each
write the two fractions ​·13 · ​ and ​·81 · ​ shown by the models piece is.
on the page. Circle the numerators in each fraction. 11
Explain that they will next explore how to compare 38
two fractions in which the numerators are the same
and the denominators are different. Compare the denominators of 1 and 1 . 3 is less than 8, so
·3· ·8·
• Write the two fractions ​·36 · ​ and ​·34 · ​ on the board and
circle the numerators. Ask students to picture a the whole is divided into fewer parts. Since there are fewer
rectangle divided into 6 equal parts and one divided
into 4 equal parts. Direct students attention to the parts, each part is bigger. So, the unit fraction 1 is greater
methaoecdhme ​·16 lo·s ​ disoefsl ​m·63 t·h ​ aaanlnlderw ​·,43 ·i s​ looln33tohoffettphhaeog ​s·14 e·e ​.s  ​·.61M · ​s  awkeiltlhceovpeorinletstshoaft ·3·
than ·18·.
• Ask students to work with a partner to answer the
Reflect question in their own words and invite Here’s another example:
students to share.
111 111
Hands-On Activity 666 444

• Ask students to use fraction tiles or bars to lay The unit fractions used to The unit fractions used to
these fractions on a blank sheet of
paper: ​·21 · ​,  ​·14 · ​,  ​·51 · ​,  ​·16 · ​,  ​·81 · ​,  and ​·1 10·  ​ .  Instruct them to make 3 are smaller. make 3 are bigger.
label each unit fraction. To help students ·6· ·4·
understand that the smaller the fractional piece,
the larger the denominator is, ask questions such 3 smaller parts are less than 3 bigger parts. So, 3 is less than 3 .
as: What do you notice about the size of the unit ·6· ·4·
fractions as the denominator gets larger? How many
tenths does it take to make one whole? One-half? Reflect
Why does it take more tenths to make one whole?
Then ask them to show ​·83 · ​ and ​·34 · ​.  Ask: Which 1 Explain how you can use unit fractions to help you compare fractions.
fraction is larger and Why? Possible answer: If the denominators of the unit fractions are the same,

you can look at how many of them you have to compare the fractions. If

the denominators of the unit fractions are different, you can figure out

which parts are bigger to compare the fractions.

L18: Understand Comparing Fractions 163
©Curriculum Associates, LLC Copying is not permitted.

Mathematical Discourse

• What is the same about the two fractions ​·63 · ​ and · ​43·  ​ ?
The numerators

• What is different?

The denominators

• When you see a fraction that is written with
numbers, why is it helpful to picture the unit
fractions in your mind?

If the denominators are large, I know that
each unit fraction is small. If the denominator
is a small number, like 2 or 3, I know the size
of the parts will be larger than if the
denominators were 5 or 6.

184 L18: Understand Comparing Fractions

©Curriculum Associates, LLC  Copying is not permitted.

Part 2: Guided Instruction Lesson 18

AT A GLANCE Part 2: Guided Instruction Lesson 18

Students use models to compare two fractions that Explore It
either have the same denominators or the same
numerators. Use the models to help you compare fractions with the same denominator.
2 Write the fraction shaded below each model.
STEP BY STEP
Circle the fraction that is greater.
• Tell students that they will have time to work
individually on the Explore It problems on this page 21
and then share their responses in groups. You may ·3· ·3·
choose to work through the first problem together as
a class. 3 Write the fraction shaded below the first

• As students work individually, circulate among them. model. Shade the second model to show
This is an opportunity to assess student
understanding and address student misconceptions. a greater fraction. Write the greater fraction. 4
Use the Mathematical Discourse questions to engage ·8·
student thinking. Possible answers: 5 , 6 , 7 , 8 ; shading will vary.
·8· ·8· ·8· ·8·
• When students have completed the page, ask them to
work with a partner and return to each problem. Use the models to help you compare fractions with the same numerator.
Instruct them to cover up the models and to picture
the fractions in their mind. Then have them practice 4 Write the fraction shaded below each model.
explaining why one of the fractions is greater. Circle the fraction that is greater.

• Discuss each problem. Ask for volunteers to explain 1 1
·6· ·2·
how they figured out which fraction was larger or
5 Write the fraction shaded below each model.
smaller. For problem 6, ask for all the possible ways Circle the fraction that is less.

students could shade the model so the fraction is less 22
than ​·56 · ​.  ·6· ·3·
• Take note of students who are still having difficulty
and wait to see if their understanding progresses as 6 Write the fraction shaded below the first 5
they work in their groups during the next part of the model. Shade the second model to show ·6·
lesson. a fraction that is less but has the same
numerator.
SMP Tip: Encourage students to picture fractions
Explain how you know the fraction is less.
in their minds based upon experiences with Possible answer: The same number of parts are shaded in both models.
fraction models. This practice helps students
conceptualize as they work more symbolically with The 1 parts are smaller than the 1 parts, so 5 is less than 5 .
fractions later on (SMP 2). ·8· ·6· ·8· ·6·

164 L18: Understand Comparing Fractions ©Curriculum Associates, LLC Copying is not permitted.

Concept Extension

• Students may not be familiar with the way each ​·18 · ​
is divided in the models for problem 3. Make two
models of the rectangles on paper and cut them
out. Cut apart the eighths in one of the models.
Show students that even though some of the
eighths may look different, they are the same.
Rotate some of the parts and place them on top of
other eighths to prove that they cover the area and
are equivalent.

Mathematical Discourse

• How can you prove that each of the eight parts is
equal in the model in number 3?
You could cut out the pieces and compare them.

L18: Understand Comparing Fractions 185
©Curriculum Associates, LLC  Copying is not permitted.

Part 2: Guided Instruction Lesson 18

AT A GLANCE Part 2: Guided Instruction Lesson 18

Students revisit the problem on page 164 and explain Talk About It
how to use unit fractions to compare the fractions. They
also explore using number lines to compare fractions. Solve the problems below as a group.
7 Look at your answers to problems 2 and 3. Explain how to use unit fractions to
STEP BY STEP
compare fractions with the same denominator. Possible answer: The unit
• Organize students into pairs or groups and ask them fractions are the same. Look at the numerators to see how many unit
to answer the Talk About It questions as a group. fractions are in each fraction. Compare these numbers.

• Walk around to each group, listen to, and join in on 8 Look at your answers to problems 4–6. What is different about the numerators
discussions at different points. Use the Mathematical and denominators in these fractions than the fractions in problems 2 and 3?
Discourse questions to help support or extend
students’ thinking. Possible answer: The fractions in problems 2 and 3 have the same

• Be sure students are able to describe how the denominators and different numerators. The fractions in problems 4–6
problems are different on the page. Fractions have
the same number of whole equal parts if the have the same numerators and different denominators.
denominators are the same.
Explain how to use unit fractions to compare the fractions with the same
• Direct the group’s attention to Try It Another Way. numerator. Possible answer: You have to look at the size of the unit
Draw the number lines on the board for problems 10 fraction, since the number of unit fractions in each is the same. The one
and 11. Have volunteers from the groups come up to
the board and use the number lines to explain how with the smaller unit fraction is less.
they figured out which fraction was greater or less.
9 Isaiah is comparing 3 and 3 . Both fractions have a numerator of 3. How can he tell
SMP Tip: Problem 8 gives students practice which ·8· ·6· 1 are smaller
Possible answer: He knows that ·8· s
looking at the structure of the fractions to be fraction is less?
compared. Students notice whether the numerators 1 1 1 1
or denominators are the same or different, and they than ·6· s. He has the same number of ·8· s and ·6· s, so the fraction made of ·8· s
identify two possible kinds of problems they will
encounter when they compare fractions (SMP 7). is smaller. 3 is less than 3 .
·8· ·6·

Try It Another Way

Work with your group to use the number lines to compare fractions.

10 Look at the fractions on the number 11 Look at the fractions on the number
lines. Circle the fraction that is less. lines. Circle the fraction that is greater.

0 3 10 3 1 0 4 1 0 3 1
8 4 6 6

L18: Understand Comparing Fractions 165
©Curriculum Associates, LLC Copying is not permitted.

Concept Extension Mathematical Discourse

• Emphasize that two fractions must have some • What do you look at first when you are comparing
part that is the same in order to compare them. two fractions?
The first thing students need to do is find which
part is the same. If the denominators are the Look to see if either the denominators or
same, that means the unit fractions that make up numerators are the same.
the whole are the same. They can look at the
numerator to see how many copies of the unit • Which fractions seem easiest for you to compare,
fractions there are. The more copies, the greater those that have the same numerators or those that
the fraction. If the numerators are the same, that have the same denominators? Explain why.
means the number of unit fraction copies is the
same. The denominator tells the size of each unit Answers will vary. Have students explain their
fraction. The more unit fractions that make up a answers.
whole, the smaller each unit fraction is.

186 L18: Understand Comparing Fractions

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Part 3: Guided Practice Lesson 18

AT A GLANCE Part 3: Guided Practice Lesson 18

Students demonstrate their understanding of comparing Connect It
two fractions that have either the same numerators or
the same denominators. Talk through these problem as a class, then write your answers below.

STEP BY STEP 12 Create: Draw an area model or number line to show 5 . Find a fraction with the
·8·
• Discuss each Connect It problem as a class using the 5
discussion points outlined below. same denominator that is less than ·8· .

Possible student work:
Create:
• You may choose to have students work in pairs to 0 35 1
88
encourage sharing ideas. Each partner draws a
different model. Explain how you found your answer. Possible answer: I know that any
• For a quick and easy assessment, have students draw 5 5 3 5
their models on small whiteboards or on paper and fraction to the left of ·8· is less than ·8· . So, ·8· is less than ·8· .
hold them up. Choose several pairs to explain their
models to the class. 13 Explain: Mario painted 2 of the wall in his bedroom. Mei Lyn painted 2 of a wall in
• Use the following to lead a class discussion: ·6· ·4·

What did you keep in mind as you thought of a fraction her bedroom. Both walls are the same size. Explain how you know who painted
to choose that was less than ·​85 · ​  ?
more. Possible answer: Each painted the same number of parts of the wall.
How can you prove that the fraction you chose is less
than ​· 85·  ​ ? Sixths are smaller than fourths, so 2 sixths is less than 2 fourths. That
Is there another way to use a number line(s) to show a
smaller fraction? [Students may use two number lines means Mei Lyn painted more.
or show both fractions on the same number line.]
14 Justify: Jace and Lianna each baked a loaf of bread. Jace cut his in halves and
Explain: Lianna cut hers in thirds.
• This problem expects students to look at the
fractions and picture the size of the unit fractions, 11
reasoning that sixths are smaller than fourths, so 23
Mei Lyn must have painted more.
• Begin the discussion by asking questions such as: Jace says they can use their loaves of bread to show that 1 is less than 1 . Lianna
What do you picture each wall looking like? ·2· ·3·
Does each wall have 2 parts painted?
Which 2 parts cover more area of the wall? How do you says they can’t. Who is correct? Explain why.
know?
Possible answer: Lianna is correct. The wholes have to be the same size to
L18: Understand Comparing Fractions
©Curriculum Associates, LLC  Copying is not permitted. compare fractions.

166 L18: Understand Comparing Fractions ©Curriculum Associates, LLC Copying is not permitted.

Justify:

• This discussion shows whether students understand
that in order to compare two fractions, the wholes
must be the same size.

• Discuss why these two models are different from the
models they’ve used to compare two fractions [They
are a different size.] Explain that the wholes must be
the same size in order to compare ​·21 · ​ and ​·31 · ​.  Be sure
that students justify their answer by giving reasons
for why Lianna is correct.

• Expect students to also notice that the little “nob” on
the model to show thirds is not on the middle piece,
so the 3 sections of bread probably aren’t equal.

SMP Tip: Students practice critiquing the

reasoning of others and constructing arguments to
explain why an answer is or isn’t correct (SMP 3).

• Ask how they could show that ​·13 · ​ is smaller than ​·12 · ​. 

187

Part 4: Common Core Performance Task Lesson 18

AT A GLANCE Part 4: Common Core Performance Task Lesson 18

Students use information in fraction form to create a Put It Together
problem that involves comparing fractions. They create 15 Mrs. Ericson made sandwiches for her 4 children. Each sandwich was the
a model to help them solve the problem.
same size. After lunch, each child had a different fraction of his or her
STEP BY STEP sandwich left. Matt had ·14· , Elisa had ·83· , Carl had ·34· , and Riley had ·87· .

• Direct students to complete the Put It Together task A Use this information to write a problem that compares two fractions with the
on their own. same numerator. Who had more left, Elisa or Carl?

• Go over the directions with students and let them B Use this information to write a problem that compares two fractions with the
know they have some choices to make. Be sure same denominator. Who had more left, Matt or Carl?
students understand that they should circle the
problem that they will model using either a fraction C Choose one of your problems to solve. Circle the question you chose. Draw a
model or number line. model or number line to help you find the answer.

• As students work on their own, walk around to Explain how you could use unit fractions to think about the problem.
assess their progress and understanding, to answer Possible answer: The unit fractions for Matt’s and Carl’s fractions were
their questions, and to give additional support, if
needed. both 1 . Matt had one 1 and Carl had three 1 s. 3 is greater than 1, so
·4· ·4· ·4·
SCORING RUBRICS
Carl had more left.
A Points Expectations
L18: Understand Comparing Fractions 167
2 The student correctly chooses two fractions ©Curriculum Associates, LLC Copying is not permitted.
to compare that either have the same
numerators or the same denominators. The B Points Expectations
question for the problem asks which
student had more or less of his or her 2 The model or number line clearly shows
sandwich left.
both fractions that are compared. The
1 The student correctly chooses two fractions e(oxrp·​ 81l·a ​ sn) aotrioenxpinlacilnusdtehsawt EhloisahaadndthCeamrl ohsatd​·41 ·t ​  she
to compare that either have the same same number of parts and that Carl’s unit
numerators or same denominators. The
question asks which fraction is larger or fractions were larger.
smaller, but does not clearly relate to the
context of the problem (students and the 1 The model or number line clearly shows
amount of sandwich left). both fractions that are compared. The
explanation does not clearly explain how
0 The student does not choose two fractions the student used unit fractions to see who
to compare that have the same numerators had the least or the most left.
or same denominators and is unable to ask
a question that relates to comparing 0 The model did not correctly compare two
fractions of sandwiches left. fractions and the explanation did not
explain the reasoning behind the use of unit
fractions to compare fractions OR The
student did not choose the correct two
fractions to compare and therefore was
unable to create a model and explain how
to use unit fractions.

188 L18: Understand Comparing Fractions

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Differentiated Instruction Lesson 18

Intervention Activity On-Level Activity

For students who have difficulty comparing two Compare fractions by playing a game.
fractions with the same numerator, give them more
practice building models for the two fractions and Materials: 10 blank note cards. Write one of these
thinking about the size of each unit fraction. Have fractions on each card: ​·42 · ​,  ​·25 · ​,  ​·62 · ​,  ​·2 8·  ​,  ​·38 · ​,  ​·63 · ​,  ​·34 · ​,  ​·54 · ​,  ​·48 · ​,  ​·44 · ​
them follow these steps for comparing ​·32 · ​ and ​·28 · ​. 
• Draw two rectangles that are the same size. Show Instruct students to play the comparing game in
pairs. Students shuffle and divide up the cards
thirds in one and eighths in the other. evenly. For one turn, a student places the card on the
• Label each unit fraction in each of the models. desk with the fraction showing. The partner does the
• Ask which unit fraction is larger. [thirds] same with his or her card. If the numerators are the
• Shade in each fraction in the models. same, they call out “same numerators” and then
• Tlahrginekr ,th“Eanac ​·82 h· ​.  ”​·13 · ​ is larger than ​·81 · ​,  so ​·23 · ​ must be compare the size of the unit fractions to determine
the largest fraction. The student with that card gets a
point. If the denominators are the same, they call out
“same denominators” and decide who has the largest
fraction and gets the point. Students record points on
a white board or on a piece of paper. If two cards do
not either have same denominators or same
numerators, students call out “draw” and put the
cards aside. After all cards have been placed down,
shuffle the cards again and repeat the process at least
4 more times to determine who has the most points.

Challenge Activity

Compare fractions equal to or greater than one by playing a game.

Materials: 10 blank note cards. Write one of these fractions on each card: ​·28 · ​,  ​·88 · ​,  ​·84 · ​,  ​·81 · ​,  ​·42 · ​,  ​·44 · ​,  ​·12 · ​,  ​·41 · ​,  ​·36 · ​,  ​·33 · ​

This activity builds upon the On-Level Activity game. Instruct students to play the comparing game in pairs.
Students shuffle and divide up the cards evenly. For one turn, a student places the card on the desk with the
fraction showing. The partner does the same with his or her card. If the numerators are the same, they call out
“same numerators” and then compare the size of the unit fractions to determine the largest fraction. The student
with that card gets a point. If the denominators are the same, they call out “same denominators” and decide
who has the largest fraction and gets the point. If students can prove that two fractions are equivalent, they
both get 2 points. Students record points on a white board or on a piece of paper. If two cards do not have same
denominators, same numerators, or are not equivalent, students call out “draw” and put the cards aside. After
all cards have been placed down, shuffle the cards again and repeat the process at least four more times to
determine who has the most points.

L18: Understand Comparing Fractions 189
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Develop Skills and Strategies

Lesson 19

Use Symbols to Compare Fractions

LESSON OBJECTIVES THE LEARNING PROGRESSION

• Use symbols to record the results of comparing In this lesson students continue to compare two
fractions. fractions by reasoning about size of unit fraction shown
in models and on number lines. The focus of this
• Read comparison statements fluently and accurately. lesson is on recording the results of comparisons by
using symbols to write comparison statements.
• Use models and number lines to explain and justify
fraction comparisons. Students see that comparing fractions also means they
must look to see if two fractions are equal. If two
PREREQUISITE SKILLS fractions are equal, they record it using the equal
symbol. Students review or learn the meaning of the ,
• Understand the meaning of fractions. and . symbols and practice using them to record their
comparisons.
• Identify fractions represented by models and number
lines. Students solve problems that involve comparisons.
Students’ concrete experiences with comparing fractions
• Understand that the size of a fractional part is and recording them using symbols prepares them to
relative to the size of the whole. move on to more abstract work with comparing fractions
and fraction operations in future grades.
• Identify equivalent fractions and explain why they
are equivalent. Teacher Toolbox Teacher-Toolbox.com

• Understand that in order to compare two fractions, Ready Lessons Prerequisite 3.NF.A.3d
students must reason about the size of unit fractions Tools for Instruction Skills
shown by the denominators and number of parts Interactive Tutorials ✓
shown in the numerator in each fraction. ✓✓
✓✓
VOCABULARY

There is no new vocabulary. Review the following key
terms.

unit fraction: a fraction with a numerator of 1; other
fractions are built from unit fractions

numerator: the number above the fraction bar in a
fraction that tells the number of equal parts out of the
whole

denominator: the number below the fraction bar in a
fraction that tells the number of equal parts in the whole

CCSS Focus

3.NF.A.3 E xplain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
d. C ompare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that

comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the
symbols ., 5, or ,, and justify the conclusions, e.g., by using a visual fraction model.

STANDARDS FOR MATHEMATICAL PRACTICE: SMP 2, 3, 4, 7 (see page A9 for full text)

190 L19: Use Symbols to Compare Fractions

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Part 1: Introduction Lesson 19

AT A GLANCE Lesson 19 Part 1: Introduction Develop Skills and Strategies

Students read a comparison word problem and answer a Use Symbols to Compare Fractions CCSS
series of questions that help them connect comparing
fractions using models to recording comparisons using 3.NF.A.3d
symbols.
In Lesson 18, you learned how to compare fractions. Take a look at this problem.
STEP BY STEP Erica’s cup is · 64· full. Ethan’s cup is ·56· full. Use ,, ., or 5 to compare ·64· and ·56· .

• Tell students that this page reviews how to use Explore It
models to compare two fractions and demonstrates Use the math you already know to solve the problem.
how to use symbols to record the comparisons.
The fractions have the same denominator. What do you need to think about to
• Have students read the problem at the top of the compare the two fractions? Possible answer: You need to think about how
page. Remind students that they have had practice many unit fractions it takes to make each fraction.
with solving this kind of comparing problem
previously. How many sixths does Erica have? 4

• Work through the Explore It questions and directives How many sixths does Ethan have? 5
as a class.
Use a symbol to compare those two whole numbers. 4 , 5
• Write the symbol , on the board and underneath
write the words “less than.” Write the symbol . on Is the amount in Erica’s cup less than, greater than, or equal to the amount in
the board and underneath write the words “more Ethan’s’ cup? less than
than.” Make sure that students understand that the
answers for the third and fourth bulleted question Explain how you can use a symbol to compare the two fractions.
are whole numbers that tell the number of sixths that 4 5
Erica and Ethan have. Since 4 sixths is less than 5 sixths, ·6· , ·6· .

• Ask students to practice with a partner how they 168 L19: Use Symbols to Compare Fractions ©Curriculum Associates, LLC Copying is not permitted.

would help another student write fractions and Mathematical Discourse
symbols to compare ​·64 · ​ and ​·65 · ​.  Ask for volunteers to
model what they would say. • How did you remember what the symbol , and .
mean when you used them to compare whole
Concept Extension numbers?

• Try to give students many opportunities to think Have students explain their answers.
about new concepts independently and practice
expressing them in their own words. This is an • Why might you need to use an equals sign when
important step in the learning process. This is comparing two fractions.
also helpful for students who need more practice
with language skills and math vocabulary. The fractions might be equal.

L19: Use Symbols to Compare Fractions 191
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Part 1: Introduction Lesson 19

AT A GLANCE Part 1: Introduction Lesson 19

Students review ways to remember the meaning for the Find Out More
, and . symbols and use the symbols to record
comparisons. You have already learned how to figure out if one fraction is less than, greater than, or
equal to another. Now you will use the symbols ,, ., or 5 to compare fractions.
STEP BY STEP
, means less than . means greater than 5 means equal to
• Read Find Out More as a class. Write the , and .
Think of the , and . symbols as the mouth of an alligator. The alligator’s mouth will
symbols on the board. Explain how to visualize the always be open to eat the greater fraction.

two symbols by thinking that the symbols look like Think about the fractions 1 and 1 . 1 is greater. 1 is less.
·2· ·8· ·2· ·8·
the mouth of an alligator that points to the greater
number. Write ​·12 · ​ . ​·18 · ​ and ​·18 · ​ , ​·12 · ​ on the board. Circle 1 1 or 1 1
the . sign in the first problem. Tell the class that if ·2· ·8· ·8· ·2·

the first thing you see is the open mouth, then you 1 is greater than 1 or 1 is less than 1 .
·2· ·8· ·8· ·2·
know it means “greater than.” If the first thing you
You can switch the order of the fractions. Just be careful which symbol you use. If the
see is the point or smaller part of the symbol, then greater fraction is first, you use .. If the greater fraction is last, you use ,.

the symbol means “less than.” Invite students to Also, remember that sometimes one fraction is not greater than the other. Sometimes
they are equivalent. Then you use 5 to compare them.
share other strategies to help them remember the
1 5 1 and 7 5 7
meaning of the symbols. ·2· ·2· ·8· ·8·
• Write ​·12 · ​ 5 ​·12 · ​ and ​·78 · ​ 5 ​·87 · ​ on the board. Point out that
Reflect
comparing also means looking to see if fractions are
1 Use the symbols , and . to compare 7 and 3 . Explain your answers.
the same. 7 3 3 7 ·8· ·8·
·8· . ·8· and ·8· , ·8·
• For a quick assessment, dictate the fraction
comparisons from Reflect to students and have them Possible explanation: There are more 1 s in 7 than in 3 . So, 7 is greater
practice writing what they hear using symbols. Have ·8· ·8· ·8· ·8·
them write their responses on a piece of paper or 3 3 7
small white board they can hold up. Have students than ·8· and ·8· is less than ·8· .
share reasoning for why one fraction is less than or
greater than the other. L19: Use Symbols to Compare Fractions 169
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• Ask students to work again with a partner to write
two ways to compare the fractions in Reflect and Concept Extension
practice explaining why the first fraction listed is
greater or less than the second fraction. Ask for • Students often need help with instant recall to
volunteers to share their explanations. cement what the , and . symbols mean and to
promote quick and easy reading of comparisons.
SMP Tip: Students look at the , and . symbols To help students, remind them that if they see the
large or open part of the mouth first when reading
to connect the structure of the symbols to their the symbol, they know to say “greater than.” If they
meaning in mathematical expressions. Students see the small or closed part of the sign first when
who connect the visual representation to the reading the symbol, they know to say “less than.”
symbolic meaning learn to read comparison
sentences with ease (SMP 7). Mathematical Discourse

• What do you need to think about when comparing
the two fractions in the Reflect problem? Why?

It’s important to look closely at the two fractions
to see if you are comparing fractions with the
same denominators or numerators. Once you
know whether you are comparing fractions with
the same denominators or fractions with the
same numerators, you will know if you should
think about the size of unit fractions or how
many parts are in each numerator.

192 L19: Use Symbols to Compare Fractions

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Part 2: Modeled Instruction Lesson 19

AT A GLANCE Part 2: Modeled Instruction Lesson 19

Students review how to compare fractions that have the Read the problem below. Then explore different ways to compare fractions.
same numerators. They use symbols to record the Use ,, ., or 5 to compare · 84· and ·64· .
comparisons.

STEP BY STEP Picture It

• Explain to students that they will now use models You can use models to help you compare fractions.
and number lines to compare two fractions that have
the same numerators. This model shows 4 . This model shows 4 .
·8· ·6·
• Read the problem at the top of the page as a class.
Model It
Direct students’ attention to the models showing
·​84 · ​ and ​·46 · ​ and ask them to figure out which fraction You can also use number lines to help you compare fractions.
is greater.
This number line is divided into eighths. It shows 4 .
• For a quick review, ask students to raise their thumb ·8·
up if their answer is yes, put their thumb down if the
answer is no and sideways if unsure. Then ask these 0 4 1
questions: 8
Are both models the same size? [yes]
Are the unit fractions the same size in each model? [no] This number line is divided into sixths. It shows 4 .
Is ·​ 48· ​larger than ​·46 ·  ​ ? [yes] Ask students how they know. ·6·
Is ·​ 64· ​larger than ​·64 · ​  ? [no] Ask students how they know.
0 4 1
• Use the same process to review using the number 6
line to compare the two fractions. Ask these
questions: 170 L19: Use Symbols to Compare Fractions ©Curriculum Associates, LLC Copying is not permitted.
Are both number lines the same length? [yes]
Do both number lines show the same fractions? [no] Concept Extension
Is the fraction ​·48 · ​in the center of the number line? [yes]
Ask students how they know. Practice reading fraction comparisons.

Is the fraction ​·64 · ​closer to zero than ·​48 · ​  ? [no] Ask • Write several comparisons on the board, such as
students how they know. ​·56 · ​ . ​·26 · ​,  ​·34 · ​ . ​·14 · ​,  ​·38 · ​  , ​·78 · ​,  ​·31 · ​  , ​·33 · ​.  Instruct students to
work with a partner to practice reading the
Is ·​ 46· ​larger than ​·48 · ​? [yes] Ask students how they know. comparison number sentences correctly. Then
have students switch the fractions and symbols
Real-World Connection and practice saying each with a partner. As
students practice in pairs, walk around the room
Ask students to think of a situation in the real world and give support as needed.

when they might need to compare two fractions.

Have them share. An example: One store says the

sale price for psrhioceesisi s​·41  ·​·12  ​ ·o ​ offfffufullllpprircicee. . Another store
says the sale

L19: Use Symbols to Compare Fractions 193
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Part 2: Guided Instruction Lesson 19

AT A GLANCE Part 2: Guided Instruction Lesson 19

Students revisit the problem on page 170 and use Connect It
symbols to compare the fractions in the problem.
Now you will solve the problem from the previous page using symbols.
STEP BY STEP
2 Explain how you can use the model to compare the fractions.
• Read and answer the Connect questions as a class. Possible answer: The models are the same size. The area shaded on
After students write their answer to problems 4
and 5, ask them explain how they knew which the 4 model is less than the area shaded on the 4 model.
symbol to write. Invite students to share their ·8· ·6·
explanations for each answer. Ask questions such as:
Can anyone add to what     said? Do you agree 3 Explain how you can use the number lines to show how the fractions compare.
with what      just said? Explain why. Does that 4 4
make sense? Why? How do you know that is correct? Possible answer: ·8· is closer to 0 than ·6· is, so it is less.

• Organize students into small groups. Ask them to 4 Write the comparison:
complete the Try It problems. Assign each group one
or two of the problems to present to the class to using words: 4 eighths is less than than 4 sixths.
share how they compared each fraction and decided 4
on which symbol to use. using symbols: 4 , ·6·
·8·
SMP Tip: Try to give students many opportunities
5 Now switch the order of the fractions. Write the comparison:
to critique or comment on other students’ thinking
or explanations. In addition, ask questions that using words: 4 sixths is greater than than 4 eighths.
promote students’ justifying their own explanations
(SMP 3). using symbols: 4 . 4
·6· ·8·

6 Explain how to use symbols to compare two fractions.

Possible answer: You can use a model to find out which fraction is greater.

If you write the greater fraction first, use the . symbol. If you write the

greater fraction last, use the , symbol.

Try It

Use what you just learned about using symbols to compare fractions to solve
these problems. You can draw models on a separate piece of paper.

7 Use ,, ., or 5 to compare each set of fractions. Each symbol will be used once.

4.2 2, 2 15 1
·6· ·6· ·4· ·3· ·2· ·2·

8 Use ,, ., or 5 to compare each set of fractions. Each symbol will be used once.

353 2, 2 2. 1
·4· ·4· ·8· ·2· ·3· ·3·

L19: Use Symbols to Compare Fractions 171
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TRY IT SOLUTIONS

7 Solution: ​·46 · ​ . ​·62 · ​,  ​·24 · ​ , ​·32 · ​,  ​·12 · ​ 5 ​·21 · ​;  Students may write
the correct symbol, but read it incorrectly.

8 Solution: ​·34 · ​ 5 ​·43 · ​,  ​·28 · ​ , ​·22 · ​,  ​·23 · ​ . ​·31 · ​;  Students may have
drawn models on a separate piece of paper or used
reasoning (and no models) to compare the fractions.

194 L19: Use Symbols to Compare Fractions

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Part 3: Guided Practice Lesson 19

Part 3: Guided Practice Lesson 19 Part 3: Guided Practice Lesson 19

Study the models below. Then solve problems 9–11. 10 David and Rob each got the same snack pack of crackers. David I think drawing a model
might help. Be sure the
Student Model ate 3 of his snack pack. Rob ate 3 of his snack pack. Who ate wholes are the same size.
·6· ·4·
Su and Anthony live the same distance from school. Su biked
The fractions have the · 34· of the way to school. In the same amount of time, Anthony more? Compare the fractions using a symbol.
same denominator, so walked · 41· of the way to school. Who went the greater distance?
they are easy to compare Compare the fractions using a symbol. Show your work.
on the same number line.
Look at how you could show your work using a number line. Possible student work using models:
Pair/Share
0 1 3 1
How do you find the ·4· ·4·
greater number on a
number line? Solution: Su went a greater distance. 3 . 1
·4· ·4·

What do you need to 9 Julia and Mackenzie have the same number of homework Rob ate more. 3 . 3 Pair/Share
think about when you ·4· ·6·
compare fractions that problems. Julia has done 1 of her problems. Mackenzie has Solution: Which fraction is made
have different ·3· of bigger unit fractions?
denominators? Why?

done 1 of her problems. Which student has done less of her
·2·
11 What number could go in the blank to make the comparison Itsh·58a·nletshsethfraanctoior ngrtehaatter
homework? Compare the fractions using a symbol. goes in the blank?
true? Circle the letter of the correct answer.
Show your work. Pair/Share
5 ,
·8· Does Blake’s answer
Possible student work using models: make sense?
A 5
·8·

B 4
·8·

C 6
·8·

D 1
·8·

Blake chose A as the correct answer. How did he get that answer?
5
Pair/Share He chose the one that was equal to ·8· instead of looking for

How did you know one that was greater than 5 .
which fraction was ·8·
smaller? 1 1
Solution: Julia has done less homework. ·3· , ·2·

172 L19: Use Symbols to Compare Fractions ©Curriculum Associates, LLC Copying is not permitted. L19: Use Symbols to Compare Fractions 173
©Curriculum Associates, LLC Copying is not permitted.

AT A GLANCE 9 Solution: Julie has done less homework. ​·13 · ​ , ​·12 · ​; 
Students may draw two equal wholes and shade
Students use symbols to compare fractions in problems in ​·31 · ​ and ​·12 · ​ on the models or draw a number line and
and show their thinking using models or number lines. show both fractions. (DOK 1)

STEP BY STEP 10 tSwolou teiqonu:aRl wobhoatleesmaonrde.s ​h·43 ·a ​ .de ​·36 ·i ​n;  S​·36 t· ​u adnedn ​·43t ·s ​ omnatyhedraw
models or draw a number line and show the
• Ask students to solve the problems individually. Be
sure to remind students to read the hints on the sides two fractions. (DOK 1)
of each page. Ask students to underline the question
in each problem to help them keep in mind whether 11 Solution: C; Since the denominators are all the same,
the problem asks for who or what is more or less. identify the fraction that has a numerator greater
Point out that students should draw a model or use than 5.
number lines to show their work or thinking. Explain to students why the other two answer
choices are not correct:
• When students have completed each problem, have
them Pair/Share to discuss their solutions with a B and D are not correct because they are both less
partner or in a group. gthreaant e​·58 ·r ​.  tThhaen q​·58 ·u ​.  e(sDtiOonKa3s)ks for a fraction that is

SOLUTIONS 195

Ex A number line is shown as one way to solve the
problem. Students could also compare the
numerators because the denominators are the same.
3 . 1, so ​·43 · ​ . ​·41 · .​

L19: Use Symbols to Compare Fractions
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Part 4: Common Core Practice Lesson 19

Part 4: Common Core Practice Lesson 19 Part 4: Common Core Practice Lesson 19
Solve the problems.
4 Look at the comparison below.

, 3
·4·
1 Which fraction could go in the blank to make the comparison true?
Tyrone wrote a fraction in the blank to make the comparison true. His fraction had an
. 1 8 in the denominator. What is one fraction that Tyrone could have used?
·2·
Possible student work using models:

A 2 Show your work.
·4·

B 4
·8·

C 2 1 , 2 , 3 , 4 , 5 are all possible answers.
·3· ·8· ·8· ·8· ·8· ·8·

D 2
·6·

Answer Tyrone could have put 5 in the blank to make the comparison true.
·8·

2 Shade the rectangles below to represent the given fractions. Then use your diagrams
to help you complete the statement below with ,, ., or 5.

1 5 Tran and Noah were each given the same amount of clay in art class. Tran divided his
4 clay into 3 equal pieces. He used 2 of the pieces to make a bowl. Noah divided his clay

2 into 4 equal pieces. He also used 2 of the pieces to make a bowl. Tran said that he had
8
more clay left over than Noah. Is Tran correct? Explain.

No, Tran is not correct. Tran used 2 of his clay and Noah used 2 of his clay. Tran
·3· ·4·
1 2
4 = 8 divided his clay into fewer pieces, so his 2 pieces were bigger than Noah’s 2 pieces.

That means that he used more clay than Noah, so he has less clay left over.

3 Use the numbers below to build fractions that make the statement true. There is more
than one correct answer.

68134

3 6 Sample answer shown.
8,8

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L19: Use Symbols to Compare Fractions
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AT A GLANCE 4 Solution: Possible answers are ​·81 · ​,  ​·82 · ​,  ​·83 · ​,  ​·84 · ​,  or ​·58 · ​
because, when shaded, these fractions cover less
Students use symbols to compare fractions to answer
questions that might appear on a mathematics test. tttshhhpeeaa nc​·34m e·  ​​·68  ao·t ​nh, dwdaenlhc a​hi·43 c·no ​h ddoowsweeosoauuilnfldrdatchcboteeviomeenrqottudhoaeelsl;hstaoSamt,duneedoteathnmlatetossuhssnhatstahdalaesens ​·43,is · n​ ​·34  o· ​. n
(DOK 2)
SOLUTIONS

1 Solution: C; Students understand that they are

looking for a fraction greater than ​·21 · ​.  They see that 5 Solution: No; cTlraayn. ​·32 u· ​ .se d​·24 ·  ​​·32;  · ​E oxf phliasncaltaiyonasndwNillovaahry.
the other fractions are either equal to or less than ​·21 · ​.  used ​·24 · ​ of his

(DOK 1)

2 Solution: See completed diagram on student page (DOK 2)
above; ​·14 · ​5 ​·82 · ​  because 1 shaded section in the top
rectangle covers the same area as 2 shaded sections

in the bottom rectangle. (DOK 1)
3 Solution: Possible answer: ​·38 · ​ , ​·68 · ​;  The denominators

are the same, so the first fraction must have a

numerator that is less than the numerator in the

second fraction for the statement to be true. (DOK 2)

196 L19: Use Symbols to Compare Fractions

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Differentiated Instruction Lesson 19

Assessment and Remediation

• Ask students to solve this problem using symbols to compare the fractions: Ty and Luke drank the same size

cartons of juice when playing a game. By the end of the game, Ty had drunk ​·83 · ​ of his juice and Luke had
drunk ​·36 · ​ of his juice. Who drank the most juice? Use symbols to show who drank more.

• For students who are struggling, use the chart below to guide remediation.

• After providing remediation, check students’ understanding. Ask students to solve this problem using
symbols to compare the fractions: The female bird ate ​·62 · ​ of a worm. The male bird ate ​·82 · ​ of the same size
worm. Which bird ate more? Use symbols to compare the fractions.

If the error is . . . Students may . . . To remediate . . .

the student said be confused about the Help students figure out a strategy for remembering what each
that Ty drank more
and wrote ​·38 · ​ > · ​63· ​ meaning of the . and , symbol means that makes sense to them. Give them practice writing

symbols. comparisons using symbols and reading comparisons.

Hands-On Activity Challenge Activity

Practice using symbols to compare. Practice using symbols to compare.

Materials: Ten note cards with one of these fractions Materials: Ten note cards with one of these fractions
written on each card ​·46 · ​,  ​·24 · ​,  ​·28 · ​,  ​·38 · ​,  ​·36 · ​,  ​·84 · ​,  ​·62 · ​,  ​·43 · ​,  ​·44 · ​,  ​·33 · ​,  written on each card ​·28 · ​,  ​·88 · ​,  ​·84 · ​,  ​·41 · ​,  ​·44 · ​,  ​·42 · ​,  ​·42 · ​,  ​·12 · ​,  ​·22 · ​,  ​·28 · ​, 
three note cards with one of these symbols on three note cards with one of these symbols on each:
each: ,, ., 5, and pencils and paper for each
student. Give each pair one set of fraction cards and ,, ., 5, and pencils and paper for each student.
one set of symbol cards.
Give each pair one set of fraction cards and one set
The students shuffle the deck of fraction cards, and
of symbol cards.
each student takes a card and lays it down with the
This activity builds upon the Hands-On Activity and
fraction face up. Students choose a symbol card to gives student practice comparing fractions equal to
and greater than one whole in addition to practice
place between the two fraction cards to make a true comparing fractions less than one whole. Students
use the same procedure as the hands on activity:
statement. They write the statement on their paper they use the symbol cards to create a comparison
between the fraction cards, and they write the
and read it aloud. They then switch the order of the comparison in two ways by switching the order of
the fractions and completing the new comparison
two fractions and use a different symbol to correctly with the correct symbol. In addition to writing down
each comparison statement, students read each
compare the fractions. They write the comparison statement aloud.

and read it out loud. Students use the same process

to choose and compare the next two fraction cards in

the deck. If the fractions are equivalent, such as ​·33 · ​
and ​·44 · ​,  students can still switch the cards, but the

symbol card will remain the same.

L19: Use Symbols to Compare Fractions 197
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