find part of a group using unit fractions

LESSON AT A GLANCE

487A Chapter 8

About the MathProfessional Development

LESSON AT A GLANCE

Find Part of a Group Using Unit Fractions

LESSON 8.8

Professional Development Videos

Modeling Part of a GroupFinding part of a group is a skill that will be used when multiplying a whole number by a fraction. Students will not learn how to multiply fractions in this grade, but the concepts in this lesson provide background for that procedure.

In this lesson, the problem of finding 1 _ 4 of a group of 12 can be modeled by a group of 12 objects that has been separated into 4 smaller groups. Each of the smaller groups has 3 objects. Students may see the connection between the arrays and equal groups they made when multiplying and dividing whole numbers.

Place 12 counters in 4 equal groups.

There are 3 counters in each group, so 1 __ 4 of 12 equals 3.

number of groups to count

number of groups

1 __ 4 of 12

total number of objects

Learning ObjectiveFind fractional parts of a group using unit fractions.

Language ObjectiveStudents model and explain how a fraction can tell how many are in part of a group.

MaterialsMathBoard, two-color counters

F C R Focus:Common Core State Standards3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b

as the quantity formed by a parts of size 1/b.

MATHEMATICAL PRACTICESMP4 Model with mathematics.MP5 Use appropriate tools strategically.

F C R Coherence:Standards Across the GradesBefore2.G.A.3

Grade 33.NF.A.1

After4.NF.A.1

F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency.......On Your OwnLevel 3: Applications..................................Think Smarter and Go Deeper

F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 441J.

FOCUS COHERENCE RIGOR

ENGAGE1

Lesson 8.8 487B

Daily RoutinesCommon CoreDaily RoutinesCommon Core

How can a fraction tell how many are in

part of a group?

with the Interactive Student Edition

Essential QuestionHow can a fraction tell how many are in part of a group?

Making ConnectionsAsk students to draw 6 flowers on a sheet of paper and color 1 flower purple, 3 flowers red, and 2 flowers yellow. Then ask the following questions:

What fraction of the flowers did you color purple? 1 __ 6    Why is the bottom number of your fraction 6? There are six flowers in all. What fraction of the flowers did you color red? 3 _ 6  

Learning ActivityWhat is the problem the students are trying to solve? Connect the story to the problem.

Ask the following questions.

• What does the fraction one third mean? There are 3 equal groups and 1 group is being counted. 

• If you separate 12 items into 3 equal groups, what fraction represents 1 group? 1 __ 3   

Literacy and MathematicsView the lesson opener with the students. Then, do one or more of the following activities at the end of the lesson:

• Have students write a problem involving using a unit fraction to find part of a group. Ask students to draw a picture that models their problem.

• Have students write the number 24 at the top of a sheet of paper. Have them use counters to find 1 _ 2 , 1 _ 3 , 1 _ 4 , 1 _ 6 and 1 _ 8 of 24. Have students record the results in a chart. Ask them to describe how they found each value.

Problem of the Day 8.8April’s class is collecting pennies for charity. They collect 574 pennies. What is the number of pennies they collect rounded to the nearest hundred?

Vocabulary• Interactive Student Edition• Multimedia Glossary e

about 600 pennies

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MATHEMATICAL PRACTICES 3MathTalk

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Chapter 8 487

Lesson 8.8

Find Part of a Group Using Unit FractionsEssential Question How can a fraction tell how many are in part of a group?

Audrey buys a bouquet of 12 flowers. One third of them are red. How many of the flowers are red?

ActivityMaterials ■ two-color counters ■ MathBoard

• Put 12 counters on your MathBoard.

• Since you want to find 1 _ 3 of the group, there should

be _ equal groups. Draw the counters below.

• Circle one of the groups to show _.

Then count the number of counters in that group.

There are _ counters in 1 group. 1 _ 3 of 12 = _

So, _ of the flowers are red.

• What if Audrey buys a bouquet of 9 flowers and one third of them are yellow? Use your MathBoard and counters to find how many of the flowers are yellow.

____

• How many flowers does Audrey

buy in all? ____• What fraction of the flowers are

red? ____

Number and Operations—Fractions—3.NF.A.1

MATHEMATICAL PRACTICESMP4, MP5

Apply How can you use the numerator and denominator in a fraction to find part of a group?

12 fl owers

1 _ 3

4

3 fl owers

4

1_3

3

4

Possible answer: the denominator

tells how many groups I divide the

total into. The numerator tells how

many of the groups I count.

Name

Circle equal groups to solve. Count the number of shapes in 1 group.

1. 1 __ 4

of 8 5

2. 1 __ 3

of 9 5

3. 1 __ 4

of 16 5

4. 1 __ 6

of 18 5


Lauren bought 12 stamps for postcards. She gave Brianna 1 _ 6 of them.

How many stamps did Lauren give to Brianna?

Step 1 Find the total number of stamps. 12 stamps

Step 2 Since you want to find 1 _ 6 of the group, there should be

6 equal groups. Circle one of the groups to show 1 _ 6

.

Step 3 Find 1 _ 6 of the stamps. How many stamps are in 1 group? 2 stamps

So, Lauren gave Brianna 2 stamps. 1 _ 6 of 12 5 2

Lesson 8.8Reteach

Check students’ circles.

3

3

4

2

8-19 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

Name

Name Number of Tickets Left

Evan

Omar

Kate

Jenny

Part of the Group

Evan and his friends go to a theme park. Each friend buys

24 tickets and rides only 1 ride. Solve the problem.

1. Evan uses 1 __ 3

of his tickets to ride

the Loop-D-Loop. How many tickets does he use?

2. Omar uses 1 __ 6

of his tickets to ride

the water slide. How many tickets does he use?

3. Kate uses 1 __ 2

of her tickets to ride

the roller coaster. How many tickets does she use?

4. Jenny uses 1 __ 4

of her tickets to ride

the merry-go-round. How many tickets does she use?

5. Stretch Your Thinking Use the information in 1–4 to find the number of tickets each friend has left.

6. The friends now want to go on the Loop-D-Loop and the roller coaster. Explain why only 1 of the friends can go on both of these rides.

Lesson 8.8Enrich

18

12

2016

Possible explanation: to go on the Loop-D-Loop and roller

coaster, you need 20 tickets.

6 tickets

8 tickets

12 tickets

4 tickets

8-20 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

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487 Chapter 8

Reteach 8.8 Enrich 8.8

LESSON 8.8

Unlock the Problem MATHEMATICAL PRACTICES

Check that students understand that they need to find 1 _ 3 of 12.

ActivityMP4 Model with mathematics. Have students use their materials and work through the activity together. Then ask:

• Why were the counters divided into three groups? Possible answer: because 3 is the denominator and you want to find how many counters are in 1 of the 3 groups.

• How is each group alike? Each group has the same number of counters.

• What would you do differently if you wanted to find 1 _ 4 of 12? Possible answer: I would divide the counters into 4 equal groups.

• How many groups of counters did you make to find 1 _ 3 of 9? Why? 3; Since the denominator is 3, I make 3 equal groups.

MathTalk Use Math Talk to focus on students’

understanding of the numerator and denominator.

• Explain what the numerator and denominator of a fraction tell you. The denominator is the total number of items or groups and the numerator is the part being counted.

MP6 Attend to precision. Help students connect multiplication and division as they make equal groups.

• How is finding 1 _ 3 of 12 like division? I divide 12 by 3 to make 3 equal groups.

• How can you check that 4 counters are 1 _ 3 of 12? I can multiply 3 × 4. 3 × 4 = 12

• Why can you check division using multiplication? Because multiplication and division are inverse operations.

MP2 Reason abstractly and quantitatively. • How can you use a model to find 1 _ 3 of 9?

Separate 9 counters into 3 equal groups. The number of counters in each group is the answer.

3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

HandsOn

Quick Check

If

Rt I 1

2

3

Then

COMMON ERRORS COMMON ERRORS

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Share and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and Show MATHBOARDMATHBOARDMATHBOARDMATHBOARDMATHMATHMATHMATHBOARDBOARDBOARDBOARD Math

Talk MATHEMATICAL PRACTICES 6

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1. Use the model to find 1 _ 2 of 8. _

Think: How many counters are in 1 of the 2 equal groups?

Circle equal groups to solve. Count the number of flowers in 1 group.

2. 1 _ 4 of 8 = _ 3. 1 _ 3 of 6 = _ 4. 1 _ 6 of 12 = _

Try This! Find part of a group.

Raul picks 20 flowers from his mother’s garden. One fourth of them are purple. How many of the flowers are purple?

STEP 1 Draw a row of 4 counters.

Think: To find 1 _ 4 , make 4 equal groups.

STEP 2 Continue to draw as many rows of 4 counters as you can until you have 20 counters.

STEP 3 Then circle _ equal groups.

Think: Each group represents 1 _ 4 of the flowers.

There are _ counters in 1 group.

1 _ 4 of 20 = _

So, _ of the flowers are purple.

Describe why you count the number of counters in just one of the groups when finding 1 _ 2 of any number.

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4


Possible explanation: the numerator in 1 _ 2 is 1, so I counted the counters in 1 of the 2 groups.

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3_MNLESE342149_C08L08.indd 488 2/18/14 5:40 PM

Advanced LearnersAdvanced LearnersAdvanced Learners

Lesson 8.8 488

ELL Strategy: Cooperative Grouping

Pair students of mixed language proficiency levels to build vocabulary for fractions.

•Students make sets of index cards with the fractions written in number and picture form on one side and written in word form on the other.

•In pairs, have students select a card and name the fraction.

Try This!Read the problem with students.

• In the fraction 1 _ 4 , what does the denominator represent? the number of groups

• How do you know how many groups to count? The numerator is 1, so I will count 1 group.

a student misses the checked exercises

Differentiate Instruction with • Reteach 8.8

• Personal Math Trainer 3.NF.A.1

• RtI Tier 1 Activity (online)

Error Students may use the denominator as the number in each group rather than the number of equal groups.

Example Students might circle 4 groups of 2 in Exercise 1.Springboard to Learning For these exercises, have students practice saying the fractions as “[numerator] out of [denominator] groups.”

Share and Show The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

EXPLAIN3MATHBOARDMATHBOARD

Visual Small Group

•Write the following problems on the board and challenge students to complete them. Provide students with hints such as 1 week = 7 days and 1 year = 12 months.

1 _ 4 of a year= 3 months 1 _ 7 of a week = 1 day

1 _ 6 of a day = 4 hours 1 _ 5 of the month of April = 6 days

1 _ 6 of an hour = 10 minutes 1 __ 10 of a minute = 6 seconds

•Then have students write and solve their own fraction problems about time.

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Chapter 8 • Lesson 8 489

Circle equal groups to solve. Count the number of flowers in 1 group.

SMARTER Draw counters. Then circle equal groups to solve.

11. 1 _ 8 of 16 = _ 12. 1 _ 6 of 24 = _

5. 1 _ 4 of 12 = _ 6. 1 _ 3 of 15 = _ 7. 1 _ 4 of 16 = _

8. 1 _ 6 of 30 = _ 9. 1 _ 3 of 12 = _ 10. SMARTER

1 _ 2 of 6 = _

13. DEEPER Gerry has 50 sports trading cards. Of those cards, 1 _ 5 of them are baseball cards, 1 __ 10 of them are football cards, and the rest are basketball cards. How many more basketball cards than baseball cards does Gerry have?

14. DEEPER Barbara has a mixed garden that has 16 rows of different flowers and vegetables. One-fourth of the rows are lettuce, 1 _ 8 of the rows are pumpkins, and 1 _ 2 of the rows are red tulips. The other rows are carrots. How many rows of carrots are in Barbara’s garden?

42

3

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Possible drawings

are shown.

25 more cards 2 rows

489 Chapter 8

On Your Own If students complete the checked exercises correctly, they may continue with the On Your Own section.Help students make connections to the arrays that they used in multiplication. Encourage students to look for the row or column that illustrates the denominator of the fraction.

• How did you know how many equal groups to circle in Exercises 5–10? The denominator tells the number of equal groups to circle.

• After you circled the equal groups, what did you do next? Possible answer: I counted the number of objects in 1 group.

SMARTER

Exercise 10 has the flowers arranged in a different way than the rest of the exercises to ensure students are not by habit circling the columns of flowers.

SMARTER

Exercises 11 and 12 require students to decide how many counters to draw and how to arrange them in equal groups before answering the question. If students have difficulty, remind them that they should have equal groups that reflect the denominator, but the total number of counters should equal the whole number.

MP4 Model with mathematics. To extend their thinking, have students tell another fraction that names one circled group in Exercise 10. 3 _ 6

ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

EVALUATE5 Formative Assessment

MATHEMATICAL PRACTICES COMMUNICA E CONSTRUCT ARGUMENTS

Flower Seeds Bought

Name Number of Packs

Brooke

Cole

Ryan 8

12

20

WRITE MathShow Your Work

Personal Math Trainer

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Use the table for 15–16.

15. MATHEMATICALPRACTICE 4 Use Diagrams One fourth of

the seed packs Ryan bought are violet seeds. How many packs of violet seeds did Ryan buy? Draw counters to solve.

16. DEEPER One third of Brooke’s seed packs and one fourth of Cole’s seed packs are daisy seeds. How many packs of daisy seeds did they buy altogether? Explain how you know.

17. SMARTER Sense or Nonsense? Sophia bought 12 pots. One sixth of them are green. Sophia said she bought 2 green pots. Does her answer make sense? Explain how you know.

18. SMARTER A florist has 24 sunflowers in a container. Mrs. Mason buys 1 _ 4 of the flowers. Mr. Kim buys 1 _ 3 of the flowers. How many sunflowers are left? Explain how you solved the problem.

2 packs

Check students’

drawings.

9 packs; possible explanation: 1 _ 3 of 12 = 4 and 1 _

4 of

20 = 5. 4 + 5 = 9, so Brooke and Cole bought 9 packs of

daisy seeds altogether.

10 sunfl owers: Possible explanation: I divided the array

into 4 parts and found 1 _ 4 of 24 = 6. Then I circled thirds in

the array and found 1 _ 3 of 24 = 8. I added 6 + 8 = 14 and

subtracted the sum from 24.

Yes; possible explanation: I divided 12 counters into

6 equal groups. Then I counted the number in 1 of the

6 groups. There were 2 counters in 1 group, so 1 _ 6 of 12 = 2.

Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.

Lesson 8.8 490

LiteratureThe Whole Picture

Students read the book and model fractional parts.

Students complete blue Activity Card 11 by finding

fractional parts of a group of pattern blocks.

ActivitiesFraction Action

Essential QuestionUsing the Language ObjectiveReflect Have students model and explain to answer the Essential Question.How can a fraction tell how many are in part of a group? Use the denominator to find how many smaller, equal groups to divide the total number in the group into. Then use the numerator to find how many groups to count. Count the total number of objects in those groups.

Math Journal WRITE MathExplain how to find which is greater: 1 _ 4 of 12 or 1 _ 3 of 12.

MATHEMATICAL PRACTICES

MP4 Model with mathematics. How do you decide how many groups to draw? The fraction one fourth tells me to draw 4 equal groups.

DEEPER

Exercise 16 requires students to interpret data for two sets of numbers. Then they solve a multistep problem to find the answer.

SMARTER

Personal Math Trainer SMARTER

Be sure to assign this problem to students in the Personal Math Trainer. It features a video to help them model and answer the problem. Students must be able to identify 1 _ 4 of a group, identify 1 _ 3 of the same whole group, and recognize how to combine this information to answer all steps of the problem. Students who answer 6, 8, or 14 only completed part of the problem. Make sure students correctly identified these partial answers and then ask them to reread the problem to find out whether they have answered the question.

Math on the Spot Video TutorUse this video to help students model and solve this type of Think Smarter problem.

Problem Solving • Applications

Extend the Math Activity

Problem SolvingProblem Solving

Name

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Chapter 8 491


Circle equal groups to solve. Count the number of items in 1 group.

1. 1 _ 4

of 12 = _

2. 1 _ 8

of 16 = _

5. Marco drew 24 pictures. He drew 1 _ 6

of

them in art class. How many pictures

did Marco draw in art class?

6. Caroline has 16 marbles. One eighth of them are blue. How many of Caroline’s marbles are blue?

3. 1 _ 3

of 12 = _

4. 1 _ 3

of 9 = _

3

Lesson 8.8Practice and Homework

COMMON CORE STANDARD—3.NF.A.1 Develop understanding of fractions as numbers.

7. WRITE Math Explain how to find which is greater: 1 _

4 of 12 or 1 _

3 of 12.


2

34

4 pictures 2 marbles

Check students' work.

Practice and HomeworkUse the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student’s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers.

491 Chapter 8

Multiple Representations of FractionsMaterials Fraction Circles, Fraction Strips (see eTeacher Resources), MathBoard

In this chapter, students have used number lines, regions, and groups to model fractions. This activity will help students better understand fraction concepts by connecting different representations of the same fraction.

Investigate Students will work with a partner and use manipulatives, drawings, paper folding, or other representations to show fractions.

• One player selects a fraction piece from the fraction circles or strips. The other player represents the fraction in a different way. The first player then chooses another representation for the same fraction.

• For example, if the first player selects a 1 __ 3 piece from the fraction circles, the second player might choose the fraction strip for 1 __ 3 . The first player might write 1 __ 3 on his or her MathBoard, then the second player might show 1 __ 3 on a number line.

• Play continues until one of the players cannot think of a different way to represent the fraction.

Summarize Ask students to tell how some of the ways they chose to represent fractions are alike or different. You may wish to ask them why it is important to know different ways to represent fractions. They should recognize that different representations help them model different situations. For example, use number lines for distance problems; counters for problems about groups; and area models for region problems.

Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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Lesson Check (3.NF.A.1)

1. Ms. Davis made 12 blankets for her grandchildren. One third of the blankets are blue. How many blue blankets did she make?

2. Jackson mowed 16 lawns. One fourth of the lawns are on Main Street. How many lawns on Main Street did Jackson mow?

Spiral Review (3.OA.A.7, 3.NBT.A.1, 3.NBT.A.2)

3. Find the difference.

509

2175

__

4. Find the quotient.

6 q ww 54

5. There are 226 pets entered in the pet show. What is 226 rounded to the nearest hundred?

6. Ladonne made 36 muffins. She put the same number of muffins on each of 4 plates. How many muffins did she put on each plate?

334

4 blue blankets

200

4 lawns

9 muffins

9

Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section.

Lesson 8.8 492

find part of a group using unit fractions

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