93A Chapter 2

About the MathProfessional Development

LESSON AT A GLANCE

Learning ObjectiveDivide 3- and 4-digit dividends by 1-digit divisors.

Language ObjectiveStudents share their best advice on how to solve and check division problems.

MaterialsMathBoard

F C R Focus:Common Core State Standards5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies

based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

MATHEMATICAL PRACTICESMP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP8 Look for and express regularity in repeated reasoning.

F C R Coherence:Standards Across the GradesBefore4.NBT.B.6

Grade 55.NBT.B.6

After6.NS.A.16.NS.B.2

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 85J.

FOCUS COHERENCE RIGOR

LESSON 2.2

Interactive Student Edition

Personal Math Trainer

Math on the Spot

Animated Math Models

iTools: Base-Ten Blocks

HMH Mega Math

Professional Development Videos

Teaching for DepthPrior to this lesson, students checked their work by using compatible numbers to make estimates of quotients. In this lesson, they use inverse operations in the form of related number sentences.

Multiplication and division are inverse operations, so the related number sentences students use to check their work involve multiplication and division. For example:

10 ÷ 2 = 5

5 × 2 = 10

To check the division 10 ÷ 2 = 5, we multiply the quotient (5) by the divisor (2) and compare the product to the dividend (10). Since 5 × 2 = 10, the division checks.

Divide by 1-Digit Divisors

Multimedia and Technology

ENGAGE1Daily Routines

Common Core

inverseoperations

How do you solve and check division

problems?

Lesson 2.2 93B

with the Interactive Student Edition

Essential QuestionHow do you solve and check division problems?

Making ConnectionsInvite students to tell you what they know about relating multiplication and division. Ask the following questions.

What are inverse operations? Possible answer: Inverse operations are operations that use the same numbers to do the opposite. What are two examples of inverse operations? addition and subtraction; multiplication and division

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

• How do you use subtraction to check the result of addition? I subtract one of the addends from the sum. The result should be the other addend.

• How can you use multiplication to check division? I can multiply the quotient by the divisor. The result should be the dividend.

Literacy and MathematicsView the lesson opener with the students. Then, choose one or more of the following activities.

• Have students write an explanation of the relationship between two basic multiplication facts and their inverse division facts.

• Have students write a story that involves two strategies they could use to solve 5,618 ÷ 9.

Vocabulary BuilderMaterials Vocabulary Cards (see eTeacher Resources)

Use vocabulary cards to help students remember definitions for review words.

Ask students to respond to a question or statement such as:

In what sort of problem might you use.....?

How would you know when to use.....?

Problem of the Day 2.2Holly has 38 pecans. She has 5 friends coming to visit. If Holly and her 5 friends each eat an equal number of pecans, about how many pecans will each person eat? about 6 pecans

VocabularyInteractive Student EditionMultimedia Glossary e

n

EXPLORE25.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

92 7

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Unlock the ProblemUnlock the Problem

MathTalk MATHEMATICAL PRACTICES 2

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Lesson 2.2

Divide by 1-Digit DivisorsEssential Question How do you solve and check division problems?

• Underline the sentence that tells you what you are trying to find.

• Circle the numbers you need to use.

Jenna’s family is planning a trip to Oceanside,

California. They will begin their trip in Scranton,

Pennsylvania, and will travel 2,754 miles over

9 days. If the family travels an equal number of

miles every day, how far will they travel each day?

Divide. 2,754 ÷ 9

STEP 1

Use an estimate to place the first digit in the quotient.

Estimate. 2,700 ÷ 9 = ___

The first digit of the quotient is in

the ___ place.

STEP 2

Divide the hundreds.

STEP 3

Divide the tens.

STEP 4

Divide the ones.

Since ___ is close to the estimate of ___ ,

the answer is reasonable.

So, Jenna’s family will travel ___ miles each day.

Number and Operations in Base Ten—5.NBT.B.6

MATHEMATICAL PRACTICESMP1, MP2, MP8

Reasoning Explain how you know the quotient is 306 and not 36.

300

hundreds

306

306

Possible explanation: I know that I need to place

the fi rst digit of the quotient in the hundreds

place, so the quotient should be a 3-digit number.

300

Name

So, 757 4 4 is 189 r1.

quotient

divisor

remainder

dividend

Divide by 1-Digit Divisors

You can use compatible numbers to help you place the first digit in the quotient. Then you can divide and check your answer.

Divide. 4 q w 757

Step 1 Estimate with compatible numbers to decide where to place the first digit.

757 4 4

800 4 4 5 200

The first digit of the quotient is in the hundreds place.

Step 2 Divide.

4 q w 757 24 35

232

37

36

1

189 r1

2

Step 3 Check your answer.

189

3 4

756

1 1

757

Since 189 is close to the estimate of 200, the answer is reasonable.

Divide. Check your answer.

1. 8 q w 136 2. 7 q w 297 3. 5 q w 8,126

4. 7 q w 4,973 5. 3 q w 741 6. 7 q w 456

Lesson 2.2

Reteach

Check students’ work.

65 r1247710 r3

1,625 r142 r317

2-7 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

Name

1. 4

7 q w 287

4. 1 r4

9 q w 11

7. 4 r5

7 q w

4 1

10. 1, 5 2 9 r2

4 q w 6, 1

13. 1, 48 r1

6 q w

,93 5

2. 2

3 q w 18

5. 4 0 r5

q w 3 5

8. 3 6 r1

q w 7 3 5

11. 2,5 4

q w 7,6 6 2

14. 245

q w 1,225

3. 5 5

q w 44

6. 2,2 0 r1

2 q w 4,4

9

9. 1, 9 7 r

3 q w 5, 8 7 2

12. 5 3 r3

4 q w

5

15. 8 r3

7 q w 5 9

16. Explain the strategy

you used to solve Exercise 1.

17. Stretch Your Thinking Explain

how you would solve a division

problem with an unknown divisor.

Division Detective

For each exercise below, fi nd the unknown number

that belongs in each box. Not all boxes will need a number.

Lesson 2.2

Enrich

5

2

6

61

1

Possible answer:

I would estimate

possible divisors.

Then I would multiply

each possible divisor

by the quotient.

Possible answer:

I divided the tens and

then the ones. Then I

checked my answer

by multiplying the

quotient by the divisor.

8 4

79

8 3 2 1

5

3 2

1578

2 8 0

42

08

2-8 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

DifferentiatedInstruction1

2

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93 Chapter 2

Unlock the Problem MATHEMATICAL PRACTICES

MP1 Make sense of problems and persevere in solving them.What is your estimate of the driving distance in miles from Scranton, Pennsylvania, to Oceanside, California? Read the problem to learn the distance.After students underline and circle the relevant information, ask:

• Why do we use division to solve this problem? Possible answer: Division is used to find the number of equal groups or the number in each group. I know that in this problem, 2,754 is the whole, and 9 is the number of groups. So, I divide 2,754 by 9 to find the number of miles they will travel each day.

In Step 1, ask students to name the basic division fact that is used to place the first digit in the quotient, and have them explain how that fact is used to estimate the quotient. 27 ÷ 9 is the basic fact; since the dividend has 4 digits, 2,700 ÷ 9 is used to produce an estimate of the quotient, or 300.

MathTalk Use Math Talk to focus on students’

understanding of using zero as a placeholder in a quotient.

• How does your estimate help you know that the quotient is 306 and not 36? Possible answer: My estimate of the quotient is 300. I know the answer is going to be close to 300, which makes a quotient of 36 unreasonable.

ELL Strategy: Frontload

Frontload the language and vocabulary of division by color-coding division problems to ensure comprehension of the concept.

• Have students make a color key on their paper; Quotient is red, Product is blue, Dividend is orange, and Divisor is purple.

• Model a division problem, writing the numbers in the corresponding colors while using the vocabulary and pointing to the related part of the problem.

LESSON 2.2

Enrich 2.2Reteach 2.2

Meeting Individual NeedsDifferentiatedInstruction

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connect Division and multiplication are inverse operations. Inverse

operations are opposite operations that undo each other. You can

use multiplication to check your answer to a division problem.

You can use what you know about checking division to find an

unknown value.

To check your answer to a division problem, multiply the quotient by the divisor. If there is a remainder, add it to the product. The result should equal the dividend.

Since the result of the check is equal to the dividend,

the division is correct.

So, 614 ÷ 6 is __.

Try This! Find the unknown number by finding the value of

n in the related equation.

+ 2

102× 6

quotient divisor

remainder

dividend

Example Divide. Check your answer.

102 r2

6 q w 614 –6

01 –0

14 –12

2

dividend divisor quotient

Multiply the divisor and the quotient.

n = __

63 7 q w

n = 7 × 63

125 r 6 q w 752

B

752 5 6 3 125 1 n

dividend divisor quotient remainder

Multiply the divisor and the quotient.

752 = 750 + n

Think: What number added to 750 equals 752?

n = __

102 r2

441

441

612

614

2

2

COMMON ERRORS

85 6

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r7 9 5

6 3 7

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6 ,4

1 2

11

r3 2 1

2, 5 9 32 49

1 3

1

8

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Advanced Learners

Lesson 2.2 94

CONNECT It is important for students to understand that they can multiply the quotient they found by the divisor and the product should be the dividend.

Example MP1 Make sense of problems and persevere in solving them.If the results of checking the answer and the dividend are identical, the answer is correct. If they differ, point out that students should look for an error instead of performing the computation a second time. (When students find and recognize their own errors, they are less likely to make those same errors a second time.)

Try This!• How can you use what you know about

checking division to find the value of n in the equation for A? To check division, I multiply the quotient by the divisor to see if I get the dividend. The letter n represents the dividend, so I can multiply the quotient by the divisor to find the value of n.

• For B, why do you first find 6 × 125? Possible answer: when I check division, I multiply the divisor by the quotient before I add the remainder.

Error When dividing, subtraction produces a number that is greater than or equal to the divisor.

Example 63 ) — 2,3 0 7 − 1 8 ___ 5

Springboard to Learning Remind students to compare the result of every subtraction to the divisor. Point out that the division cannot continue if the result of the subtraction is greater than or equal to the divisor. If the remainder is greater than the divisor, another group of the same size as the divisor can be made.

• Provide students with the following problems.

• Explain that each shaded box represents an unknown digit. Have students find each unknown digit.

• Students may wish to write similar problems for their classmates to solve.

KinestheticIndividual/Partners

DifferentiatedInstruction

EXPLAIN3

Quick Check

If

Rt I 1

2

3

Quick Check

If

Rt I 1

2

3

Then

On Your OwnOn Your Own

Share and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowShare and ShowMATHBOARDMATHBOARDMATHBOARDMATHBOARDMATHMATHMATHMATHBOARDBOARDBOARDBOARD

MathTalk MATHEMATICAL PRACTICES 8

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Chapter 2 • Lesson 2 95

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Divide. Check your answer.

1. 8 q w 624 Check. 2. 4 q w 3,220 Check. 3. 4 q w 1,027 Check.

4. 6 q w 938 5. 4 q w 762 6. 3 q w 5,654 7. 8 q w 475

Divide.

8. 4 q w 671 9. 9 q w 2,023 10. 3 q w 4,685 11. 8 q w 948

Practice: Copy and Solve Divide.

12. 1,326 ÷ 4 13. 5,868 ÷ 6 14. 566 ÷ 3 15. 3,283 ÷ 9

16. n = 4 × 58

n = ___

17. 589 = 7 × 84 + n

n = ___

18. n = 5 × 67 + 3

n = ___

MATHEMATICALPRACTICE 2 Use Reasoning Algebra Find the value of n in each equation.

Write what n represents in the related division problem.

Generalize Explain how multiplication can help you check a quotient.

78

Possible explanation: I can multiply the quotient by the divisor. If there is no remainder, the product should equal the dividend. If there is a remainder, add it to the product to get the dividend.

167 r3 224 r7 1,561 r2 118 r4

331 r2 978 188 r2 364 r7

232; dividend 1; remainder 338; dividend

805 256 r3

156 r2 190 r2 1,884 r2 59 r3

78 3 8

_

624

805 3 4

_

3,220

256 3 4

__

1,024 1 3

__

1,027

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95  Chapter 2

On Your Own If students complete the checked exercises correctly, they may continue with the remaining exercises.MP2 Reason abstractly and quantitatively. For Exercises 16–18, you may want to point out that each shows the check of a division problem. Encourage students to write the related division problem.

a student misses the checked exercises

Differentiate Instruction with • Reteach2.2

• PersonalMathTrainer5.NBT.B.6

• RtITier1Activity(online)

Share and Show MATHBOARDMATHBOARD

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.

MathTalk Use Math Talk to focus on students’

understanding of checking a quotient.

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES EVALUATE5 Formative Assessment

ELABORATE4

GamesWhat’s Left?

Students write and solve division problems to make a specific remainder.

LiteratureNiagara Falls Numbers

Games

Large Gold Nuggets Found

Welcome Stranger

Welcome

Willard

2,284 troy ounces

2,217 troy ounces

788 troy ounces California

Australia

Australia

Name Weight Location

elcome

Willard

2,217 troy ounces

788 troy ounces

WRITE Math • Show Your Work

MATHEMATICAL PRACTICES MODEL • REASON • MAKE SENSE

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Problem Solving • ApplicationsProblem Solving • Applications

23. SMARTER Lily's teacher wrote the division problem on the board.Using the vocabulary box, label the parts of the division problem. Then, using the vocabulary, explain how Lily can check whether her teacher's quotient is correct.

82 9 q w 738

quotient divisor dividend

Use the table to solve 19–21.

19. If the Welcome gold nugget were turned into 3 equal-sized gold bricks, how many troy ounces would each brick weigh?

20. Pose a Problem Look back at Problem 19. Write a similar problem by changing the nugget and the number of bricks. Then solve the problem.

21. DEEPER Suppose the Willard gold nugget was turned into 4 equal-sized gold bricks. If one of the bricks was sold, how many troy ounces of the Willard nugget would be left?

22. SMARTER There are 246 students going on a field trip to pan for gold. If they are going in vans that hold 9 students each, how many vans are needed? How many students will ride in the van that isn’t full?

739 troy ounces

Possible answer: If the Welcome Stranger nugget

were turned into 4 equal-sized gold bricks, how

many troy ounces would each brick weigh? 571

Lily can check her teacher’s work by multiplying the divisor and the quotient. If

the quotient is correct, the product should equal the dividend.

591 troy ounces

28 vans; 3 students

troy ounces

quotient

dividenddivisor

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

Lesson 2.2 96

Essential QuestionUsing the Language ObjectiveReflect Have students share their best advice on how to answer the essential question. How do you solve and check division problems? Possible answer: I multiply the quotient by the divisor, and then add the remainder to the product. If the result is identical to the dividend, the problem checks.

Math Journal WRITE Math

Use a map to plan a trip in the United States. Find the number of miles between your current location and your destination, and divide the mileage by the number of days or hours that you wish to travel.

MATHEMATICAL PRACTICES

A troy ounce is a unit of weight used for precious metals. One troy ounce is about the same as 1.1 customary ounces.For Problem 20, invite volunteers to share their problems and answers with the class.

SMARTER

Problem 22 requires students to interpret the remainder.• What is the problem asking you to find?

How many vans are needed and how many students ride in a van that isn't full.

SMARTER

Problem 23 assesses a student's ability to solve and check a division problem. Using the vocabulary words, students identify a divisor, dividend, and quotient. Then, students explain how to check a division problem using a divisor, dividend, and quotient.

Students read the book and solve division problems about Niagara Falls.

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

DO NOT EDIT--Changes must be made through "File info" CorrectionKey=B

Problem Solving • Thinking

Meeting Individual Needs

Problem Solving • Applications

Cross-Curricular

Problem SolvingProblem Solving

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Divide by 1-Digit Divisors

Divide.

5. n = 3 × 45

____

6. 643 = 4 × 160 + n

____

7. n = 6 × 35 + 4

____

8. Randy has 128 ounces of dog food. He feeds his

dog 8 ounces of food each day. How many days

will the dog food last?

______

9. Angelina bought a 64-ounce can of lemonade

mix. She uses 4 ounces of mix for each pitcher of

lemonade. How many pitchers of lemonade can

Angelina make from the can of mix?

______

Chapter 2 97

Find the value of n in each equation. Write what n represents

in the related division problem.

1. 4 q w 724

___

2. 5 q w 312

___

3. 278 ÷ 2

___

4. 336 ÷ 7

___

10. WRITE Math Use a map to plan a trip in the United States. Find the

number of miles between your current location and your destination, and

divide the mileage by the number of days or hours that you wish to travel.

4 q w 724 181

−432

−32

04− 4

0

COMMON CORE STANDARD—5.NBT.B.6Perform operations with multi-digit whole numbers and with decimals to hundredths.

Lesson 2.2Practice and Homework

16 days

Check students’ work.

16 pitchers

n = 3; remaindern = 135; dividend n = 214; dividend

62 r2 139 48181

97 Chapter 2

SCIENCE SOCIAL STUDIES

• The gain of heat energy causes water to change phase from solid (ice) to liquid to gas (water vapor). When the temperature reaches ice’s melting point of 32°F, it begins to form a liquid. At sea level, when water is heated to its boiling point of 212°F, it begins to change into a gas. Every substance has its own melting and boiling points. For example, iron’s melting point is 2,795°F and its boiling point is 4,982°F.

• At sea level, Toshi puts a container of 59°F water on a hot stove. The water temperature increases 9°F every minute. How long will it take for the water to reach its boiling point of 212ºF? 17 minutes

• Before gasoline-powered tractors, steam-powered threshing machines were used to harvest wheat. John Froelich invented the first gasoline-powered tractor in 1892. The tractor could move forward and backward. Only four of these tractors were ever made, but they were the stepping stones to today’s tractors.

• John Froelich used his tractor to harvest 72,000 bushels of wheat in 52 days, or about 1,384 bushels a day. If he worked 8 hours to harvest 1,384 bushels, how many bushels did he harvest each hour? 173 bushels

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.

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98

Lesson Check (5.NBT.B.6)

1. A color printer will print 8 pages per minute.

How many minutes will it take to print a report

that has 136 pages?

2. A postcard collector has 1,230 postcards.

If she displays them on pages that hold

6 cards each, how many pages does she need?

Spiral Review (5.NBT.A.1, 5.NBT.B.5, 5.NBT.B.6)

3. Francis is buying a stereo system for $196.

She wants to pay for it in four equal monthly

installments. What is the amount she will pay

each month?

4. A bakery bakes 184 loaves of bread in 4 hours.

How many loaves does the bakery bake in

1 hour?

5. Marvin collects trading cards. He stores

them in boxes that hold 235 cards each.

If Marvin has 4 boxes full of cards, how many

cards does he have in his collection?

6. What is the value of the digit 7 in 870,541?

17 minutes

$49

940 cards

205 pages

46 loaves

70,000

Lesson 2.2 98

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.