problem solving • find the whole group using unit fractions · f c r focus: common core state...

$: Problem Solving • Find the Whole Group Using Unit Fractions · F C R Focus: Common Core State Standards 3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when$
493A Chapter 8

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Problem Solving • Find the Whole Group Using Unit Fractions

LESSON 8.9

Why Teach ThisThere are many different strategies that can be used to solve problems. The different strategies should be modeled and practiced so that they become familiar to students. Then students can apply their knowledge of problem-solving strategies to determine which strategy is best to solve a particular problem.

Visualizing a problem with objects or pictures can make a solution to a problem more evident. Drawing a diagram is similar to making a picture.

In this lesson, students will solve problems that involve a fraction of a group and the number of objects in that fractional part. They will draw the number of fractional parts in a whole group. Then they will use what they know about the number of objects in one fractional part to draw the items in all the parts. The total number of objects they draw is the number of objects in the whole group.

Professional Development Videos

Learning ObjectiveSolve fraction problems by using the strategy draw a diagram.

Language ObjectiveStudents write in their Math Journal an example of the strategy draw a diagram and explain how it is used to solve fraction problems.

MaterialsMathBoard

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 PRACTICESMP1 Make sense of problems and persevere in solving them. MP4 Model with mathematics.MP5 Use appropriate tools strategically. MP6 Attend to precision.

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 8.9 493B

Daily RoutinesCommon Core

Daily RoutinesCommon Core

How can you use the strategy draw a diagram to solve fraction problems?

Fluency BuilderMaterials index cards numbered 0–9

Mental Math Give each pair of students a set of cards numbered 0–9. Mix the cards and place them facedown. Students take turns choosing 2 cards, naming the sum, the other related addition fact, and the two related subtraction facts.

Problem of the Day 8.9Jordan has 32 trophies. He wants to put his trophies on a bookcase that has 4 shelves. How many trophies should he put on each shelf to have an equal number of trophies on each shelf?

______

Vocabulary• Interactive Student Edition• Multimedia Glossary e

with the Interactive Student Edition

Essential QuestionHow can you use the strategy draw a diagram to solve fraction problems?

Making ConnectionsInvite students to share what they know about fractions and drawing diagrams. Display a drawing of twelve fish. Circle four of the fish. Then ask:

• What fraction of the fish is circled? 1 __ 4   

• Describe how you could draw a picture that shows one half of the fish are striped? Separate the fish into two equal groups. Draw stripes on one of the groups.

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

• One third of the airplanes Pedro saw were large. He saw 8 large airplanes. If the rest are small, what information is missing? the total number of airplanes Pedro saw

• Use circles to represent the large airplanes Pedro saw. How many circles should you draw to show 1 __ 3 of the whole group? 8

Literacy and Mathematics• Have students write a problem involving finding the total number

of items given a unit fraction and the number of items in one part.

8 trophies

Common Core Fluency Standard 3.NBT.A.2

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EXPLORE2

Name

1. Max has 3 beta fish in his fish tank. One half of his fish are beta fish. How many fish does Max have in his tank?

2. Two boys are standing in line. One sixth of the students in line are boys. How many students are standing in line?


There are 3 apple juice boxes in the cooler. One fourth of the juice boxes in the cooler are apple juice. How many juice boxes are in the cooler?

Read the Problem Solve the Problem

What do I need to find?

I need to find are in the cooler.

Describe how to draw a diagram to solve.

The denominator in 1 _ 4

tells you that there are parts in the whole group. Draw 4 circles to show parts.

Since 3 juice boxes are 1 _ 4

of the group, draw counters in the first circle.

Since there are counters in the first circle, draw counters in each of the remaining circles. Then count all of the counters.

So, there are juice boxes in the cooler.

What information do I need to use?

There are apple juice boxes.

of the juice boxes are apple juice.

How will I use the information?

I will use the information in the problem to draw a diagram.

Lesson 8.9Reteach

12

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3

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4how many juice boxes

12 students6 fish

One fourth

3

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8-21 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

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Name

What Part of the Group?

Draw a quick picture to solve.

1. Lisa’s dog has 4 squeaky toys. Two thirds of the dog’s toys are squeaky toys. How many dog toys does Lisa’s dog have in all?

2. Sam has 9 yellow pencils in his desk. Three fourths of his pencils are yellow. How many pencils does Sam have in his desk?

3. Julia has 8 red barrettes. Two fourths of her barrettes are red. How many barrettes does Julia have?

4. Antonio has 15 pennies in his pocket. Five eighths of his coins are pennies. How many coins does Antonio have in his pocket?

5. Stretch Your Thinking One half of the birds at a pet store are yellow. Tara buys one of the yellow birds. Then one third of the birds at the store are yellow. How many yellow birds were at the pet store before Tara bought one? Explain how you know.

Lesson 8.9Enrich

Check students’ work.

2; Possible explanation: when you buy 1

yellow bird, the fraction of yellow birds

changes from 1 _ 2 to 1 _ 3 . So, there were 4 birds

to start with, 2 of which were yellow.

24 coins16 barrettes

12 pencils6 dog toys

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8-22 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

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Unlock the ProblemUnlock the Problem

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Read the Problem

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Solve the Problem

Cameron has 4 clown fish in his fish tank. One third of the fish in the tank are clown fish. How many fish does Cameron have in his tank?

Use the graphic organizer to help you solve the problem.


I need to find ___ are in Cameron’s fish tank.


Cameron has _ clown fish.

__ of the fish in the tank are clown fish.


I will use the information in the problem

to draw a __.

Describe how to draw a diagram to solve.

The denominator in 1 _ 3 tells you

that there are _ equal parts in the whole group. Draw 3 circles to

show _ equal parts.

Since 4 fish are 1 _ 3 of the whole group,

draw _ counters in the first circle.

Since there are _ counters

in the first circle, draw _ counters in each of the remaining circles. Then find the total number of counters.

Chapter 8 493

Problem Solving • Find the Whole Group Using Unit FractionsEssential Question How can you use the strategy draw a diagram to solve fraction problems?

So, Cameron has _ fish in his tank.

PROBLEM SOLVINGLesson 8.9

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

MATHEMATICAL PRACTICESMP1, MP2, MP8

how many � sh

4

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12

One third

diagram

Possible drawing:

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DifferentiatedInstruction1

2

3

493 Chapter 8

Enrich 8.9Reteach 8.9

Unlock the Problem MATHEMATICAL PRACTICES

To introduce the lesson, have students watch the Real World Video, Feeding Big Cats at the Zoo. Have a class discussion about how zookeepers care for the animals at the zoo, and how it is similar to having pets, such as fish in an aquarium.MP1 Make sense of problems and persevere in solving them. Guide students to read each question in the graphic organizer and answer it before they solve the problem.

• Why do you draw 3 circles? Because the 4 fish are one third of the fish in the tank. Draw 3 circles to show the 3 equal parts of all the fish.

• Why do you draw 4 counters in each of the 3 circles? Four fish are one third of the fish in the tank, so there are 4 fish in each third.

• How could you check your answer with addition or multiplication? Possible answers: I can check with addition by adding 4 three times— 4 + 4 + 4 = 12. I can check with multiplication by multiplying 3 × 4 to get the answer 12.

MP4 Model with mathematics.•Cameron said that 1 __ 6 of his fish are angel

fish. How can you use the answer of 12 fish and a diagram to determine the number of angel fish Cameron has in his tank? Since 1 __

6

of the fish are angel fish, the denominator tells you to divide the 12 counters into 6 even groups. Each group will have 2 counters. So, Cameron has 2 angel fish.

ELL Strategy: Illustrate Understanding

Students can use drawings to solve fraction problems.

•Select a problem from the lesson.

•Explain how to use the denominator to know how many circles to draw and model it for the group.

•Model and explain how to read the problem to see how many counters to draw inside the circle. Have students draw them.

•Then, model how to use the drawing to figure out how many items there are in total and have students draw them.

LESSON 8.93.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.


Meeting Individual NeedsDifferentiated

Instruction

COMMON ERRORS

Try Another Problem

MathTalk MATHEMATICAL PRACTICES 1

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494

Read the Problem Solve the Problem

A pet store has 2 gray rabbits. One eighth of the rabbits at the pet store are gray. How many rabbits does the pet store have?




1. MATHEMATICALPRACTICE 8 Draw Conclusions How do you know that your

answer is reasonable?

2. How did your diagram help you solve the problem? ____

Make Sense of Problems Suppose 1 _ 2 of the rabbits are gray. How can you find the number of rabbits at the pet store?

I need to fi nd how many rabbits are at the pet

store.

Since the denominator in 1 _ 8 is 8, there are

8 equal parts in the whole group. Draw

8 circles to show 8 equal parts.

Since 2 rabbits are 1 _ 8 of the whole group,

draw 2 counters in the fi rst circle.

Since there are 2 counters in the fi rst circle,

draw 2 counters in each of the remaining

circles. Then fi nd the total number of

counters.

Possible explanation: since half of the rabbits

are gray, there are two groups with 2 rabbits

in them. So, I can multiply the 2 rabbits by

2 groups to fi nd 4 rabbits in the whole group.

So, the pet store has 16 rabbits.

The pet store has 2 gray rabbits. One eighth

of all the rabbits are gray.

Possible answer: since the whole group is divided into 8 equal parts, I can multiply

8 groups of 2 to fi nd there are 16 rabbits in all. 16 = 16, so my answer is reasonable.

me see how many counters I needed to draw to fi nd the total number of rabbits.

Possible answer: it helped

I will use the information in the problem to

draw a diagram.

Advanced Learners

Lesson 8.9 494

Error Students might not know how many groups to draw.

Example I drew 2 groups with 2 counters each. So, the pet store has 4 rabbits.Springboard to Learning Have students circle the word that describes the denominator in the fraction. Then have students draw that many groups. The denominator is 8, so there should be 8 groups of 2 counters each, or 16 counters in all.

Try Another ProblemHave students read the problem about the rabbits. Then have students answer the questions in the graphic organizer and solve the problem. Invite students to share their diagrams and explanations.

• How many equal groups did you show with your counters? Explain. Possible answer: I made 8 equal groups because one eighth of the rabbits are gray.

• How many counters did you draw in all? Explain. 16; because 2 rabbits are one eighth of the rabbits, I showed 2 rabbits in each of the 8 groups, and 8 × 2 = 16.

MP8 Look for and express regularity in repeated reasoning.• How many of the rabbits are not gray?

How do you know? 14; I found that there are 16 rabbits in all. If 2 are gray, then 16 − 2, or 14, rabbits are not gray.

MathTalk Use Math Talk to focus on

students’ understanding of the relationship between the denominator and the number of equal groups in a whole.

• If you know the fraction of the rabbits that are gray, what other information do you need to know to find the total number of rabbits? the number of rabbits that are gray

You may suggest that students place completed Try Another Problem graphic organizers in their portfolios.

• Present students with the following problem:

If you know that 1 __ 8 of a number is 3, how can you find the number?

• Have students solve the problem. 24

• Then have students write a rule they can use to find a whole given a unit fraction of it. Have students share their rules. Possible answer: Multiply the denominator of the unit fraction by the number part given.

Visual / KinestheticIndividual / Partners

EXPLAIN3

Quick Check

If

Rt I 1

2

3

Quick Check

If

Rt I 1

2

3

Then

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

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√ Circle the question.

√ Underline important facts.

√ Put the problem in your own words.

√ Choose a strategy you know.

1. Lily has 3 dog toys that are red. One fourth of all her dog toys are red. How many dog toys does Lily have?

First, draw _ circles to show _ equal parts.

Next, draw _ toys in _ circle since

_ circle represents the number of red toys.

Last, draw _ toys in each of the remaining circles. Find the total number of toys.

So, Lily has _ dog toys.

2. SMARTER What if Lily has 4 toys that are red? How many dog toys would she have?

3. The pet store sells bags of pet food. There are 4 bags of cat food. One sixth of the bags of food are cat food. How many bags of pet food does the pet store have?

4. Rachel owns 2 parakeets. One fourth of all her birds are parakeets. How many birds does Rachel own?

Chapter 8 • Lesson 9 495

Unlock the Problem

4

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16 dog toys

24 bags

8 birds

1

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495 Chapter 8

MP2 Reason abstractly and quantitatively. To extend their thinking, have students tell about the unidentified parts in each problem. For example, if one fourth of the dog toys are red, what can you tell about the remaining toys? Possible answer: the remaining 9 toys are three fourths of the group and are a color different than red.

Share and Show MATHBOARDMATHBOARD

The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.

SMARTER

Exercise 2 gives students an opportunity to analyze how a change in the number of objects in a part affects the whole number of objects in the group.Use the checked exercises for Quick Check.

a student misses the checked exercises

Differentiate Instruction with • Reteach 8.9

• Personal Math Trainer 3.NF.A.1

• RtI Tier 1 Activity (online)


ELABORATE4

EVALUATE5 Formative Assessment

WRITE Math Show Your Work

MATHEMATICAL PRACTICES COMMUNICA E CONSTRUCT ARGUMENTS

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On Your OwnOn Your Own

_ hats

8. SMARTER Two hats are 1 _ 3 of the group. How many hats are in the whole group?

5. SMARTER Before lunchtime, Abigail and Teresa each read some pages from different books. Abigail read 5, or one fifth, of the pages in her book. Teresa read 6, or one sixth, of the pages in her book. Whose book had more pages? How many more pages?

6. MATHEMATICALPRACTICE 2 Represent a Problem Six

friends share 5 meat pies. Each friend first eats half of a meat pie. How much more meat pie does each friend need to eat to finish all the meat pies and share them equally? Draw a quick picture to solve.

7. DEEPER Braden bought 4 packs of dog treats. He gave 4 treats to his neighbor’s dog. Now Braden has 24 treats left for his dog. How many dog treats were in each pack? Explain how you know.

Teresa’s book; 11 more pages

Has been eaten

Need to eat

6

Possible drawing for 6:

Possible answer: 1 third, or 2 sixths

7 dog treats; possible explanation: I began

with the 24 treats left for Braden’s dog.

I added the number of treats he gave

away: 24 + 4 = 28. Then I divided that

number by 4 packs: 28 ÷ 4 = 7.

1 2 3 4 5 6

1 2

3

4 5

6Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD

Lesson 8.9 496

Essential QuestionUsing the Language ObjectiveReflect Have students write an example in their Math Journal to answer the Essential Question.How can you use the strategy draw a diagram to solve fraction problems? Possible answer: I can draw a diagram to represent all the equal groups in a whole. Then I can draw the number of objects in each group. Finally, I can add or multiply to find how many objects are in the whole.

Math Journal WRITE MathWrite a problem about a group of objects in your classroom. Tell how many are in one equal part of the group. Solve your problem. Draw a diagram to help you.

On Your Own MATHEMATICAL PRACTICES

If students complete the checked exercises correctly, they may continue with the On Your Own section.

SMARTER

Students complete blue Activity Card 11 by finding

fractional parts of a group of pattern blocks.t

Students read the book and model fractional parts.

LiteratureThe Whole Picture

ActivitiesFraction Action

MP2 Reason abstractly and quantitatively. Exercise 6 requires students to compare the number of wholes and the number of shares in order to find equal shares.

SMARTER

This item assesses a student’s ability to find a whole group given a unit fraction. Students should understand that the model represents three equal parts of the whole group, with two hats making up one of the parts. Students who give two as the answer likely confused the part of the group with the whole.

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

Cross-Curricular

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Chapter 8 497


Draw a quick picture to solve.

1. Katrina has 2 blue ribbons for her hair. One fourth of all her ribbons are blue. How many ribbons does Katrina have in all?

_____

2. One eighth of Tony’s books are mystery books. He has 3 mystery books. How many books does Tony have in all?

_____

3. Brianna has 4 pink bracelets. One third of all her bracelets are pink. How many bracelets does Brianna have?

_____

4. Ramal filled 3 pages in a stamp album. This is one sixth of the pages in the album. How many pages are there in Ramal’s stamp album?

_____

5. Jeff helped repair one half of the bicycles in a bike shop last week. If Jeff worked on 5 bicycles, how many bicycles did the shop repair last week?

_____

6. WRITE Math Write a problem about a group of objects in your classroom. Tell how many are in one equal part of the group. Solve your problem. Draw a diagram to help you.

8 ribbons

Lesson 8.9Practice and Homework

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

24 books

12 bracelets

18 pages


10 bikes


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.

497 Chapter 8

SCIENCE SOCIAL STUDIES

• Take a class walk through a local park. Have students classify the animals they see according to birds, reptiles, and amphibians. Remind students that animals like alligators and turtles are reptiles, and animals like frogs and salamanders are amphibians.

• Suppose 1 __ 6 of the animals they saw were reptiles. If they saw 2 reptiles, what was the total number of animals they saw? 12 animals

• Discuss how people use money to purchase goods and services. Goods are items you can see and touch, like food, clothing, or a book. Services are provided by other people like the doctor, the person who cuts your hair, or the person who fixes the washing machine.

• Ask students how they buy items they want.

• Lindsay saved $5 to buy a game. She has 1 __ 4 the amount of money she needs. How much money does the game cost? $20

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498

Lesson Check (3.NF.A.1)

1. A zoo has 2 male lions. One sixth of the lions are male lions. How many lions are there at the zoo?

2. Max has 5 red model cars. One third of his model cars are red. How many model cars does Max have?

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

5. Ken has 6 different colors of marbles. He has 9 marbles of each color. How many marbles does Ken have in all?

6. Eight friends share two pizzas equally. How much of a pizza does each friend get?

3. There are 382 trees in the local park. What is the number of trees rounded to the nearest hundred?

4. The Jones family is driving 458 miles on their vacation. So far, they have driven 267 miles. How many miles do they have left to drive?

458 − 267

__

12 lions

400 trees

54 marbles

15 model cars

191 miles

2 eighths

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 8.9 498

problem solving • find the whole group using unit fractions · f c r focus: common core state...

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