421A Chapter 7

About the MathProfessional Development

LESSON AT A GLANCE

Interactive Student Edition

Personal Math Trainer

Math on the Spot

iTools: Counters

Professional Development Videos

Why Teach ThisUsing a graphic organizer to scaffold a problem helps students analyze the problem to understand what they need to find, identify what information they need to use, and develop or select a strategy.

In this lesson, students use the act it out strategy. Students use counters, write equations, and work backward to solve two-step problems.

Students will represent the situations in the steps of a problem with equations containing a letter that stands for the unknown quantity. Writing equations with an unknown quantity can help students to relate the situations in problems to operations. Carrying out each operation and finding the unknown helps students to solve the two-step problems.

LESSON 7.10

Problem Solving • Two-Step Problems

Learning ObjectiveSolve two-step problems by using the strategy act it out.

Language ObjectiveStudent teams role play to show the class how you can use the strategy act it out to solve two-step problems.

MaterialsMathBoard

F C R Focus:Common Core State Standards3.OA.D.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the

unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

MATHEMATICAL PRACTICESMP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision.

F C R Coherence:Standards Across the GradesBefore2.OA.C.4

Grade 33.OA.D.8

After4.OA.A.3

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 363J.

FOCUS COHERENCE RIGOR

ENGAGE1

Lesson 7.10 421B

Daily RoutinesCommon Core

Daily RoutinesCommon Core

How can you use the strategy act it out to solve two-step

problems?

with the Interactive Student Edition

Essential QuestionHow can you use the strategy act it out to solve two-step problems?

Making ConnectionsInvite students to tell you what they know about division.

If 12 beetles are put into 2 equal groups, how many are in each group? 6 If 6 beetles are separated into 3 equal groups, how many are in each group? 2

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

• How many rabbit holes did the first set of beetles hide in? 4

• How many beetles were seen first? 28

• How many beetles were seen after the first set hid in the rabbit holes? 8

• What is the question asking for? how many beetles will be in each hole

Literacy and MathematicsChoose one or more of the following activities.

• Have students visualize the beetles separating and hiding in the holes.

• Have students draw a picture of four holes with the beetles hiding.

• Have students write a short story about what it would be like to live for 1 day as a beetle.

Fluency BuilderMaterials  K.I.M. Chart (see eTeacher Resources)

Graphic Organizer Have students work in groups to brainstorm all the division words they have learned. Then have students complete a K.I.M. chart for the words.

Students write the term or key idea in the left column. Then they write the information that goes with the term, the definition, in the center column. Then students draw a picture of the idea, a memory clue, in the right column.

KKey Idea

IInformation

MMemory Clue

divisor

dividend

quotient

Problem of the Day 7.10The product of two factors is 27. One factor is 9. What is the other factor? 3

Vocabulary• Interactive Student Edition• Multimedia Glossary e

Common Core Fluency Standard 3.OA.C.7

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Unlock the ProblemUnlock the Problem

MATHEMATICAL PRACTICES 1MathTalk

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Chapter 7 421

Problem Solving • Two-Step ProblemsEssential Question How can you use the strategy act it out to solve two-step problems?

Read the Problem

Madilyn bought 2 packs of pens and a notebook for $11. The notebook cost $3. Each pack of pens cost the same amount. What is the price of 1 pack of pens?

Name

Solve the Problem

What do I need to find?

I need to find the price of

1 pack of __.

What information do I need to use?

Madilyn spent _ in all.

She bought _ packs of

pens and _ notebook.

The notebook cost _.

How will I use the information?

I will use the information to

_ out the problem.

Describe how to act out the problem.

Start with 11 counters. Take away 3 counters.

total cost of p, cost of cost notebook 2 packs of pens ↓ ↓ ↓

_ − _ = p

_ = p

Now I know that 2 packs of pens cost _.

Next, make _ equal groups with the 8 remaining counters.

p, cost of number c, cost of 2 packs of pens of packs 1 pack of pens ↓ ↓ ↓

$8 ÷ _ = c

_ = c

So, the price of 1 pack of pens is _.

PROBLEM SOLVINGLesson 7.10Operations and Algebraic Thinking—3.OA.D.8

MATHEMATICAL PRACTICESMP1, MP3, MP6

Make Sense of Problems Why do you need to use two operations to solve the problem?

pens

act

1

2

$3

$11

$3$11

$8

$8

2

2

$4

$4

Possible explanation: I need to subtract the cost of the notebook to know what the pens cost. Then I need to divide to � nd the cost of 1 pack of pens.

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EXPLORE2

Name

Division Steps

Solve. Show your steps to nd the answer.

1. Veronica bought a pack of 50 CDs. She gave 8 to her friend, Leslie. Then she made 6 equal sets of CDs. How many CDs are in each set?

2. Sid has 2 boxes of markers with the same number of markers in each box. He gives 3 markers to his sister. Now Sid has 13 markers. How many markers were in each box?

3. Casey bought 30 basketball trading cards. He gave 6 to his sister and then put the rest in an album. If 6 cards fit on one page of the album, how many pages did Casey use?

4. Barron bought a 25-pound bag of dog food. He still had 3 pounds from an older bag. If he feeds his dogs 4 pounds of food each week, how many weeks until all the dog food is gone?

5. Manny ordered 16 skateboard wheels. If he sold 4 wheels to Brad, how many skateboards can he put wheels on if each board uses 4 wheels?

6. Kiera has 27 balls of yarn. Her mother gave her 5 more balls. If she makes scarves that use 4 balls each, how many scarves can she make?

7. Thomas starts with 36 photos and throwsaway 6 that are too dark. Then he organizes the rest ofthem on scrapbook pages so that there are 5 photos oneach page. How many pages will he use? Draw to explain.

Lesson 7.10Enrich

50 2 8 5 42; 42 4 6 5 7; 7 CDs

13 + 3 = 16; 16 ÷ 2 = 8; 8 markers

6 pages; Possible drawing is shown.

30 2 6 5 24; 24 4 6 5 4; 4 pages

25 1 3 5 28; 28 4 4 5 7; 7 weeks

16 2 4 5 12; 12 4 4 5 3; 3 skateboards

27 1 5 5 32; 32 4 4 5 8; 8 scarves

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7-24 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

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Name 

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7

ProblemSolving•Two-StepProblems

Chloe bought 5 sets of books. Each set had the same number of books. She donated 9 books to her school. Now she has 26 books left. How many books were in each set that Chloe bought?

Read the Problem Solve the Problem

WhatdoIneedtofind?

I need to find how many ___were in each ___.

First, begin with the number of books left. Add the number of books donated.

Then divide to find the number of books in each set.

So, _ books were in each set.

WhatinformationdoIneedtouse?

I need to use the information given:

Chloe bought _ sets of books.

She donated _ books.

She has _ books left.

HowwillIusetheinformation?

I will use the information to ___ the problem.

Solvetheproblem.

1. Jackie had 6 equal packs of pencils. Her friend gave her 4 more pencils. Now she has 52 pencils. How many pencils were in each pack?

2. Tony had 4 equal sets of sports cards. He gave his friends 5 cards. Now he has 31 cards. How many cards were in each set?

Lesson 7.10Reteach

8pencils 9cards

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DifferentiatedInstruction1

2

3

421  Chapter 7

Enrich 7.10

Unlock the Problem MATHEMATICAL PRACTICES

MP1 Make sense of problems and persevere in solving them.  Have students explain how the description to act out the problem helps them to solve the problem. • Why do you subtract $3 from $11? Possible

answer: the total cost for the pens and notebook is $11. You have to subtract the cost of the notebook to find out how much the pens cost.

• Why do you divide by 2 in the last step? The cost of 2 packs of pens is $8. To find the cost of 1 pack, you divide by 2.

Discuss the letter that stands for the unknown in each equation. Tell students that a letter is similar to the answer lines or the squares that they have seen in equations in the position of the answer they need to find.• How can you use multiplication and 

addition to check your answer? I can multiply the cost of 1 pack of pens, $4 by 2. That gives me $8. Then add the notebook price $3, to $8 to get $11. So, my answer is correct.

MathTalk Use Math Talk to focus on solving  

  multistep problems. MP2 Reason abstractly and quantitatively.• How would the problem change if Madilyn 

bought 3 packs of pens and a notebook for $15? You would still subtract the price of the notebook from $15, but now you would divide the remaining amount by 3 instead of 2.

ELL Strategy:   Model Language

Go over four strategies—act it out, draw a diagram, make a table, and find a pattern.•Write the name of each strategy and put an 

icon or visual to represent the strategy on an index card. 

•Have students draw a card and act out the strategy. Allow students to use props as needed. The other students guess which strategy is being performed. 

•Discuss the strategy using sentence frames. You can use the _______ strategy to solve problems by _______.  

LESSON 7.103.OA.D.8  Solve  two-step word problems using  the  four operations. Represent  these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.

Reteach 7.10

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Meeting Individual NeedsDifferentiated

Instruction

Try Another Problem

MATHEMATICAL PRACTICES 3MathTalk

Apply What is another strategy you could use to solve this problem?

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Chad bought 4 packs of T-shirts. He gave 5 T-shirts to his brother. Now Chad has 19 shirts. How many T-shirts were in each pack?

What do I need to find?

What information do I need to use?

How will I use the information?

Describe how to act out the problem.

• How can you use multiplication and subtraction to check your answer?N

Read the Problem Solve the Problem

I need to fi nd the number of T-shirts that

were in each pack.

Chad bought 4 packs of T-shirts and

gave 5 T-shirts to his brother. Chad

has 19 shirts left.

I will use the information to act out the

problem.

Possible answer: I can start with the number of T-shirts in each pack and work

through the problem. Multiply 6 shirts by 4 packs: 4 × 6 = 24. Then subtract 5 T-shirts:

24 − 5 = 19. So, my answer is correct.

Possible explanation: I could draw a diagram adding

5 tiles to 19 tiles. Then I would make an array by

dividing the 24 tiles into 4 equal rows.

Possible description is given.

Start with 19 counters. Add 5 counters.

fi nal number T-shirts t, T-shirts in

of T-shirts given away 4 packs

↓ ↓ ↓ 19 + 5 = t

24 = t

Now I know there were 24 T-shirts in

the packs.

Next, divide the counters into 4 equal

groups to fi nd the number of T-shirts in

each pack.

t, T-shirts in number p, number in

4 packs of packs each pack

↓ ↓ ↓ 24 ÷ 4 = p

6 = p

So, there were 6 T-shirts in each pack.

COMMON ERRORS Advanced Learners

Lesson 7.10 422

Error When solving some multistep problems, students may not understand they can use an inverse operation.

Example In this problem, students might want to subtract 5 T-shirts instead of adding them, using the reasoning that they were given away.

Springboard to Learning Tell students that when working a problem from the end to the beginning, each operation should be an inverse operation.

Try Another ProblemHave students answer the questions in the graphic organizer and solve the problem. • Suppose you used counters to solve the

problem. How many counters would you start with and why? Possible answer: 19; because that is the number of T-shirts Chad has left

• What would you do next, and why? Possible answer: add 5 counters; Since Chad gave away 5 T-shirts, you need to add 5 back to find the number of T-shirts he had to start with.

• What does 24 represent? the number of T-shirts Chad started with

• Why did you divide by 4? to find how many T-shirts were in each pack

Discuss students’ solutions and analyze whether they are reasonable.• How might you use mental math to find

if your answer is reasonable? Possible answer: use a multiplication fact: 4 × 6 = 24, and count back 5 to 19.

• How might you use estimation to find if your answer is reasonable? Possible answer: think of a multiplication fact for 4 that has a product of about 19. 4 × 5 = 20 and 4 × 6 = 24, so the answer is about 5 or 6.

• How would your strategy change if Chad bought 3 packs of T-shirts, gave away 5, and had 19 T-shirts leftover? I would still add 19 and 5, but would divide 24 by 3 instead of 4.

MathTalk Use Math Talk to focus on

using different strategies to solve problems.

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

portfolios.

Logical / MathematicalIndividual / Partners

Materials play coins

• Write the following problem on the board.

Cathy had 6 coins. She used 2 coins to buy an apple for 50¢. Then her friend gave her 35¢. Now she has 75¢. How much money did Cathy have to start? What are the coins Cathy had before buying the apple?

• Have students solve the problem. Provide students with play coins if they need to act out the problem. 90¢; possible answers: 2 quarters and 4 dimes, or 3 quarters and 3 nickels

• Have students write out the steps they used to solve the problem.

• Challenge students to write more problems on their own and exchange papers to solve them.

√ Circle the question.

√ Underline the important facts.

√ Choose a strategy you know.

On Your OwnOn Your Own

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

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1. Mac bought 4 packs of toy cars. Then his friend gave him 9 cars. Now Mac has 21 cars. How many cars were in each pack?

Act out the problem by using counters or the picture and by writing equations.

First, subtract the cars Mac’s friend gave him. total cars given c, cars in cars to Mac 4 packs

↓ ↓ ↓

21 − _ = c

_ = c

Then, divide to find the number of cars in each pack.c, cars in number p, number in

4 packs of packs each pack↓ ↓ ↓

12 ÷ _ = p

_ = p

So, there were _ cars in each pack.

Chapter 7 • Lesson 10 423

2. SMARTER What if Mac bought 8 packs of toy boats, and then he gave his friend 3 boats? If Mac has 13 boats now, how many boats were in each pack?

3. SMARTER Ryan gave 7 of his model cars to a friend. Then he bought 6 more cars. Now Ryan has 13 cars. How many cars did Ryan start with?

Unlock the Problem

2 boats

14 cars

3

9

12

3

4

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Quick Check

If

Rt I 1

2

3

Then

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

423  Chapter 7

a student misses the checked excercises

Differentiate Instruction with•Reteach 7.10

•Personal Math Trainer 3.OA.D.8

•RtI Tier 1 Activity (online)

ShareandShow MATHBOARDMATHBOARD

The first problem connects to the learning model.

SMARTER

Exercise 2 requires students to use logical reasoning to solve a multistep problem. Have students review their answers and decide whether they are reasonable.MP1Makesenseofproblemsandpersevereinsolvingthem. To extend students’ thinking about the steps of a multistep problem, have students think of the hidden question. For example, in Exercise 2 the hidden question is How many boats did Mac have before he gave his friend 3 boats?Use the checked excercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

EXPLAIN3

OnYourOwnIf students complete the checked excercises correctly, they may continue with the On Your Own section. Have students review their answers and decide whether they are reasonable.

SMARTER

Exercise 3 is a multistep problem that uses addition and subtraction. Have students share their methods for solving the problem.

MathontheSpotVideoTutorUse this video to help students model and solve this type of Think Smarter problem.

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Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES

ELABORATE4

Personal Math Trainer

MATHEMATICAL PRACTICES ANALYZE • LOOK FOR STRUCTURE • PRECISION

WRITE Math Show Your Work

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4. DEEPER Chloe bought 5 sets of books. Each set has the same number of books. She donated 9 of her books to her school. Now she has 26 books. How many books were in each set?

5. Hilda cuts a ribbon into 2 equal pieces. Then she cuts 4 inches off one piece. That piece is now 5 inches long. What was the length of the original ribbon?

6. DEEPER Teanna has 2 boxes of color pencils. One box has 20 color pencils and the other box has 16 color pencils. She gives her brother 3 of the color pencils. She wants to put the color pencils that she has left into 3 equal groups. How many color pencils will Teanna put in each group?

7. MATHEMATICALPRACTICE 6 Rose saw a movie, shopped, and ate at a

restaurant. She did not see the movie first. She shopped right after she ate. In what order did Rose do these activities? Explain how you know.

8. SMARTER Eleni bought 3 packs of crayons. Each pack contains the same number of crayons. She then found 3 crayons in her desk. Eleni now has 24 crayons. How many crayons were in each pack she bought? Explain how you solved the problem.

ate, shopped, saw a movie; possible explanation: since

Rose did not see the movie rst, she saw it second or third.

Since she shopped right after she ate, she saw the movie

third, which means she ate rst and shopped second.

Possible explanation: First, I subtracted 24 − 3 = 21. Then I

divided 21 by 3 and got 7, so there were 7 crayons in each pack.

7 books

18 inches

11 color pencils

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Games

EVALUATE5 Formative Assessment

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES

Lesson 7.10 424

Essential QuestionUsing the Language ObjectiveReflect Have student teams role play to answer the Essential Question. How can you use the strategy act it out to solve two-step problems? Possible answer: I can act out the problem by using counters, writing descriptions, making drawings, or writing equations. It organizes what I am doing and helps me see the steps. It also makes checking my work easy.

Math Journal WRITE Math

Write a division word problem and explain how to solve it by acting it out.

MATHEMATICAL PRACTICES

DEEPER

Work with students to determine the hidden question in Exercise 4. In this exercise, the hidden question is: How many books did Chloe have before she donated books to her school?MP6 Attend to precision. Exercise 7 requires students to use logical reasoning to determine the order of events and then explain their thinking.

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. If students make errors, encourage them to act out the problem. Students may act out buying 3 packs of crayons and then picking up 3 more crayons. Repeating the steps in reverse order will help them visualize the solution steps.

GamesDivision Cover-Up

LiteratureThe Garden Fence

Students practice division facts to place counters on the gameboard.

Students read the book and use division facts to find how much wood they need to build a fence.

Students complete purple Activity Card 19 by relating division facts to known multiplication facts.

ActivitiesDivision Dilemmas

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Meeting Individual Needs

Extend the Math Activity

27 4 3 , 18 2 5

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Chapter 7 425

Problem Solving • Two-Step Problems

1. Jack has 3 boxes of pencils with the same number of pencils in each box. His mother gives him 4 more pencils. Now Jack has 28 pencils. How many pencils are in each box?

_____

2. The art teacher has 48 paintbrushes. She puts8 paintbrushes on each table in her classroom.How many tables are in her classroom? _____

3. Ricardo has 2 cases of video games with the same number of games in each case. He gives 4 games to his brother. Ricardo has 10 games left. How many video games were in each case?

_____

4. Patty has $20 to spend on gifts for her friends. Her mother gives her $5 more. If each gift costs $5, how many gifts can she buy?

_____

5. Joe has a collection of 35 DVD movies. He received 8 of them as gifts. Joe bought the rest of his movies over 3 years. If he bought the same number of movies each year, how many movies did Joe buy last year?

_____

6. WRITE Math Write a division word problem and explain how to solve it by acting it out.

Solve the problem.

8 pencilsThink: I can start with 28 counters and act out the problem.

Lesson 7.10Practice and Homework

COMMON CORE STANDARD—3.OA.D.8 Solve problems involving the four operations, and identify and explain patterns in arithmetic.

6 tables

7 video games

5 gifts

9 movies

Check students’ work.

425 Chapter 7

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.

InequalitiesMaterials index cards

Investigate In this activity, students compare quantities using inequality signs, , or ..

• Prepare index cards by writing problems that are not equal in value (27 4 3, 12 1 4, 18 2 5, and so on) and one index card for each of the signs , and ..

• Write the signs , and . on the board. Explain each sign to students; , is less than, . is greater than.

• Explain that an inequality is made using two amounts that are not equal. The signs , and . show that the two sides are not equal.

• Write the following examples on the board.

2 3 3 , 10 4 1

45 4 5 . 12 2 7• Have groups of students take turns writing

inequalities. Two students each choose an index card. They place the cards on the table. Then another student chooses the sign card that makes the inequality true.

Summarize Ask students to explain how they decided which sign made the inequality true.

Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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Spiral Review (3.OA.A.1, 3.OA.A.3, 3.OA.A.4, 3.OA.D.8)

5. Meredith practices the piano for3 hours each week. How many hours will she practice in 8 weeks?

6. Find the unknown factor.

9 × ■ = 36

3. Each foot has 5 toes. How many toes do 6 feet have?

4. Each month for 5 months, Sophie makes 2 quilts. How many more quilts does she need to make before she has made 16 quilts?

1. Gavin saved $16 to buy packs of baseball cards. His father gives him $4 more. If each pack of cards costs $5, how many packs can Gavin buy?

2. Chelsea buys 8 packs of markers. Each pack contains the same number of markers. Chelsea gives 10 markers to her brother. Then, she has 54 markers left. How many markers were in each pack?

Lesson Check (3.OA.D.8)

30 toes

4 packs

6 quilts

8 markers

24 hours 4

Lesson 7.10 426

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.