Ready Common Core Mathematics Teacher Resource ... AWS
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 Content Emphasis in the Common Core Standards Connecting with the Ready® Teacher Toolbox Using i-Ready® Diagnostic with Ready Common Core Features of Ready Common Core Instruction Supporting Research
A14 A16 A18 A20 A22 A38
Correlation Charts Common Core State Standards Coverage by Ready Instruction Interim Assessment Correlations
A42 A46
Lesson Plans (with Answers) CCSS Emphasis Unit 1: Operations and Algebraic Thinking, Part 1 Lesson 1
Understand the Meaning of Multiplication
1 3
M
11
M
23
M
33
M
41
M
49
M
59
M
CCSS Focus - 3.OA.A.1 Embedded SMPs - 2–4
Lesson 2
Use Order and Grouping to Multiply
CCSS Focus - 3.OA.B.5 Embedded SMPs - 2–4, 7
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
67
Unit 2: Number and Operations in Base Ten Lesson 8
Use Place Value to Round Numbers
CCSS Emphasis 70 72
S/A
82
S/A
94
S/A
CCSS Focus - 3.NBT.A.1 Embedded SMPs - 1, 5–8
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 Lesson 11 Solve One-Step Word Problems Using Multiplication and Division
103 106 109
M
121
M
131
M
CCSS Focus - 3.OA.A.3 Embedded SMPs - 1–4, 7
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 Lesson 14 Understand What a Fraction Is
141 144 146
M
154
M
162
M
170
M
182
M
190
M
CCSS Focus - 3NF.A.1 Embedded SMPs - 2–4, 6
Lesson 15 Understand Fractions on a Number Line CCSS Focus - 3.NF.A.2a, 2b Embedded SMPs - 2, 3, 7
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
199
CCSS Emphasis
Unit 5: Measurement and Data
202
Lesson 20 Tell and Write Time
207
M
215
M
225
M
CCSS Focus - 3.MD.A.1 Embedded SMPs - 1, 3, 4, 6
Lesson 21 Solve Problems About Time CCSS Focus - 3.MD.A.1 Embedded SMPs - 1, 3–6
Lesson 22 Liquid Volume CCSS Focus - 3.MD.A.2 Embedded SMPs - 2, 4, 6
Lesson 23 Mass
235 M
CCSS Focus - 3.MD.A.2 Embedded SMPs - 1, 2, 4–6
Lesson 24 Solve Problems Using Scaled Graphs
245
S/A
255
S/A
265
S/A
275
M
283
M
293
M
303
S/A
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
315 318 320
S/A
328
S/A
338
S/A
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
346
Focus on Math Concepts
Lesson 18
Understand Comparing Fractions LESSON OBJECTIVES
THE LEARNING PROGRESSION
• Understand that in order to compare two fractions, students must reason about the size of unit fractions shown by the denominators and number of parts shown in the numerator in each fraction.
In this lesson, students apply their understanding of fractions to compare two fractions. They use fraction models and number lines to help them reason about the size of unit fractions.
• Analyze the numerators and denominators in fractions to be compared to determine if the fractions have the same numerators or denominators.
Students learn to look carefully at the numerators and denominators of both fractions they are comparing to determine if the numerators or the denominators are the same. If the denominators are the same, the students know the fractions are built from the same size unit fraction and the fraction with the most parts in the numerator is larger. If the numerators are same, then students reason about the size of the unit fractions used to make each of the two fractions.
• Explain why one fraction is smaller or larger when comparing two fractions using models or number lines.
PREREQUISITE SKILLS • Understand the meaning of fractions. • Identify fractions represented by models and number lines. • Understand that the size of a fractional part is relative to the size of the whole.
Students apply this understanding to solve problems that involve comparing fractions and explain why one fraction is larger or smaller. Teacher Toolbox
• Identify equivalent fractions and explain why they are equivalent.
Prerequisite Skills
VOCABULARY
Ready Lessons
There is no new vocabulary. Review the following key terms.
Tools for Instruction
unit fraction: a fraction with a numerator of 1; other fractions are built from unit fractions
Teacher-Toolbox.com
Interactive Tutorials
3.NF.A.3d
✓ ✓✓ ✓✓
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 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. d. Compare 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)
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L18: Understand Comparing Fractions
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Part 1: Introduction
Lesson 18
AT A GLANCE
Focus on Math Concepts
Students explore what it means to compare fractions. They compare fractions with the same denominators using models and number lines.
Lesson 18
Part 1: Introduction
CCSS 3.NF.A.3d
Understand Comparing Fractions
How do we compare fractions?
STEP BY STEP
When you compare fractions, you figure out which is smaller, which is larger, or if they are the same size.
• Remind students that they’ve compared whole numbers such as 345 and 509 to see which is smaller, larger, or if the two numbers are the same.
You can use models or number lines to help you compare two fractions. 1 4
• Introduce the Question at the top of the page. Ask 4 ··
1
4 ··
If not, it might look like 1 is
[
greater than 2 .
4 ··
4 ··
Think Sometimes when you compare fractions, the denominators are the same.
[
The models below show two wholes that are the same size divided into sixths.
Circle the model of the fraction that is less.
Think about how many unit fractions it takes to make each fraction you are comparing.
and is farther away from 0. Direct students’ attention to the two different size wholes that show 1 and 2 .
1 6
4 ··
Emphasize that when you compare two fractions, the
1 6
1 6
1 6
1 6
1 6
1 6
It takes two 1 s to make 2.
It takes five 1s to make 5. 6 6 6 6 ·· ·· ·· ·· 2 is made of fewer unit fractions than 5 . So, 2 is less than 5 . 6 6 6 6 ·· ·· ·· ··
size of the wholes must be the same. • Have a student to read the Think statement aloud.
3 4
the same to compare fractions.
explain how they know. 2 covers more of the square 4 ·· than 1 . Asks students to use the number line to 4 ·· explain why 2 is larger than 1 . 2 has more fourths 4 4 ·· 4 ·· ·· 4 ··
2 4
The size of the wholes must be
two square fraction models, tell which is larger, and
]
1 4
Both of these show that 1 is less than 2 .
students to identify the two fractions shown by the
]
0
2 4
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L18: Understand Comparing Fractions
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Ask students what two fractions are shown by the models. 3 2 and 5 4 Write the two fractions on the 6 ··
6 ··
board. Remind students what a unit fraction is and explain that each of the fractions was made up of the unit fraction 1 . Explain that you can use unit 6 ··
fractions and fraction models to show if a fraction is larger or smaller. The fraction model shows that
Hands-On Activity Use fraction bars to compare two fractions.
it takes more copies of 1 to make 5 than to make
Materials: fraction bars or circles, blank paper
only 2 .
Have students work in pairs and use a blank paper
6 ··
6 ··
6 ··
Mathematical Discourse • What is the same about the two fractions? The denominators are the same size. • What is different about the two fractions? They are built with a different number of unit fractions or they have a different numerator.
L18: Understand Comparing Fractions
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as their work space. Ask each pair to put the one whole fraction bar at the top of their paper. Instruct one student in the pair to build the fractions 2 and 4 5 ··
and have the other point to the larger fraction and
5 ··
explain why it is larger. Ask a student to share why. Do the same to compare the fractions 3 and 5 , 3 and 6 .
8 ··
8 ·· 10 ··
10 ··
183
Part 1: Introduction
Lesson 18
AT A GLANCE Students explore ways to compare fractions that have the same numerators, but different denominators.
Part 1: Introduction
Lesson 18
Think You can compare fractions with like numerators and different denominators. Think about two different unit fractions from the same
STEP BY STEP
It’s like cutting up a piece of paper. The more pieces you cut the paper into, the smaller each piece is.
whole, such as 1 and 1 . 3 ··
• Read the Think statement together. On the board, write the two fractions 1 and 1 shown by the models 3 ··
8 ··
on the page. Circle the numerators in each fraction.
1 3
8 ··
1 8
Compare the denominators of 1 and 1 . 3 is less than 8, so 3 ··
parts, each part is bigger. So, the unit fraction 1 is greater 3 ··
Explain that they will next explore how to compare
than 1.
two fractions in which the numerators are the same
Here’s another example:
and the denominators are different. • Write the two fractions 3 and 3 on the board and 6 ··
4 ··
8 ··
the whole is divided into fewer parts. Since there are fewer
8 ··
1 6
1 6
1 6
1 4
1 4
1 4
The unit fractions used to
The unit fractions used to
make 3 are smaller.
make 3 are bigger.
6 ··
4 ··
circle the numerators. Ask students to picture a
3 smaller parts are less than 3 bigger parts. So, 3 is less than 3 .
rectangle divided into 6 equal parts and one divided
Reflect
into 4 equal parts. Direct students attention to the models of 3 and 3 on the page. Make the point that 6 ··
4 ··
each 1 is smaller, so 3 of those 1 s will cover less of 6 ··
6 ·· the model than will 3 of the 1 s. 4 ··
• Ask students to work with a partner to answer the Reflect question in their own words and invite students to share.
Hands-On Activity these fractions on a blank sheet of paper: 1 , 1 , 1 , 1 , 1 , and 1 . Instruct them to 10 ··
label each unit fraction. To help students understand that the smaller the fractional piece, the larger the denominator is, ask questions such as: What do you notice about the size of the unit fractions as the denominator gets larger? How many tenths does it take to make one whole? One-half? Why does it take more tenths to make one whole? Then ask them to show 3 and 3 . Ask: Which 8 ··
fraction is larger and Why?
184
4 ··
1 Explain how you can use unit fractions to help you compare fractions.
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
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163
Mathematical Discourse
• Ask students to use fraction tiles or bars to lay
2 ·· 4 ·· 5 ·· 6 ·· 8 ··
6 ··
• What is the same about the two fractions 3 and 3 ? 6 4 ·· ·· 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.
4 ··
L18: Understand Comparing Fractions
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Part 2: Guided Instruction
Lesson 18
AT A GLANCE Part 2: Guided Instruction
Students use models to compare two fractions that either have the same denominators or the same numerators.
Lesson 18
Explore It Use the models to help you compare fractions with the same denominator. 2 Write the fraction shaded below each model.
Circle the fraction that is greater.
STEP BY STEP
2
3 ··
3 Write the fraction shaded below the first
model. Shade the second model to show a greater fraction. Write the greater fraction.
4
Possible answers: 5 , 6 , 7 , 8 ; shading will vary.
8 ··
8 ·· 8 ·· 8 ·· 8 ··
Use the models to help you compare fractions with the same numerator. 4 Write the fraction shaded below each model.
Circle the fraction that is greater.
• As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking.
1
1
6 ··
2 ··
5 Write the fraction shaded below each model.
Circle the fraction that is less. 2
2
6 ··
3 ··
6 Write the fraction shaded below the first
• When students have completed the page, ask them to work with a partner and return to each problem. Instruct them to cover up the models and to picture the fractions in their mind. Then have them practice explaining why one of the fractions is greater. • Discuss each problem. Ask for volunteers to explain
1
3 ··
• Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through the first problem together as a class.
model. Shade the second model to show a fraction that is less but has the same numerator.
5
6 ··
Explain how you know the fraction is less. Possible answer: The same number of parts are shaded in both models. The 1 parts are smaller than the 1 parts, so 5 is less than 5 . 8 ··
164
6 ··
8 ··
6 ··
L18: Understand Comparing Fractions
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how they figured out which fraction was larger or smaller. For problem 6, ask for all the possible ways students could shade the model so the fraction is less than 5 . 6 ··
• Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson.
SMP Tip: Encourage students to picture fractions in their minds based upon experiences with fraction models. This practice helps students conceptualize as they work more symbolically with fractions later on (SMP 2).
Concept Extension • Students may not be familiar with the way each 1 8 ··
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
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185
Part 2: Guided Instruction
Lesson 18
AT A GLANCE Students revisit the problem on page 164 and explain how to use unit fractions to compare the fractions. They also explore using number lines to compare fractions.
Part 2: Guided Instruction
Lesson 18
Talk About It 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 • Organize students into pairs or groups and ask them to answer the Talk About It questions as a group. • Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse questions to help support or extend students’ thinking. • Be sure students are able to describe how the problems are different on the page. Fractions have the same number of whole equal parts if the denominators are the same. • Direct the group’s attention to Try It Another Way. Draw the number lines on the board for problems 10 and 11. Have volunteers from the groups come up to the board and use the number lines to explain how they figured out which fraction was greater or less.
compare fractions with the same denominator. Possible answer: The unit fractions are the same. Look at the numerators to see how many unit fractions are in each fraction. Compare these numbers.
8 Look at your answers to problems 4–6. What is different about the numerators
and denominators in these fractions than the fractions in problems 2 and 3? Possible answer: The fractions in problems 2 and 3 have the same denominators and different numerators. The fractions in problems 4–6 have the same numerators and different denominators. Explain how to use unit fractions to compare the fractions with the same numerator. Possible answer: You have to look at the size of the unit fraction, since the number of unit fractions in each is the same. The one with the smaller unit fraction is less. 9 Isaiah is comparing 3 and 3 . Both fractions have a numerator of 3. How can he tell 8 6 ·· ·· Possible answer: He knows that 1 s are smaller which fraction is less? 8 ··
than 1 s. He has the same number of 1 s and 1 s, so the fraction made of 1 s 6 ··
8 ··
is smaller. 3 is less than 3 . 8 ··
6 ··
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
lines. Circle the fraction that is less. 0
3 8
1
0
3 4
11 Look at the fractions on the number
lines. Circle the fraction that is greater. 1
0
4 6
1
L18: Understand Comparing Fractions
SMP Tip: Problem 8 gives students practice
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0
3 6
1
165
looking at the structure of the fractions to be compared. Students notice whether the numerators or denominators are the same or different, and they identify two possible kinds of problems they will encounter when they compare fractions (SMP 7).
Concept Extension • Emphasize that two fractions must have some part that is the same in order to compare them. The first thing students need to do is find which part is the same. If the denominators are the same, that means the unit fractions that make up the whole are the same. They can look at the numerator to see how many copies of the unit fractions there are. The more copies, the greater the fraction. If the numerators are the same, that means the number of unit fraction copies is the same. The denominator tells the size of each unit fraction. The more unit fractions that make up a whole, the smaller each unit fraction is.
186
Mathematical Discourse • What do you look at first when you are comparing two fractions? Look to see if either the denominators or numerators are the same. • Which fractions seem easiest for you to compare, those that have the same numerators or those that have the same denominators? Explain why. Answers will vary. Have students explain their answers.
L18: Understand Comparing Fractions
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Part 3: Guided Practice
Lesson 18
AT A GLANCE Part 3: Guided Practice
Students demonstrate their understanding of comparing two fractions that have either the same numerators or the same denominators.
Lesson 18
Connect It Talk through these problem as a class, then write your answers below. 12 Create: Draw an area model or number line to show
same denominator that is less than 5 .
STEP BY STEP
5 . Find a fraction with the
8 ··
8 ··
Possible student work:
• Discuss each Connect It problem as a class using the discussion points outlined below.
3 8
0
Create:
5 8
1
Explain how you found your answer. Possible answer: I know that any fraction to the left of 5 is less than 5 . So, 3 is less than 5 . 8 ··
• You may choose to have students work in pairs to encourage sharing ideas. Each partner draws a different model.
8 ··
8 ··
8 ··
13 Explain: Mario painted 2 of the wall in his bedroom. Mei Lyn painted 2 of a wall in 6 4 ·· ··
her bedroom. Both walls are the same size. Explain how you know who painted more. Possible answer: Each painted the same number of parts of the wall. Sixths are smaller than fourths, so 2 sixths is less than 2 fourths. That
• For a quick and easy assessment, have students draw their models on small whiteboards or on paper and hold them up. Choose several pairs to explain their models to the class.
means Mei Lyn painted more. 14 Justify: Jace and Lianna each baked a loaf of bread. Jace cut his in halves and
Lianna cut hers in thirds. 1 2
• Use the following to lead a class discussion:
1 3
Jace says they can use their loaves of bread to show that 1 is less than 1 . Lianna 2 ··
3 ··
says they can’t. Who is correct? Explain why.
What did you keep in mind as you thought of a fraction to choose that was less than 5 ? 8 ··
Possible answer: Lianna is correct. The wholes have to be the same size to compare fractions.
166
L18: Understand Comparing Fractions
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How can you prove that the fraction you chose is less than 5 ?
Justify:
Is there another way to use a number line(s) to show a smaller fraction? [Students may use two number lines or show both fractions on the same number line.]
• This discussion shows whether students understand that in order to compare two fractions, the wholes must be the same size.
8 ··
Explain: • This problem expects students to look at the fractions and picture the size of the unit fractions, reasoning that sixths are smaller than fourths, so Mei Lyn must have painted more. • Begin the discussion by asking questions such as: What do you picture each wall looking like? Does each wall have 2 parts painted? Which 2 parts cover more area of the wall? How do you know?
• 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 1 and 1 . Be sure 2 ··
3 ··
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 1 is smaller than 1 . 3 ··
L18: Understand Comparing Fractions
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2 ··
187
Part 4: Common Core Performance Task
Lesson 18
AT A GLANCE Part 4: Common Core Performance Task
Students use information in fraction form to create a problem that involves comparing fractions. They create a model to help them solve the problem.
Lesson 18
Put It Together 15
Mrs. Ericson made sandwiches for her 4 children. Each sandwich was the same size. After lunch, each child had a different fraction of his or her
STEP BY STEP
sandwich left. Matt had 1 , Elisa had 3 , Carl had 3 , and Riley had 7 . 4 ··
• Direct students to complete the Put It Together task on their own.
8 ··
4 ··
8 ··
A Use this information to write a problem that compares two fractions with the same numerator. Who had more left, Elisa or Carl?
• Go over the directions with students and let them know they have some choices to make. Be sure students understand that they should circle the problem that they will model using either a fraction model or number line.
B Use this information to write a problem that compares two fractions with the same denominator. Who had more left, Matt or Carl?
C Choose one of your problems to solve. Circle the question you chose. Draw a model or number line to help you find the answer.
• As students work on their own, walk around to assess their progress and understanding, to answer their questions, and to give additional support, if needed.
Explain how you could use unit fractions to think about the problem. Possible answer: The unit fractions for Matt’s and Carl’s fractions were both 1 . Matt had one 1 and Carl had three 1 s. 3 is greater than 1, so 4 ··
4 ··
4 ··
Carl had more left.
SCORING RUBRICS A
Points Expectations 2
188
The student correctly chooses two fractions to compare that either have the same numerators or the same denominators. The question for the problem asks which student had more or less of his or her sandwich left.
1
The student correctly chooses two fractions to compare that either have the same numerators or same denominators. The question asks which fraction is larger or smaller, but does not clearly relate to the context of the problem (students and the amount of sandwich left).
0
The student does not choose two fractions to compare that have the same numerators or same denominators and is unable to ask a question that relates to comparing fractions of sandwiches left.
L18: Understand Comparing Fractions
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B
167
Points Expectations 2
The model or number line clearly shows both fractions that are compared. The explanation includes who had the most 1 s 4 ·· (or 1 s) or explains that Elisa and Carl had the 8 ·· same number of parts and that Carl’s unit fractions were larger.
1
The model or number line clearly shows both fractions that are compared. The explanation does not clearly explain how the student used unit fractions to see who had the least or the most left.
0
The model did not correctly compare two 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.
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 thinking about the size of each unit fraction. Have them follow these steps for comparing 2 and 2 . 3 ··
8 ··
• Draw two rectangles that are the same size. Show thirds in one and eighths in the other. • Label each unit fraction in each of the models. • Ask which unit fraction is larger. [thirds] • Shade in each fraction in the models. • Think, “Each 1 is larger than 1 , so 2 must be 3 ··
larger than 2 .”
8 ··
3 ··
8 ··
Materials: 10 blank note cards. Write one of these fractions on each card: 2 , 2 , 2 , 2 , 3 , 3 , 3 , 4 , 4 , 4 4 ·· 5 ·· 6 ·· 8 ·· 8 ·· 6 ·· 4 ·· 5 ·· 8 ·· 4 ··
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. 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: 8 , 8 , 8 , 8 , 4 , 4 , 2 , 4 , 6 , 3 2 ·· 8 ·· 4 ·· 1 ·· 2 ·· 4 ·· 1 ·· 1 ·· 3 ·· 3 ··
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
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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 fractions.
In this lesson students continue to compare two fractions by reasoning about size of unit fraction shown in models and on number lines. The focus of this lesson is on recording the results of comparisons by using symbols to write comparison statements.
• Read comparison statements fluently and accurately. • Use models and number lines to explain and justify fraction comparisons.
PREREQUISITE SKILLS • Understand the meaning of fractions. • Identify fractions represented by models and number lines. • Understand that the size of a fractional part is relative to the size of the whole. • Identify equivalent fractions and explain why they are equivalent.
Students see that comparing fractions also means they must look to see if two fractions are equal. If two fractions are equal, they record it using the equal symbol. Students review or learn the meaning of the , and . symbols and practice using them to record their comparisons. Students solve problems that involve comparisons. Students’ concrete experiences with comparing fractions and recording them using symbols prepares them to move on to more abstract work with comparing fractions and fraction operations in future grades.
• Understand that in order to compare two fractions, students must reason about the size of unit fractions shown by the denominators and number of parts shown in the numerator in each fraction.
Teacher Toolbox Prerequisite Skills
VOCABULARY
Ready Lessons
There is no new vocabulary. Review the following key terms.
Tools for Instruction
unit fraction: a fraction with a numerator of 1; other fractions are built from unit fractions
Teacher-Toolbox.com
Interactive Tutorials
3.NF.A.3d
✓ ✓✓ ✓✓
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 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. d. Compare 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)
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Part 1: Introduction
Lesson 19
AT A GLANCE
Develop Skills and Strategies
Students read a comparison word problem and answer a series of questions that help them connect comparing fractions using models to recording comparisons using symbols.
Lesson 19
Part 1: Introduction
CCSS 3.NF.A.3d
Use Symbols to Compare Fractions
In Lesson 18, you learned how to compare fractions. Take a look at this problem. Erica’s cup is 4 full. Ethan’s cup is 5 full. Use ,, ., or 5 to compare 4 and 5 . 6 ··
STEP BY STEP • Tell students that this page reviews how to use models to compare two fractions and demonstrates how to use symbols to record the comparisons.
6 ··
6 ··
Explore It Use the math you already know to solve the problem.
• Have students read the problem at the top of the page. Remind students that they have had practice with solving this kind of comparing problem previously.
The fractions have the same denominator. What do you need to think about to compare the two fractions? Possible answer: You need to think about how many unit fractions it takes to make each fraction.
4
How many sixths does Erica have?
• Work through the Explore It questions and directives as a class. • Write the symbol , on the board and underneath write the words “less than.” Write the symbol . on the board and underneath write the words “more than.” Make sure that students understand that the answers for the third and fourth bulleted question are whole numbers that tell the number of sixths that Erica and Ethan have.
6 ··
5
How many sixths does Ethan have?
Use a symbol to compare those two whole numbers. 4 , 5 Is the amount in Erica’s cup less than, greater than, or equal to the amount in Ethan’s’ cup? less than Explain how you can use a symbol to compare the two fractions. Since 4 sixths is less than 5 sixths, 4 , 5 . 6 ··
168
6 ··
L19: Use Symbols to Compare Fractions
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• Ask students to practice with a partner how they would help another student write fractions and symbols to compare 4 and 5 . Ask for volunteers to 6 ··
6 ··
model what they would say.
Concept Extension • Try to give students many opportunities to think about new concepts independently and practice expressing them in their own words. This is an important step in the learning process. This is also helpful for students who need more practice with language skills and math vocabulary.
L19: Use Symbols to Compare Fractions
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Mathematical Discourse • How did you remember what the symbol , and . mean when you used them to compare whole numbers? Have students explain their answers. • Why might you need to use an equals sign when comparing two fractions. The fractions might be equal.
191
Part 1: Introduction
Lesson 19
AT A GLANCE Students review ways to remember the meaning for the , and . symbols and use the symbols to record comparisons.
STEP BY STEP
Part 1: Introduction
Lesson 19
Find Out More 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. , means less than
• Read Find Out More as a class. Write the , and . symbols on the board. Explain how to visualize the
. means greater than
5 means equal to
Think of the , and . symbols as the mouth of an alligator. The alligator’s mouth will always be open to eat the greater fraction. Think about the fractions 12 and 18 . 12 is greater. 18 is less. ··
1
1
or
1 is greater than 1 8 ··
or
two symbols by thinking that the symbols look like
2 ··
8 ··
·· ··
1
··
1
8 ··
2 ··
1 is less than 1 . 2 ··
the mouth of an alligator that points to the greater
2 ··
number. Write 1 . 1 and 1 , 1 on the board. Circle
You can switch the order of the fractions. Just be careful which symbol you use. If the greater fraction is first, you use .. If the greater fraction is last, you use ,.
2 ··
8 ··
8 ··
2 ··
the . sign in the first problem. Tell the class that if the first thing you see is the open mouth, then you know it means “greater than.” If the first thing you see is the point or smaller part of the symbol, then the symbol means “less than.” Invite students to share other strategies to help them remember the
8 ··
Also, remember that sometimes one fraction is not greater than the other. Sometimes they are equivalent. Then you use 5 to compare them. 151 2 ··
757 8 ··
and
2 ··
8 ··
Reflect 1 Use the symbols , and . to compare 7 and 3 . Explain your answers. 8 8 ·· ·· 7 3 3 7
8 ··
. and , 8 ··
8 ··
8 ··
Possible explanation: There are more 1 s in 7 than in 3 . So, 7 is greater than 3 and 3 is less than 7 . 8 ··
8 ··
8 ··
8 ··
8 ··
8 ··
8 ··
meaning of the symbols. • Write 1 5 1 and 7 5 7 on the board. Point out that 2 ··
2 ··
8 ··
8 ··
comparing also means looking to see if fractions are the same. • For a quick assessment, dictate the fraction comparisons from Reflect to students and have them practice writing what they hear using symbols. Have them write their responses on a piece of paper or small white board they can hold up. Have students share reasoning for why one fraction is less than or greater than the other. • Ask students to work again with a partner to write two ways to compare the fractions in Reflect and practice explaining why the first fraction listed is greater or less than the second fraction. Ask for volunteers to share their explanations.
SMP Tip: Students look at the , and . symbols to connect the structure of the symbols to their meaning in mathematical expressions. Students who connect the visual representation to the symbolic meaning learn to read comparison sentences with ease (SMP 7).
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169
Concept Extension • Students often need help with instant recall to cement what the , and . symbols mean and to promote quick and easy reading of comparisons. To help students, remind them that if they see the large or open part of the mouth first when reading the symbol, they know to say “greater than.” If they see the small or closed part of the sign first when reading the symbol, they know to say “less than.”
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.
L19: Use Symbols to Compare Fractions
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Part 2: Modeled Instruction
Lesson 19
AT A GLANCE Part 2: Modeled Instruction
Students review how to compare fractions that have the same numerators. They use symbols to record the comparisons.
Lesson 19
Read the problem below. Then explore different ways to compare fractions. Use ,, ., or 5 to compare 4 and 4 . 8 ··
STEP BY STEP
6 ··
Picture It You can use models to help you compare fractions.
• Explain to students that they will now use models 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
You can also use number lines to help you compare fractions.
4 and 4 and ask them to figure out which fraction 8 ··
This number line is divided into eighths. It shows 4 .
6 ··
8 ··
is greater.
0
4 8
1
This number line is divided into sixths. It shows 4 .
• For a quick review, ask students to raise their thumb up if their answer is yes, put their thumb down if the answer is no and sideways if unsure. Then ask these questions: Are both models the same size? [yes] Are the unit fractions the same size in each model? [no]
6 ··
0
4 6
1
Is 4 larger than 4 ? [yes] Ask students how they know. 8 ··
6 ··
Is 4 larger than 4 ? [no] Ask students how they know. 6 ··
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6 ··
• Use the same process to review using the number line to compare the two fractions. Ask these questions: Are both number lines the same length? [yes] Do both number lines show the same fractions? [no] Is the fraction 4 in the center of the number line? [yes] 8 ··
Ask students how they know. Is the fraction 4 closer to zero than 4 ? [no] Ask 6 ··
students how they know.
8 ··
Is 4 larger than 4 ? [yes] Ask students how they know. 6 ··
8 ··
Real-World Connection Ask students to think of a situation in the real world when they might need to compare two fractions. Have them share. An example: One store says the sale price for shoes is 1 off full price. Another store 2 ··
says the sale price is 1 off full price. 4 ··
L19: Use Symbols to Compare Fractions
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Concept Extension Practice reading fraction comparisons. • Write several comparisons on the board, such as 5 . 2 , 3 . 1 , 3 , 7 , 1 , 3 . Instruct students to 6 ··
6 ·· 4 ··
4 ·· 8 ··
8 ·· 3 ··
3 ··
work with a partner to practice reading the comparison number sentences correctly. Then have students switch the fractions and symbols and practice saying each with a partner. As students practice in pairs, walk around the room and give support as needed.
193
Part 2: Guided Instruction
Lesson 19
AT A GLANCE Students revisit the problem on page 170 and use symbols to compare the fractions in the problem.
STEP BY STEP • Read and answer the Connect questions as a class. After students write their answer to problems 4 and 5, ask them explain how they knew which symbol to write. Invite students to share their explanations for each answer. Ask questions such as: Can anyone add to what said? Do you agree with what just said? Explain why. Does that make sense? Why? How do you know that is correct? • Organize students into small groups. Ask them to complete the Try It problems. Assign each group one or two of the problems to present to the class to share how they compared each fraction and decided on which symbol to use.
Part 2: Guided Instruction
Lesson 19
Connect It Now you will solve the problem from the previous page using symbols. 2 Explain how you can use the model to compare the fractions.
Possible answer: The models are the same size. The area shaded on the 4 model is less than the area shaded on the 4 model. 8 ··
6 ··
3 Explain how you can use the number lines to show how the fractions compare.
Possible answer: 4 is closer to 0 than 4 is, so it is less. 8 ··
6 ··
4 Write the comparison:
less than
using words: 4 eighths is , using symbols: 4
than 4 sixths.
4
8 ··
6 ··
5 Now switch the order of the fractions. Write the comparison:
using words: 4 sixths is greater than than 4 eighths. using symbols: 4
6 ··
4
.
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.
SMP Tip: Try to give students many opportunities to critique or comment on other students’ thinking or explanations. In addition, ask questions that promote students’ justifying their own explanations (SMP 3).
4
6 ··
.
2
6 ··
2
4 ··
,
2
1
3 ··
2 ··
5
1
2 ··
8 Use ,, ., or 5 to compare each set of fractions. Each symbol will be used once.
3
4 ··
5
3
2
4 ··
8 ··
,
2
2
2 ··
3 ··
.
1
3 ··
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TRY IT SOLUTIONS 7 Solution: 4 . 2 , 2 , 2 , 1 5 1 ; Students may write 6 ··
6 ·· 4 ··
3 ·· 2 ··
2 ··
the correct symbol, but read it incorrectly. 8 Solution: 3 5 3 , 2 , 2 , 2 . 1 ; Students may have 4 ··
4 ·· 8 ··
2 ·· 3 ··
3 ··
drawn models on a separate piece of paper or used reasoning (and no models) to compare the fractions.
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L19: Use Symbols to Compare Fractions
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Part 3: Guided Practice
Lesson 19
Part 3: Guided Practice
Lesson 19
Study the models below. Then solve problems 9–11. Student Model
The fractions have the same denominator, so they are easy to compare on the same number line.
Part 3: Guided Practice
Lesson 19
10 David and Rob each got the same snack pack of crackers. David
ate 3 of his snack pack. Rob ate 3 of his snack pack. Who ate 6 ··
4 ··
Su and Anthony live the same distance from school. Su biked
more? Compare the fractions using a symbol.
3 of the way to school. In the same amount of time, Anthony
Show your work.
walked 1 of the way to school. Who went the greater distance?
Possible student work using models:
4 ··
4 ··
I think drawing a model might help. Be sure the wholes are the same size.
Compare the fractions using a symbol. Look at how you could show your work using a number line.
Pair/Share How do you find the greater number on a number line?
What do you need to think about when you compare fractions that have different denominators?
0
Solution:
1
4 ··
3
1
4 ··
Su went a greater distance. 3 . 1 4 ··
4 ··
Pair/Share
9 Julia and Mackenzie have the same number of homework
problems. Julia has done 1 of her problems. Mackenzie has 3 ··
Solution:
Which fraction is made of bigger unit fractions? Why?
Rob ate more. 3 . 3 4 ··
6 ··
done 1 of her problems. Which student has done less of her 2 ··
homework? Compare the fractions using a symbol. Show your work.
11 What number could go in the blank to make the comparison
true? Circle the letter of the correct answer. 5,
8 ··
Possible student work using models:
Is 5 less than or greater 8 ·· than the fraction that goes in the blank?
A 5
8 ··
B C
4
8 ·· 6
8 ··
D 1
8 ··
Blake chose A as the correct answer. How did he get that answer? He chose the one that was equal to 5 instead of looking for
Pair/Share How did you know which fraction was smaller?
172
one that was greater than 5 . Solution:
Julia has done less homework. 1 , 1 3 ··
8 ··
8 ··
2 ··
L19: Use Symbols to Compare Fractions
Pair/Share Does Blake’s answer make sense?
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L19: Use Symbols to Compare Fractions
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9 Solution: Julie has done less homework. 1 , 1 ;
AT A GLANCE
3 ··
Students use symbols to compare fractions in problems and show their thinking using models or number lines.
2 ··
Students may draw two equal wholes and shade in 1 and 1 on the models or draw a number line and 3 ··
2 ··
show both fractions. (DOK 1)
STEP BY STEP • Ask students to solve the problems individually. Be sure to remind students to read the hints on the sides of each page. Ask students to underline the question in each problem to help them keep in mind whether the problem asks for who or what is more or less. Point out that students should draw a model or use number lines to show their work or thinking. • When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.
10 Solution: Rob ate more. 3 . 3 ; Students may draw 4 ··
6 ··
two equal wholes and shade in 3 and 3 on the 6 ··
4 ··
models or draw a number line and show the two fractions. (DOK 1) 11 Solution: C; Since the denominators are all the same, identify the fraction that has a numerator greater than 5. Explain to students why the other two answer choices are not correct: B and D are not correct because they are both less
SOLUTIONS
than 5 . The question asks for a fraction that is
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 3 . 1 .
greater than 5 . (DOK 3)
4 ··
8 ··
8 ··
4 ··
L19: Use Symbols to Compare Fractions
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Part 4: Common Core Practice
Part 4: Common Core Practice
Lesson 19
Lesson 19
Part 4: Common Core Practice
Solve the problems.
4
Lesson 19
Look at the comparison below. ,3
1
4 ··
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 8 in the denominator. What is one fraction that Tyrone could have used?
.1
Possible student work using models:
2 ··
A B C D
Show your work.
2 4 ·· 4
8 ·· 2
3 ··
1 , 2 , 3 , 4 , 5 are all possible answers.
8 ·· 8 ·· 8 ·· 8 ·· 8 ··
2
6 ··
Answer
2
Tyrone could have put 5 in the blank to make the comparison true. 8 ··
Shade the rectangles below to represent the given fractions. Then use your diagrams to help you complete the statement below with ,, ., or 5.
5
1 4 2 8
Tran and Noah were each given the same amount of clay in art class. Tran divided his clay into 3 equal pieces. He used 2 of the pieces to make a bowl. Noah divided his clay into 4 equal pieces. He also used 2 of the pieces to make a bowl. Tran said that he had 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 1 4
3 ··
2 8
=
4 ··
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. 6
8
1
3 8
,
3
6
4
Sample answer shown.
8
Self Check Go back and see what you can check off on the Self Check on page 131. 174
L19: Use Symbols to Compare Fractions
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4 Solution: Possible answers are 1 , 2 , 3 , 4 , or 5
AT A GLANCE
8 ·· 8 ·· 8 ·· 8 ··
Students use symbols to compare fractions to answer questions that might appear on a mathematics test.
8 ··
because, when shaded, these fractions cover less space than 3 does in the model; Students shade in 4 ··
the 3 and choose a fraction to shade that has less 4 ··
SOLUTIONS
than 6 , which would cover the same amount as 3 on 8 ··
1 Solution: C; Students understand that they are looking for a fraction greater than 1 . They see that 2 ··
the other fractions are either equal to or less than 1 . 2 ··
(DOK 1)
4 ··
the model and would be equal to, not less than, 3 . 4 ··
(DOK 2) 5 Solution: No; Tran used 2 of his clay and Noah 3 ··
used 2 of his clay. 2 . 2 ; Explanations will vary.
2 Solution: See completed diagram on student page above; 1 5 2 because 1 shaded section in the top 4 ··
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4 ··
3 ··
4 ··
(DOK 2)
8 ··
rectangle covers the same area as 2 shaded sections in the bottom rectangle. (DOK 1) 3 Solution: Possible answer: 3 , 6 ; The denominators 8 ··
8 ··
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)
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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 3 of his juice and Luke had 8 ··
drunk 3 of his juice. Who drank the most juice? Use symbols to show who drank more. 6 ··
• 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 2 of a worm. The male bird ate 2 of the same size 6 ··
worm. Which bird ate more? Use symbols to compare the fractions.
8 ··
If the error is . . .
Students may . . .
To remediate . . .
the student said
be confused about the meaning of the . and , symbols.
Help students figure out a strategy for remembering what each symbol means that makes sense to them. Give them practice writing comparisons using symbols and reading comparisons.
that Ty drank more and wrote 3 > 3 8 ··
6 ··
Hands-On Activity
Challenge Activity
Practice using symbols to compare.
Practice using symbols to compare.
Materials: Ten note cards with one of these fractions written on each card 4 , 2 , 2 , 3 , 3 , 4 , 2 , 3 , 4 , 3 , 6 ·· 4 ·· 8 ·· 8 ·· 6 ·· 8 ·· 6 ·· 4 ·· 4 ·· 3 ·· three note cards with one of these symbols on each: ,, ., 5, and pencils and paper for each student. Give each pair one set of fraction cards and one set of symbol cards.
Materials: Ten note cards with one of these fractions written on each card 8 , 8 , 8 , 4 , 4 , 4 , 2 , 2 , 2 , 2 , 2 ·· 8 ·· 4 ·· 1 ·· 4 ·· 2 ·· 4 ·· 1 ·· 2 ·· 8 ·· three note cards with one of these symbols on each: ,, ., 5, and pencils and paper for each student. Give each pair one set of fraction cards and one set of symbol cards.
The students shuffle the deck of fraction cards, and each student takes a card and lays it down with the fraction face up. Students choose a symbol card to place between the two fraction cards to make a true statement. They write the statement on their paper and read it aloud. They then switch the order of the two fractions and use a different symbol to correctly compare the fractions. They write the comparison 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 3 3 ·· and 4 , students can still switch the cards, but the 4 ·· symbol card will remain the same.
This activity builds upon the Hands-On Activity and gives student practice comparing fractions equal to and greater than one whole in addition to practice comparing fractions less than one whole. Students use the same procedure as the hands on activity: they use the symbol cards to create a comparison between the fraction cards, and they write the comparison in two ways by switching the order of the fractions and completing the new comparison with the correct symbol. In addition to writing down each comparison statement, students read each statement aloud.
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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 Content Emphasis in the Common Core Standards Connecting with the Ready® Teacher Toolbox Using i-Ready® Diagnostic with Ready Common Core Features of Ready Common Core Instruction Supporting Research
A14 A16 A18 A20 A22 A38
Correlation Charts Common Core State Standards Coverage by Ready Instruction Interim Assessment Correlations
A42 A46
Lesson Plans (with Answers) CCSS Emphasis Unit 1: Operations and Algebraic Thinking, Part 1 Lesson 1
Understand the Meaning of Multiplication
1 3
M
11
M
23
M
33
M
41
M
49
M
59
M
CCSS Focus - 3.OA.A.1 Embedded SMPs - 2–4
Lesson 2
Use Order and Grouping to Multiply
CCSS Focus - 3.OA.B.5 Embedded SMPs - 2–4, 7
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
67
Unit 2: Number and Operations in Base Ten Lesson 8
Use Place Value to Round Numbers
CCSS Emphasis 70 72
S/A
82
S/A
94
S/A
CCSS Focus - 3.NBT.A.1 Embedded SMPs - 1, 5–8
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 Lesson 11 Solve One-Step Word Problems Using Multiplication and Division
103 106 109
M
121
M
131
M
CCSS Focus - 3.OA.A.3 Embedded SMPs - 1–4, 7
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 Lesson 14 Understand What a Fraction Is
141 144 146
M
154
M
162
M
170
M
182
M
190
M
CCSS Focus - 3NF.A.1 Embedded SMPs - 2–4, 6
Lesson 15 Understand Fractions on a Number Line CCSS Focus - 3.NF.A.2a, 2b Embedded SMPs - 2, 3, 7
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
199
CCSS Emphasis
Unit 5: Measurement and Data
202
Lesson 20 Tell and Write Time
207
M
215
M
225
M
CCSS Focus - 3.MD.A.1 Embedded SMPs - 1, 3, 4, 6
Lesson 21 Solve Problems About Time CCSS Focus - 3.MD.A.1 Embedded SMPs - 1, 3–6
Lesson 22 Liquid Volume CCSS Focus - 3.MD.A.2 Embedded SMPs - 2, 4, 6
Lesson 23 Mass
235 M
CCSS Focus - 3.MD.A.2 Embedded SMPs - 1, 2, 4–6
Lesson 24 Solve Problems Using Scaled Graphs
245
S/A
255
S/A
265
S/A
275
M
283
M
293
M
303
S/A
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
315 318 320
S/A
328
S/A
338
S/A
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
346
Focus on Math Concepts
Lesson 18
Understand Comparing Fractions LESSON OBJECTIVES
THE LEARNING PROGRESSION
• Understand that in order to compare two fractions, students must reason about the size of unit fractions shown by the denominators and number of parts shown in the numerator in each fraction.
In this lesson, students apply their understanding of fractions to compare two fractions. They use fraction models and number lines to help them reason about the size of unit fractions.
• Analyze the numerators and denominators in fractions to be compared to determine if the fractions have the same numerators or denominators.
Students learn to look carefully at the numerators and denominators of both fractions they are comparing to determine if the numerators or the denominators are the same. If the denominators are the same, the students know the fractions are built from the same size unit fraction and the fraction with the most parts in the numerator is larger. If the numerators are same, then students reason about the size of the unit fractions used to make each of the two fractions.
• Explain why one fraction is smaller or larger when comparing two fractions using models or number lines.
PREREQUISITE SKILLS • Understand the meaning of fractions. • Identify fractions represented by models and number lines. • Understand that the size of a fractional part is relative to the size of the whole.
Students apply this understanding to solve problems that involve comparing fractions and explain why one fraction is larger or smaller. Teacher Toolbox
• Identify equivalent fractions and explain why they are equivalent.
Prerequisite Skills
VOCABULARY
Ready Lessons
There is no new vocabulary. Review the following key terms.
Tools for Instruction
unit fraction: a fraction with a numerator of 1; other fractions are built from unit fractions
Teacher-Toolbox.com
Interactive Tutorials
3.NF.A.3d
✓ ✓✓ ✓✓
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 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. d. Compare 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)
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L18: Understand Comparing Fractions
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Part 1: Introduction
Lesson 18
AT A GLANCE
Focus on Math Concepts
Students explore what it means to compare fractions. They compare fractions with the same denominators using models and number lines.
Lesson 18
Part 1: Introduction
CCSS 3.NF.A.3d
Understand Comparing Fractions
How do we compare fractions?
STEP BY STEP
When you compare fractions, you figure out which is smaller, which is larger, or if they are the same size.
• Remind students that they’ve compared whole numbers such as 345 and 509 to see which is smaller, larger, or if the two numbers are the same.
You can use models or number lines to help you compare two fractions. 1 4
• Introduce the Question at the top of the page. Ask 4 ··
1
4 ··
If not, it might look like 1 is
[
greater than 2 .
4 ··
4 ··
Think Sometimes when you compare fractions, the denominators are the same.
[
The models below show two wholes that are the same size divided into sixths.
Circle the model of the fraction that is less.
Think about how many unit fractions it takes to make each fraction you are comparing.
and is farther away from 0. Direct students’ attention to the two different size wholes that show 1 and 2 .
1 6
4 ··
Emphasize that when you compare two fractions, the
1 6
1 6
1 6
1 6
1 6
1 6
It takes two 1 s to make 2.
It takes five 1s to make 5. 6 6 6 6 ·· ·· ·· ·· 2 is made of fewer unit fractions than 5 . So, 2 is less than 5 . 6 6 6 6 ·· ·· ·· ··
size of the wholes must be the same. • Have a student to read the Think statement aloud.
3 4
the same to compare fractions.
explain how they know. 2 covers more of the square 4 ·· than 1 . Asks students to use the number line to 4 ·· explain why 2 is larger than 1 . 2 has more fourths 4 4 ·· 4 ·· ·· 4 ··
2 4
The size of the wholes must be
two square fraction models, tell which is larger, and
]
1 4
Both of these show that 1 is less than 2 .
students to identify the two fractions shown by the
]
0
2 4
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L18: Understand Comparing Fractions
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Ask students what two fractions are shown by the models. 3 2 and 5 4 Write the two fractions on the 6 ··
6 ··
board. Remind students what a unit fraction is and explain that each of the fractions was made up of the unit fraction 1 . Explain that you can use unit 6 ··
fractions and fraction models to show if a fraction is larger or smaller. The fraction model shows that
Hands-On Activity Use fraction bars to compare two fractions.
it takes more copies of 1 to make 5 than to make
Materials: fraction bars or circles, blank paper
only 2 .
Have students work in pairs and use a blank paper
6 ··
6 ··
6 ··
Mathematical Discourse • What is the same about the two fractions? The denominators are the same size. • What is different about the two fractions? They are built with a different number of unit fractions or they have a different numerator.
L18: Understand Comparing Fractions
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as their work space. Ask each pair to put the one whole fraction bar at the top of their paper. Instruct one student in the pair to build the fractions 2 and 4 5 ··
and have the other point to the larger fraction and
5 ··
explain why it is larger. Ask a student to share why. Do the same to compare the fractions 3 and 5 , 3 and 6 .
8 ··
8 ·· 10 ··
10 ··
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Part 1: Introduction
Lesson 18
AT A GLANCE Students explore ways to compare fractions that have the same numerators, but different denominators.
Part 1: Introduction
Lesson 18
Think You can compare fractions with like numerators and different denominators. Think about two different unit fractions from the same
STEP BY STEP
It’s like cutting up a piece of paper. The more pieces you cut the paper into, the smaller each piece is.
whole, such as 1 and 1 . 3 ··
• Read the Think statement together. On the board, write the two fractions 1 and 1 shown by the models 3 ··
8 ··
on the page. Circle the numerators in each fraction.
1 3
8 ··
1 8
Compare the denominators of 1 and 1 . 3 is less than 8, so 3 ··
parts, each part is bigger. So, the unit fraction 1 is greater 3 ··
Explain that they will next explore how to compare
than 1.
two fractions in which the numerators are the same
Here’s another example:
and the denominators are different. • Write the two fractions 3 and 3 on the board and 6 ··
4 ··
8 ··
the whole is divided into fewer parts. Since there are fewer
8 ··
1 6
1 6
1 6
1 4
1 4
1 4
The unit fractions used to
The unit fractions used to
make 3 are smaller.
make 3 are bigger.
6 ··
4 ··
circle the numerators. Ask students to picture a
3 smaller parts are less than 3 bigger parts. So, 3 is less than 3 .
rectangle divided into 6 equal parts and one divided
Reflect
into 4 equal parts. Direct students attention to the models of 3 and 3 on the page. Make the point that 6 ··
4 ··
each 1 is smaller, so 3 of those 1 s will cover less of 6 ··
6 ·· the model than will 3 of the 1 s. 4 ··
• Ask students to work with a partner to answer the Reflect question in their own words and invite students to share.
Hands-On Activity these fractions on a blank sheet of paper: 1 , 1 , 1 , 1 , 1 , and 1 . Instruct them to 10 ··
label each unit fraction. To help students understand that the smaller the fractional piece, the larger the denominator is, ask questions such as: What do you notice about the size of the unit fractions as the denominator gets larger? How many tenths does it take to make one whole? One-half? Why does it take more tenths to make one whole? Then ask them to show 3 and 3 . Ask: Which 8 ··
fraction is larger and Why?
184
4 ··
1 Explain how you can use unit fractions to help you compare fractions.
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
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Mathematical Discourse
• Ask students to use fraction tiles or bars to lay
2 ·· 4 ·· 5 ·· 6 ·· 8 ··
6 ··
• What is the same about the two fractions 3 and 3 ? 6 4 ·· ·· 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.
4 ··
L18: Understand Comparing Fractions
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Part 2: Guided Instruction
Lesson 18
AT A GLANCE Part 2: Guided Instruction
Students use models to compare two fractions that either have the same denominators or the same numerators.
Lesson 18
Explore It Use the models to help you compare fractions with the same denominator. 2 Write the fraction shaded below each model.
Circle the fraction that is greater.
STEP BY STEP
2
3 ··
3 Write the fraction shaded below the first
model. Shade the second model to show a greater fraction. Write the greater fraction.
4
Possible answers: 5 , 6 , 7 , 8 ; shading will vary.
8 ··
8 ·· 8 ·· 8 ·· 8 ··
Use the models to help you compare fractions with the same numerator. 4 Write the fraction shaded below each model.
Circle the fraction that is greater.
• As students work individually, circulate among them. This is an opportunity to assess student understanding and address student misconceptions. Use the Mathematical Discourse questions to engage student thinking.
1
1
6 ··
2 ··
5 Write the fraction shaded below each model.
Circle the fraction that is less. 2
2
6 ··
3 ··
6 Write the fraction shaded below the first
• When students have completed the page, ask them to work with a partner and return to each problem. Instruct them to cover up the models and to picture the fractions in their mind. Then have them practice explaining why one of the fractions is greater. • Discuss each problem. Ask for volunteers to explain
1
3 ··
• Tell students that they will have time to work individually on the Explore It problems on this page and then share their responses in groups. You may choose to work through the first problem together as a class.
model. Shade the second model to show a fraction that is less but has the same numerator.
5
6 ··
Explain how you know the fraction is less. Possible answer: The same number of parts are shaded in both models. The 1 parts are smaller than the 1 parts, so 5 is less than 5 . 8 ··
164
6 ··
8 ··
6 ··
L18: Understand Comparing Fractions
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how they figured out which fraction was larger or smaller. For problem 6, ask for all the possible ways students could shade the model so the fraction is less than 5 . 6 ··
• Take note of students who are still having difficulty and wait to see if their understanding progresses as they work in their groups during the next part of the lesson.
SMP Tip: Encourage students to picture fractions in their minds based upon experiences with fraction models. This practice helps students conceptualize as they work more symbolically with fractions later on (SMP 2).
Concept Extension • Students may not be familiar with the way each 1 8 ··
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
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Part 2: Guided Instruction
Lesson 18
AT A GLANCE Students revisit the problem on page 164 and explain how to use unit fractions to compare the fractions. They also explore using number lines to compare fractions.
Part 2: Guided Instruction
Lesson 18
Talk About It 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 • Organize students into pairs or groups and ask them to answer the Talk About It questions as a group. • Walk around to each group, listen to, and join in on discussions at different points. Use the Mathematical Discourse questions to help support or extend students’ thinking. • Be sure students are able to describe how the problems are different on the page. Fractions have the same number of whole equal parts if the denominators are the same. • Direct the group’s attention to Try It Another Way. Draw the number lines on the board for problems 10 and 11. Have volunteers from the groups come up to the board and use the number lines to explain how they figured out which fraction was greater or less.
compare fractions with the same denominator. Possible answer: The unit fractions are the same. Look at the numerators to see how many unit fractions are in each fraction. Compare these numbers.
8 Look at your answers to problems 4–6. What is different about the numerators
and denominators in these fractions than the fractions in problems 2 and 3? Possible answer: The fractions in problems 2 and 3 have the same denominators and different numerators. The fractions in problems 4–6 have the same numerators and different denominators. Explain how to use unit fractions to compare the fractions with the same numerator. Possible answer: You have to look at the size of the unit fraction, since the number of unit fractions in each is the same. The one with the smaller unit fraction is less. 9 Isaiah is comparing 3 and 3 . Both fractions have a numerator of 3. How can he tell 8 6 ·· ·· Possible answer: He knows that 1 s are smaller which fraction is less? 8 ··
than 1 s. He has the same number of 1 s and 1 s, so the fraction made of 1 s 6 ··
8 ··
is smaller. 3 is less than 3 . 8 ··
6 ··
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
lines. Circle the fraction that is less. 0
3 8
1
0
3 4
11 Look at the fractions on the number
lines. Circle the fraction that is greater. 1
0
4 6
1
L18: Understand Comparing Fractions
SMP Tip: Problem 8 gives students practice
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0
3 6
1
165
looking at the structure of the fractions to be compared. Students notice whether the numerators or denominators are the same or different, and they identify two possible kinds of problems they will encounter when they compare fractions (SMP 7).
Concept Extension • Emphasize that two fractions must have some part that is the same in order to compare them. The first thing students need to do is find which part is the same. If the denominators are the same, that means the unit fractions that make up the whole are the same. They can look at the numerator to see how many copies of the unit fractions there are. The more copies, the greater the fraction. If the numerators are the same, that means the number of unit fraction copies is the same. The denominator tells the size of each unit fraction. The more unit fractions that make up a whole, the smaller each unit fraction is.
186
Mathematical Discourse • What do you look at first when you are comparing two fractions? Look to see if either the denominators or numerators are the same. • Which fractions seem easiest for you to compare, those that have the same numerators or those that have the same denominators? Explain why. Answers will vary. Have students explain their answers.
L18: Understand Comparing Fractions
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Part 3: Guided Practice
Lesson 18
AT A GLANCE Part 3: Guided Practice
Students demonstrate their understanding of comparing two fractions that have either the same numerators or the same denominators.
Lesson 18
Connect It Talk through these problem as a class, then write your answers below. 12 Create: Draw an area model or number line to show
same denominator that is less than 5 .
STEP BY STEP
5 . Find a fraction with the
8 ··
8 ··
Possible student work:
• Discuss each Connect It problem as a class using the discussion points outlined below.
3 8
0
Create:
5 8
1
Explain how you found your answer. Possible answer: I know that any fraction to the left of 5 is less than 5 . So, 3 is less than 5 . 8 ··
• You may choose to have students work in pairs to encourage sharing ideas. Each partner draws a different model.
8 ··
8 ··
8 ··
13 Explain: Mario painted 2 of the wall in his bedroom. Mei Lyn painted 2 of a wall in 6 4 ·· ··
her bedroom. Both walls are the same size. Explain how you know who painted more. Possible answer: Each painted the same number of parts of the wall. Sixths are smaller than fourths, so 2 sixths is less than 2 fourths. That
• For a quick and easy assessment, have students draw their models on small whiteboards or on paper and hold them up. Choose several pairs to explain their models to the class.
means Mei Lyn painted more. 14 Justify: Jace and Lianna each baked a loaf of bread. Jace cut his in halves and
Lianna cut hers in thirds. 1 2
• Use the following to lead a class discussion:
1 3
Jace says they can use their loaves of bread to show that 1 is less than 1 . Lianna 2 ··
3 ··
says they can’t. Who is correct? Explain why.
What did you keep in mind as you thought of a fraction to choose that was less than 5 ? 8 ··
Possible answer: Lianna is correct. The wholes have to be the same size to compare fractions.
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L18: Understand Comparing Fractions
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How can you prove that the fraction you chose is less than 5 ?
Justify:
Is there another way to use a number line(s) to show a smaller fraction? [Students may use two number lines or show both fractions on the same number line.]
• This discussion shows whether students understand that in order to compare two fractions, the wholes must be the same size.
8 ··
Explain: • This problem expects students to look at the fractions and picture the size of the unit fractions, reasoning that sixths are smaller than fourths, so Mei Lyn must have painted more. • Begin the discussion by asking questions such as: What do you picture each wall looking like? Does each wall have 2 parts painted? Which 2 parts cover more area of the wall? How do you know?
• 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 1 and 1 . Be sure 2 ··
3 ··
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 1 is smaller than 1 . 3 ··
L18: Understand Comparing Fractions
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2 ··
187
Part 4: Common Core Performance Task
Lesson 18
AT A GLANCE Part 4: Common Core Performance Task
Students use information in fraction form to create a problem that involves comparing fractions. They create a model to help them solve the problem.
Lesson 18
Put It Together 15
Mrs. Ericson made sandwiches for her 4 children. Each sandwich was the same size. After lunch, each child had a different fraction of his or her
STEP BY STEP
sandwich left. Matt had 1 , Elisa had 3 , Carl had 3 , and Riley had 7 . 4 ··
• Direct students to complete the Put It Together task on their own.
8 ··
4 ··
8 ··
A Use this information to write a problem that compares two fractions with the same numerator. Who had more left, Elisa or Carl?
• Go over the directions with students and let them know they have some choices to make. Be sure students understand that they should circle the problem that they will model using either a fraction model or number line.
B Use this information to write a problem that compares two fractions with the same denominator. Who had more left, Matt or Carl?
C Choose one of your problems to solve. Circle the question you chose. Draw a model or number line to help you find the answer.
• As students work on their own, walk around to assess their progress and understanding, to answer their questions, and to give additional support, if needed.
Explain how you could use unit fractions to think about the problem. Possible answer: The unit fractions for Matt’s and Carl’s fractions were both 1 . Matt had one 1 and Carl had three 1 s. 3 is greater than 1, so 4 ··
4 ··
4 ··
Carl had more left.
SCORING RUBRICS A
Points Expectations 2
188
The student correctly chooses two fractions to compare that either have the same numerators or the same denominators. The question for the problem asks which student had more or less of his or her sandwich left.
1
The student correctly chooses two fractions to compare that either have the same numerators or same denominators. The question asks which fraction is larger or smaller, but does not clearly relate to the context of the problem (students and the amount of sandwich left).
0
The student does not choose two fractions to compare that have the same numerators or same denominators and is unable to ask a question that relates to comparing fractions of sandwiches left.
L18: Understand Comparing Fractions
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B
167
Points Expectations 2
The model or number line clearly shows both fractions that are compared. The explanation includes who had the most 1 s 4 ·· (or 1 s) or explains that Elisa and Carl had the 8 ·· same number of parts and that Carl’s unit fractions were larger.
1
The model or number line clearly shows both fractions that are compared. The explanation does not clearly explain how the student used unit fractions to see who had the least or the most left.
0
The model did not correctly compare two 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.
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 thinking about the size of each unit fraction. Have them follow these steps for comparing 2 and 2 . 3 ··
8 ··
• Draw two rectangles that are the same size. Show thirds in one and eighths in the other. • Label each unit fraction in each of the models. • Ask which unit fraction is larger. [thirds] • Shade in each fraction in the models. • Think, “Each 1 is larger than 1 , so 2 must be 3 ··
larger than 2 .”
8 ··
3 ··
8 ··
Materials: 10 blank note cards. Write one of these fractions on each card: 2 , 2 , 2 , 2 , 3 , 3 , 3 , 4 , 4 , 4 4 ·· 5 ·· 6 ·· 8 ·· 8 ·· 6 ·· 4 ·· 5 ·· 8 ·· 4 ··
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. 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: 8 , 8 , 8 , 8 , 4 , 4 , 2 , 4 , 6 , 3 2 ·· 8 ·· 4 ·· 1 ·· 2 ·· 4 ·· 1 ·· 1 ·· 3 ·· 3 ··
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
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189
Develop Skills and Strategies
Lesson 19
Use Symbols to Compare Fractions LESSON OBJECTIVES
THE LEARNING PROGRESSION
• Use symbols to record the results of comparing fractions.
In this lesson students continue to compare two fractions by reasoning about size of unit fraction shown in models and on number lines. The focus of this lesson is on recording the results of comparisons by using symbols to write comparison statements.
• Read comparison statements fluently and accurately. • Use models and number lines to explain and justify fraction comparisons.
PREREQUISITE SKILLS • Understand the meaning of fractions. • Identify fractions represented by models and number lines. • Understand that the size of a fractional part is relative to the size of the whole. • Identify equivalent fractions and explain why they are equivalent.
Students see that comparing fractions also means they must look to see if two fractions are equal. If two fractions are equal, they record it using the equal symbol. Students review or learn the meaning of the , and . symbols and practice using them to record their comparisons. Students solve problems that involve comparisons. Students’ concrete experiences with comparing fractions and recording them using symbols prepares them to move on to more abstract work with comparing fractions and fraction operations in future grades.
• Understand that in order to compare two fractions, students must reason about the size of unit fractions shown by the denominators and number of parts shown in the numerator in each fraction.
Teacher Toolbox Prerequisite Skills
VOCABULARY
Ready Lessons
There is no new vocabulary. Review the following key terms.
Tools for Instruction
unit fraction: a fraction with a numerator of 1; other fractions are built from unit fractions
Teacher-Toolbox.com
Interactive Tutorials
3.NF.A.3d
✓ ✓✓ ✓✓
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 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. d. Compare 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)
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Part 1: Introduction
Lesson 19
AT A GLANCE
Develop Skills and Strategies
Students read a comparison word problem and answer a series of questions that help them connect comparing fractions using models to recording comparisons using symbols.
Lesson 19
Part 1: Introduction
CCSS 3.NF.A.3d
Use Symbols to Compare Fractions
In Lesson 18, you learned how to compare fractions. Take a look at this problem. Erica’s cup is 4 full. Ethan’s cup is 5 full. Use ,, ., or 5 to compare 4 and 5 . 6 ··
STEP BY STEP • Tell students that this page reviews how to use models to compare two fractions and demonstrates how to use symbols to record the comparisons.
6 ··
6 ··
Explore It Use the math you already know to solve the problem.
• Have students read the problem at the top of the page. Remind students that they have had practice with solving this kind of comparing problem previously.
The fractions have the same denominator. What do you need to think about to compare the two fractions? Possible answer: You need to think about how many unit fractions it takes to make each fraction.
4
How many sixths does Erica have?
• Work through the Explore It questions and directives as a class. • Write the symbol , on the board and underneath write the words “less than.” Write the symbol . on the board and underneath write the words “more than.” Make sure that students understand that the answers for the third and fourth bulleted question are whole numbers that tell the number of sixths that Erica and Ethan have.
6 ··
5
How many sixths does Ethan have?
Use a symbol to compare those two whole numbers. 4 , 5 Is the amount in Erica’s cup less than, greater than, or equal to the amount in Ethan’s’ cup? less than Explain how you can use a symbol to compare the two fractions. Since 4 sixths is less than 5 sixths, 4 , 5 . 6 ··
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6 ··
L19: Use Symbols to Compare Fractions
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• Ask students to practice with a partner how they would help another student write fractions and symbols to compare 4 and 5 . Ask for volunteers to 6 ··
6 ··
model what they would say.
Concept Extension • Try to give students many opportunities to think about new concepts independently and practice expressing them in their own words. This is an important step in the learning process. This is also helpful for students who need more practice with language skills and math vocabulary.
L19: Use Symbols to Compare Fractions
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Mathematical Discourse • How did you remember what the symbol , and . mean when you used them to compare whole numbers? Have students explain their answers. • Why might you need to use an equals sign when comparing two fractions. The fractions might be equal.
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Part 1: Introduction
Lesson 19
AT A GLANCE Students review ways to remember the meaning for the , and . symbols and use the symbols to record comparisons.
STEP BY STEP
Part 1: Introduction
Lesson 19
Find Out More 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. , means less than
• Read Find Out More as a class. Write the , and . symbols on the board. Explain how to visualize the
. means greater than
5 means equal to
Think of the , and . symbols as the mouth of an alligator. The alligator’s mouth will always be open to eat the greater fraction. Think about the fractions 12 and 18 . 12 is greater. 18 is less. ··
1
1
or
1 is greater than 1 8 ··
or
two symbols by thinking that the symbols look like
2 ··
8 ··
·· ··
1
··
1
8 ··
2 ··
1 is less than 1 . 2 ··
the mouth of an alligator that points to the greater
2 ··
number. Write 1 . 1 and 1 , 1 on the board. Circle
You can switch the order of the fractions. Just be careful which symbol you use. If the greater fraction is first, you use .. If the greater fraction is last, you use ,.
2 ··
8 ··
8 ··
2 ··
the . sign in the first problem. Tell the class that if the first thing you see is the open mouth, then you know it means “greater than.” If the first thing you see is the point or smaller part of the symbol, then the symbol means “less than.” Invite students to share other strategies to help them remember the
8 ··
Also, remember that sometimes one fraction is not greater than the other. Sometimes they are equivalent. Then you use 5 to compare them. 151 2 ··
757 8 ··
and
2 ··
8 ··
Reflect 1 Use the symbols , and . to compare 7 and 3 . Explain your answers. 8 8 ·· ·· 7 3 3 7
8 ··
. and , 8 ··
8 ··
8 ··
Possible explanation: There are more 1 s in 7 than in 3 . So, 7 is greater than 3 and 3 is less than 7 . 8 ··
8 ··
8 ··
8 ··
8 ··
8 ··
8 ··
meaning of the symbols. • Write 1 5 1 and 7 5 7 on the board. Point out that 2 ··
2 ··
8 ··
8 ··
comparing also means looking to see if fractions are the same. • For a quick assessment, dictate the fraction comparisons from Reflect to students and have them practice writing what they hear using symbols. Have them write their responses on a piece of paper or small white board they can hold up. Have students share reasoning for why one fraction is less than or greater than the other. • Ask students to work again with a partner to write two ways to compare the fractions in Reflect and practice explaining why the first fraction listed is greater or less than the second fraction. Ask for volunteers to share their explanations.
SMP Tip: Students look at the , and . symbols to connect the structure of the symbols to their meaning in mathematical expressions. Students who connect the visual representation to the symbolic meaning learn to read comparison sentences with ease (SMP 7).
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Concept Extension • Students often need help with instant recall to cement what the , and . symbols mean and to promote quick and easy reading of comparisons. To help students, remind them that if they see the large or open part of the mouth first when reading the symbol, they know to say “greater than.” If they see the small or closed part of the sign first when reading the symbol, they know to say “less than.”
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.
L19: Use Symbols to Compare Fractions
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Part 2: Modeled Instruction
Lesson 19
AT A GLANCE Part 2: Modeled Instruction
Students review how to compare fractions that have the same numerators. They use symbols to record the comparisons.
Lesson 19
Read the problem below. Then explore different ways to compare fractions. Use ,, ., or 5 to compare 4 and 4 . 8 ··
STEP BY STEP
6 ··
Picture It You can use models to help you compare fractions.
• Explain to students that they will now use models 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
You can also use number lines to help you compare fractions.
4 and 4 and ask them to figure out which fraction 8 ··
This number line is divided into eighths. It shows 4 .
6 ··
8 ··
is greater.
0
4 8
1
This number line is divided into sixths. It shows 4 .
• For a quick review, ask students to raise their thumb up if their answer is yes, put their thumb down if the answer is no and sideways if unsure. Then ask these questions: Are both models the same size? [yes] Are the unit fractions the same size in each model? [no]
6 ··
0
4 6
1
Is 4 larger than 4 ? [yes] Ask students how they know. 8 ··
6 ··
Is 4 larger than 4 ? [no] Ask students how they know. 6 ··
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6 ··
• Use the same process to review using the number line to compare the two fractions. Ask these questions: Are both number lines the same length? [yes] Do both number lines show the same fractions? [no] Is the fraction 4 in the center of the number line? [yes] 8 ··
Ask students how they know. Is the fraction 4 closer to zero than 4 ? [no] Ask 6 ··
students how they know.
8 ··
Is 4 larger than 4 ? [yes] Ask students how they know. 6 ··
8 ··
Real-World Connection Ask students to think of a situation in the real world when they might need to compare two fractions. Have them share. An example: One store says the sale price for shoes is 1 off full price. Another store 2 ··
says the sale price is 1 off full price. 4 ··
L19: Use Symbols to Compare Fractions
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Concept Extension Practice reading fraction comparisons. • Write several comparisons on the board, such as 5 . 2 , 3 . 1 , 3 , 7 , 1 , 3 . Instruct students to 6 ··
6 ·· 4 ··
4 ·· 8 ··
8 ·· 3 ··
3 ··
work with a partner to practice reading the comparison number sentences correctly. Then have students switch the fractions and symbols and practice saying each with a partner. As students practice in pairs, walk around the room and give support as needed.
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Part 2: Guided Instruction
Lesson 19
AT A GLANCE Students revisit the problem on page 170 and use symbols to compare the fractions in the problem.
STEP BY STEP • Read and answer the Connect questions as a class. After students write their answer to problems 4 and 5, ask them explain how they knew which symbol to write. Invite students to share their explanations for each answer. Ask questions such as: Can anyone add to what said? Do you agree with what just said? Explain why. Does that make sense? Why? How do you know that is correct? • Organize students into small groups. Ask them to complete the Try It problems. Assign each group one or two of the problems to present to the class to share how they compared each fraction and decided on which symbol to use.
Part 2: Guided Instruction
Lesson 19
Connect It Now you will solve the problem from the previous page using symbols. 2 Explain how you can use the model to compare the fractions.
Possible answer: The models are the same size. The area shaded on the 4 model is less than the area shaded on the 4 model. 8 ··
6 ··
3 Explain how you can use the number lines to show how the fractions compare.
Possible answer: 4 is closer to 0 than 4 is, so it is less. 8 ··
6 ··
4 Write the comparison:
less than
using words: 4 eighths is , using symbols: 4
than 4 sixths.
4
8 ··
6 ··
5 Now switch the order of the fractions. Write the comparison:
using words: 4 sixths is greater than than 4 eighths. using symbols: 4
6 ··
4
.
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.
SMP Tip: Try to give students many opportunities to critique or comment on other students’ thinking or explanations. In addition, ask questions that promote students’ justifying their own explanations (SMP 3).
4
6 ··
.
2
6 ··
2
4 ··
,
2
1
3 ··
2 ··
5
1
2 ··
8 Use ,, ., or 5 to compare each set of fractions. Each symbol will be used once.
3
4 ··
5
3
2
4 ··
8 ··
,
2
2
2 ··
3 ··
.
1
3 ··
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TRY IT SOLUTIONS 7 Solution: 4 . 2 , 2 , 2 , 1 5 1 ; Students may write 6 ··
6 ·· 4 ··
3 ·· 2 ··
2 ··
the correct symbol, but read it incorrectly. 8 Solution: 3 5 3 , 2 , 2 , 2 . 1 ; Students may have 4 ··
4 ·· 8 ··
2 ·· 3 ··
3 ··
drawn models on a separate piece of paper or used reasoning (and no models) to compare the fractions.
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Part 3: Guided Practice
Lesson 19
Part 3: Guided Practice
Lesson 19
Study the models below. Then solve problems 9–11. Student Model
The fractions have the same denominator, so they are easy to compare on the same number line.
Part 3: Guided Practice
Lesson 19
10 David and Rob each got the same snack pack of crackers. David
ate 3 of his snack pack. Rob ate 3 of his snack pack. Who ate 6 ··
4 ··
Su and Anthony live the same distance from school. Su biked
more? Compare the fractions using a symbol.
3 of the way to school. In the same amount of time, Anthony
Show your work.
walked 1 of the way to school. Who went the greater distance?
Possible student work using models:
4 ··
4 ··
I think drawing a model might help. Be sure the wholes are the same size.
Compare the fractions using a symbol. Look at how you could show your work using a number line.
Pair/Share How do you find the greater number on a number line?
What do you need to think about when you compare fractions that have different denominators?
0
Solution:
1
4 ··
3
1
4 ··
Su went a greater distance. 3 . 1 4 ··
4 ··
Pair/Share
9 Julia and Mackenzie have the same number of homework
problems. Julia has done 1 of her problems. Mackenzie has 3 ··
Solution:
Which fraction is made of bigger unit fractions? Why?
Rob ate more. 3 . 3 4 ··
6 ··
done 1 of her problems. Which student has done less of her 2 ··
homework? Compare the fractions using a symbol. Show your work.
11 What number could go in the blank to make the comparison
true? Circle the letter of the correct answer. 5,
8 ··
Possible student work using models:
Is 5 less than or greater 8 ·· than the fraction that goes in the blank?
A 5
8 ··
B C
4
8 ·· 6
8 ··
D 1
8 ··
Blake chose A as the correct answer. How did he get that answer? He chose the one that was equal to 5 instead of looking for
Pair/Share How did you know which fraction was smaller?
172
one that was greater than 5 . Solution:
Julia has done less homework. 1 , 1 3 ··
8 ··
8 ··
2 ··
L19: Use Symbols to Compare Fractions
Pair/Share Does Blake’s answer make sense?
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9 Solution: Julie has done less homework. 1 , 1 ;
AT A GLANCE
3 ··
Students use symbols to compare fractions in problems and show their thinking using models or number lines.
2 ··
Students may draw two equal wholes and shade in 1 and 1 on the models or draw a number line and 3 ··
2 ··
show both fractions. (DOK 1)
STEP BY STEP • Ask students to solve the problems individually. Be sure to remind students to read the hints on the sides of each page. Ask students to underline the question in each problem to help them keep in mind whether the problem asks for who or what is more or less. Point out that students should draw a model or use number lines to show their work or thinking. • When students have completed each problem, have them Pair/Share to discuss their solutions with a partner or in a group.
10 Solution: Rob ate more. 3 . 3 ; Students may draw 4 ··
6 ··
two equal wholes and shade in 3 and 3 on the 6 ··
4 ··
models or draw a number line and show the two fractions. (DOK 1) 11 Solution: C; Since the denominators are all the same, identify the fraction that has a numerator greater than 5. Explain to students why the other two answer choices are not correct: B and D are not correct because they are both less
SOLUTIONS
than 5 . The question asks for a fraction that is
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 3 . 1 .
greater than 5 . (DOK 3)
4 ··
8 ··
8 ··
4 ··
L19: Use Symbols to Compare Fractions
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Part 4: Common Core Practice
Part 4: Common Core Practice
Lesson 19
Lesson 19
Part 4: Common Core Practice
Solve the problems.
4
Lesson 19
Look at the comparison below. ,3
1
4 ··
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 8 in the denominator. What is one fraction that Tyrone could have used?
.1
Possible student work using models:
2 ··
A B C D
Show your work.
2 4 ·· 4
8 ·· 2
3 ··
1 , 2 , 3 , 4 , 5 are all possible answers.
8 ·· 8 ·· 8 ·· 8 ·· 8 ··
2
6 ··
Answer
2
Tyrone could have put 5 in the blank to make the comparison true. 8 ··
Shade the rectangles below to represent the given fractions. Then use your diagrams to help you complete the statement below with ,, ., or 5.
5
1 4 2 8
Tran and Noah were each given the same amount of clay in art class. Tran divided his clay into 3 equal pieces. He used 2 of the pieces to make a bowl. Noah divided his clay into 4 equal pieces. He also used 2 of the pieces to make a bowl. Tran said that he had 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 1 4
3 ··
2 8
=
4 ··
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. 6
8
1
3 8
,
3
6
4
Sample answer shown.
8
Self Check Go back and see what you can check off on the Self Check on page 131. 174
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4 Solution: Possible answers are 1 , 2 , 3 , 4 , or 5
AT A GLANCE
8 ·· 8 ·· 8 ·· 8 ··
Students use symbols to compare fractions to answer questions that might appear on a mathematics test.
8 ··
because, when shaded, these fractions cover less space than 3 does in the model; Students shade in 4 ··
the 3 and choose a fraction to shade that has less 4 ··
SOLUTIONS
than 6 , which would cover the same amount as 3 on 8 ··
1 Solution: C; Students understand that they are looking for a fraction greater than 1 . They see that 2 ··
the other fractions are either equal to or less than 1 . 2 ··
(DOK 1)
4 ··
the model and would be equal to, not less than, 3 . 4 ··
(DOK 2) 5 Solution: No; Tran used 2 of his clay and Noah 3 ··
used 2 of his clay. 2 . 2 ; Explanations will vary.
2 Solution: See completed diagram on student page above; 1 5 2 because 1 shaded section in the top 4 ··
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4 ··
3 ··
4 ··
(DOK 2)
8 ··
rectangle covers the same area as 2 shaded sections in the bottom rectangle. (DOK 1) 3 Solution: Possible answer: 3 , 6 ; The denominators 8 ··
8 ··
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)
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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 3 of his juice and Luke had 8 ··
drunk 3 of his juice. Who drank the most juice? Use symbols to show who drank more. 6 ··
• 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 2 of a worm. The male bird ate 2 of the same size 6 ··
worm. Which bird ate more? Use symbols to compare the fractions.
8 ··
If the error is . . .
Students may . . .
To remediate . . .
the student said
be confused about the meaning of the . and , symbols.
Help students figure out a strategy for remembering what each symbol means that makes sense to them. Give them practice writing comparisons using symbols and reading comparisons.
that Ty drank more and wrote 3 > 3 8 ··
6 ··
Hands-On Activity
Challenge Activity
Practice using symbols to compare.
Practice using symbols to compare.
Materials: Ten note cards with one of these fractions written on each card 4 , 2 , 2 , 3 , 3 , 4 , 2 , 3 , 4 , 3 , 6 ·· 4 ·· 8 ·· 8 ·· 6 ·· 8 ·· 6 ·· 4 ·· 4 ·· 3 ·· three note cards with one of these symbols on each: ,, ., 5, and pencils and paper for each student. Give each pair one set of fraction cards and one set of symbol cards.
Materials: Ten note cards with one of these fractions written on each card 8 , 8 , 8 , 4 , 4 , 4 , 2 , 2 , 2 , 2 , 2 ·· 8 ·· 4 ·· 1 ·· 4 ·· 2 ·· 4 ·· 1 ·· 2 ·· 8 ·· three note cards with one of these symbols on each: ,, ., 5, and pencils and paper for each student. Give each pair one set of fraction cards and one set of symbol cards.
The students shuffle the deck of fraction cards, and each student takes a card and lays it down with the fraction face up. Students choose a symbol card to place between the two fraction cards to make a true statement. They write the statement on their paper and read it aloud. They then switch the order of the two fractions and use a different symbol to correctly compare the fractions. They write the comparison 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 3 3 ·· and 4 , students can still switch the cards, but the 4 ·· symbol card will remain the same.
This activity builds upon the Hands-On Activity and gives student practice comparing fractions equal to and greater than one whole in addition to practice comparing fractions less than one whole. Students use the same procedure as the hands on activity: they use the symbol cards to create a comparison between the fraction cards, and they write the comparison in two ways by switching the order of the fractions and completing the new comparison with the correct symbol. In addition to writing down each comparison statement, students read each statement aloud.
L19: Use Symbols to Compare Fractions
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