247A Chapter 4

About the MathProfessional Development

LESSON AT A GLANCE

Interactive Student Edition

Personal Math Trainer

Math on the Spot

HMH Mega Math

LESSON 4.10

Problem Solving • Multiplication

Why Teach ThisWhen students learn to use tables to solve word problems, they become more proficient at finding the information necessary to write an equation, and in turn find the solution much easier and faster. Two important reasons for using tables in problem-solving:

• Tables provide a visual tool for organizing information. Multistep problems can become overwhelming for students if they cannot easily separate the various parts.

• Tables make it much easier to see which parts of the given information in a word problem are useful for finding the solution. Many students find it difficult to sift through the unnecessary facts in some word problems to determine the information they need.

Professional Development Videos

Learning ObjectiveSolve multiplication problems by using the strategy make a table.

Language ObjectiveStudents work with a partner to explain to another pair how you can solve multiplication problems using the make a table strategy.

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.3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.Also 3.OA.A.3, 3.OA.C.7

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

F C R Coherence:Standards Across the GradesBefore2.OA.C.4 2.OA.C.3

Grade 3 3.OA.D.8 3.OA.D.9

After4.OA.A.3 4.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 189J.

FOCUS COHERENCE RIGOR

ENGAGE1Daily Routines

Common Core

Lesson 4.10 247B

with the Interactive Student Edition

Essential QuestionHow can you use the strategy make a table to solve multiplication problems?

Making ConnectionsInvite students to tell you what they know about reading tables.

Why do we use tables? to organize information What are the parts of a table called? Rows are horizontal and columns are vertical. Have you ever used a table before? Tell us about it.

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

• How many tall cactus plants does Lucia see? 5

• What desert friends are hiding in the tall cactus plants? spiders, beetles, and ants

• How many spiders are hiding in each tall cactus plant? 3

• How many beetles are hiding in each tall cactus plant? 2

• How many ants are hiding in each tall cactus plant? 4

• What question does Lucia want to answer? how many desert friends are in all 5 tall cactus plants in total

Literacy and MathematicsChoose one or more of the following activities.

• Ask students to identify parts of the story that tell them what problem needs to be solved.

• Invite students to draw the 5 tall cactus plants and place the correct number of desert friends on each cactus.

Fluency BuilderMental Math Have students practice writing names for numbers. Ask them to write 8 different addition and subtraction facts with the answer 9. Possible answers: 1 + 8, 2 + 7, 3 + 6, 4 + 5, 10 − 1, 11 − 2 , 19 − 10, 29 − 20

Problem of the Day 4.10Delia bought 2 bags of oranges. Each bag had 5 oranges. How many oranges did Delia buy?

VocabularyInteractive Student EditionMultimedia Glossary e

10

How can you use the strategy make a table to solve multiplication

problems?

Common Core Fluency Standard 3.NBT.A.2

EXPLORE2

Pages with2 Stamps

TotalStamps

2

4

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8

10

12

14

16

Pages with1 Stamp

8

7

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4

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1

18

18

18

18

18

18

18

18

Unlock the ProblemUnlock the Problem

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Problem Solving • MultiplicationEssential Question How can you use the strategy make a table to solve multiplication problems?

Scott has a stamp album. Some pages have 1 stamp on them, and other pages have 2 stamps on them. If Scott has 18 stamps, show how many different ways he could put them in the album. Use the graphic organizer below to solve the problem.

What do I need to find?

What information do I need to use?

Scott has _ stamps. Some of the

pages have _ stamp on them, and

the other pages have _ stamps.

How will I use the information?

I will make a _ showing all the different ways of arranging the stamps in the album.

Make a table to show the number of pages with 1 stamp and with 2 stamps. Each row must equal

_ , the total number of stamps.

Read the Problem Solve the Problem

Chapter 4 247

1. What number patterns do you see in the table?

So, there are _ different ways.

PROBLEM SOLVINGLesson 4.10

Operations and Algebraic Thinking—3.OA.D.8, 3.OA.D.9 Also 3.OA.A.3, 3.OA.C.7

MATHEMATICAL PRACTICESMP1, MP4, MP5

how many different ways Scott could use

all 18 stamps with 1 stamp on some pages

and 2 stamps on others

18

1

2

18

8

Possible answers: in the fi rst column, the pattern is to count back by 1; in the second

column, the pattern is to count by twos.

table

1

4

3

2

8

6

3 4

12

9

5

20

15

16

12

Number ofPairs of Pages

Right Page

Left Page

1

1

2

1

2

2

4

2

3 4

3

6

3

4

8

4

Number ofPaintings

Blue Paint

Yellow Paint

Red Paint

Name

Using Tables

Solve. Use a table to organize the information for 1–2.

1. Marcy uses 1 bottle of blue paint, 2 bottles of yellow paint, and 1 bottle of red paint for one painting. How many bottles of paint will she need to make 4 of the same paintings?

2. Miguel puts photos in an album. The album has 5 right-hand and 5 left-hand pages. Each right-hand page holds 4 photos. Each left-hand page holds 3 photos. How many photos will fit in the album?

3. Renee earns 2 dimes each time she helps her mother. How many dimes will she earn if she helps her mother 4 times a day for a week?

4. Henry earns $6 every time he waters Mr. Young’s lawn. How much will Henry earn if he waters the lawn 4 times in July and 4 times in August?

5. Stretch Your Thinking Suppose Exercise 2 askedhow many photos fit on the first 6 pages of Miguel’s album. Explain how you could find the answer.

Lesson 4.10Enrich

Possible answer: there are 3 pairs of right- and left-hand pages in the first 6 pages. One pair of pages holds 4 1 3 5 7 photos. 3 3 7 5 21, so 6 pages would hold 21 photos.

56 dimes

35 photos

$48

16 bottles

4-24 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company

1

1

1

6

2

2 5

2

12

4 5

4

4 5

3024

Number of Pitchers

Cans of Fruit Juice

Bottles of Ginger Ale

Scoops of Sherbet

3

3

3

18

Name

6

I need to find how much of each ingredient Lucy’s mother needs to make 5 pitchers of punch.

11

table

1. Suppose Lucy’s mother decides to make 2 more pitchers of punch. How many scoops of sherbet would she need for 7 pitchers of punch? Explain your answer.

2. Jake gives his dog 4 chew bones and 1 dog toy each month. Howmany chew bones and how many toys will Jake give his dog in 5 months?

Problem Solving • Multiplication

Lucy’s mother is making punch for the students. For each pitcher, she uses 1 can of fruit juice, 1 bottle of ginger ale, and 6 scoops of sherbet. How much of each ingredient will she need to make 5 pitchers of punch?

Read the Problem Solve the ProblemWhat do I need to find? First, make a table with the information.

Next, look for information in the table that will help you solve the problem.

Look for a pattern. The cans of fruit juice and the bottles of ginger ale increase by 1. The scoops of sherbet increase by 6. Complete the table.

So, Lucy’s mother will need 5 cans of fruit juice, 5 bottles of ginger ale, and30 scoops of sherbet.

What information do I need to use?

Lucy’s mother uses can of fruit juice, bottle of ginger ale, and scoops of sherbet for each pitcher.

How will I use the information?

I will make a to show the total amounts of each ingredient Lucy’s mother needs.

Lesson 4.10Reteach

42 scoops; possible explanation:

number of scoops needed for 7 pitchers.for 1 pitcher. Multiply 7 3 6 to find the the table shows that 6 scoops are needed

20 chew bones and 5 toys

4-23 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company

DifferentiatedInstruction1

2

3

247 Chapter 4

Enrich 4.10Reteach 4.10

LESSON 4.10

Unlock the ProblemMATHEMATICAL PRACTICES

MP7 Look for and make use of structure. Be sure students understand that the column for the number of pages with 1 stamp has to start with 2 and continue by 2s because the total, 18, is an even number of stamps. Have volunteers explain how the numbers in the table are used to make a total of 18. Students should realize that they need to multiply the number in the first column by 2 because there are 2 stamps on each of the pages, and then add the number in the second column.• What do the pairs of numbers in the table

represent? All the possible number of pages containing 1 stamp or 2 stamps that will equal 18.

• How do you know your answer is reasonable? I used the table and a pattern to find every way that the stamps could be arranged. Each row equals 18 stamps, the total number of stamps Scott has.

• How would the table change if Scott had 20 stamps? The numbers in the first and third columns would change and there would be one more row.

MP2 Reason abstractly and quantitatively.• How would the numbers in the middle

column of the table change if Scott had 17 stamps? The numbers in the middle column would now be odd because the total number of stamps must be odd.

ELL Strategy: Understand Context Students can use modeling to understand unfamiliar cultural concepts or colloquial expressions.• Explain that some people collect items like

postage stamps and save them in albums.

• Show an example of a photo album or sticker or stamp album.

• Have students draw different ways to display stamps in an album. Remind them this is an array.

• How many stamps are on each page? Answers will vary. If there are 10 pages, how many stamps do you have?

3.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. 3.OA.D.9 Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations.

Try Another Problem

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248

2. What patterns do you see in this table? _______

3. How are these patterns different from the patterns in

the table on page 247? _____________

What if Scott bought 3 more stamps and now has 21 stamps? Some album pages have 1 stamp and some pages have 2 stamps. Show how many different ways he could put the odd number of stamps in the album.

What do I need to find?

What information do I need to use?

How will I use the information?

Read the Problem Solve the Problem

So, there are _ different ways.

how many ways Scott could put the stamps in the album if he bought 3 more stamps

Scott has 21 stamps. Some pages in his album have 1 stamp and some have 2 stamps.

I will make a table showing all the different ways of showing 1 or 2 stamps on a page.

Possible answer: 1st column: count back by 1;

2nd column: count by twos with odd numbers

so the pages with 1 stamp had even numbers beginning with 2. On this page, he has an

odd number of stamps, so the pages with 1 stamp are odd numbers beginning with 1.

on page 247, Scott had an even number of stamps,

Possible answer:

10

Pages with2 Stamps

Pages with1 Stamp

TotalStamps

10 1 21

9 3 21

8 5 21

7 7 21

6 9 21

5 11 21

4 13 21

3 15 21

2 17 21

1 19 21

COMMON ERRORS

Advanced Learners

Lesson 4.10 248

Error Students may have difficulty making a table from information in the word problem because they have difficulty understanding the relationship between the numbers.

Example Students may think they need to perform an operation using the numbers given in the problem.Springboard to Learning Have students underline the question. Ask them to explain how they could find the different ways that Scott could put stamps in his album. Guide them to realize that the numbers given describe a relationship. Once students understand the relationship, they can make a table to solve the problem.

Try Another ProblemHave students answer the questions in the graphic organizer and solve the problem. Invite them to share the tables and the patterns they found. Be sure students understand that since the total is odd, the number of pages with 1 stamp needs to be an odd number.• What if Scott got more stamps and now

has 24 stamps? Would the second column of the table have odd or even numbers? Explain. Even; Possible explanation: because the total is even, the pages with 1 stamp would need to be in pairs.

MP4 Model with mathematics. Have students use counters or drawings to show that the following statements are true or false. Ask students to justify their answers.

• The sum of two even numbers is always even. True; the total number of counters can be paired into groups of 2.

• The sum of two odd numbers is always odd. False; the total number of counters can be paired into groups of 2.

• The sum of an odd and even number is always even. False; the total number of counters when paired into groups of 2 will have one left over.

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

Logical / MathematicalIndividual

Display this table on the board.

?

? 1 2 3 4 5? 4 8 12 16 20

• Have students choose a title and labels for the table and challenge them to write a word problem that can be solved by using the information displayed. Possible answer: Title: Cars; Labels: Number of Cars; Number of Tires: Problem: There are 5 cars in the parking lot. Each car has 4 tires. How many tires are there in all?

• Have students share their problems with the class and have students solve the problems.

EXPLAIN3

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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1. Aaron’s mother is making lemonade. For each pitcher, she uses 1 cup of lemon juice, 1 cup of sugar, and 6 cups of water. What is the total number of cups of ingredients she will use to make 5 pitchers of lemonade?

First, make a table to show the number of cups of lemon juice, sugar, and water that are in 1 pitcher of lemonade.

Next, multiply to find the number of cups of water needed for each pitcher of lemonade.

Last, use the table to solve the problem.

So, in 5 pitchers of lemonade, there are _ cups of

lemon juice, _ cups of sugar, and _ cups of water.

This makes a total of _ cups of ingredients.

2. What if it takes 4 lemons to make 1 cup of lemon juice? How many lemons would it take to make 5 pitchers? Explain how you can use the table to help you find the answer.

3. What pattern do you see in the total number of cups of ingredients?

Chapter 4 • Lesson 10 249

Name

Think: For every pitcher, the number of cups of water increases by 6.

20 lemons; Possible explanation: the table shows that 5 cups of lemon juice are needed

for 5 pitchers. I can multiply 5 × 4 to fi nd the number of lemons needed for 5 pitchers.

5

305

Possible answer: the total number of cups increases by 8 cups for each pitcher: 8 cups

for 1 pitcher, 16 cups for 2, 24 for 3, 32 for 4, and 40 for 5.

40

Number of Pitchers 1

Cups of Lemon Juice 1

Cups of Sugar 1

Cups of Water

Total Number of Cupsof Ingredients

6

2

2

2

12

3

3

3

18

4

4

4

24

5

5

5

30

8 16 24 32 40

249 Chapter 4

a student misses the checked exercises

Differentiate Instruction with • Reteach 4.10 • Personal Math Trainer 3.OA.D.8,

3.OA.D.9 • RtI Tier 1 Activity (online)

Share and Show MATHBOARDMATHBOARD

The first problem connects to the learning model. Have students use the MathBoard to explain their thinking.Also, Exercise 1 is scaffolded to help students set up a table and understand the relationships among the numbers.Check students’ tables to be sure they have used the correct relationships between the numbers.Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

ELABORATE4

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES

EVALUATE5 Formative Assessment

Differentiated Centers Kit

DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES

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

On Your OwnOn Your Own

MATHEMATICAL PRACTICES ANALYZE

Personal Math Trainer

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4. Julie saw 3 eagles each day she went bird-watching. How many eagles did Julie see in 6 days?

5. MATHEMATICALPRACTICE 2 Use Reasoning Greg has a dollar bill,

quarters, and dimes. How many ways can he make $1.75?

Name the ways. _______

6. SMARTER Cammi needs 36 postcards. She buys 4 packages of 10 postcards. How many postcards will Cammi have left over? Explain.

7. DEEPER Phillip has 8 books on each of 3 bookshelves. His aunt gives him 3 new books. How many books does Phillip have now?

8. SMARTER Stuart has some 2-ounce, 3-ounce, and 4-ounce weights. How many different ways can Stuart combine the weights to make a total of 12 ounces? List the ways.

4 postcards; Possible explanation: Cammi buys 4 × 10 = 40

postcards. She only needs 36. 40 − 36 = 4, so she has

4 postcards left over.

18 eagles

Possible answer: 6 ways: 1 $1 bill,

27 books

3 quarters; 1 $1 bill, 1 quarter, 5 dimes; 7 quarters; 5 quarters,

5 dimes; 3 quarters, 10 dimes; 1 quarter, 15 dimes

7 different ways; six 2-oz weights; four 3-oz weights; three 4-oz weights;

one 2-oz weight, two 3-oz weights and one 4-oz weight; two 2-oz weights

and two 4-oz weights; three 2-oz weights and two 3-oz weights; four 2-oz

weights and one 4-oz weight

Lesson 4.10 250

Students practice multiplication facts between 1 and 9 by guessing the factors to make a given product.

Students read about how Amanda uses multiplication to find the number of wooden stool legs she needs to carve.

Students complete purple Activity Card 15 by practicing multiplication facts through 10 by 10.

ActivitiesSum Sentences

LiteratureThe Workshop

GamesGuess My Numbers

Essential QuestionUsing the Language ObjectiveReflect Have students work with a partner and explain to another pair to answer the Essential Question.How can you use the strategy make a table to solve multiplication problems? It organizes the information, makes patterns easier to see, and ensures I have found all the possible solutions.

Math Journal WRITE MathWrite a problem you can use the make a table strategy to solve. Then solve the problem.

On Your Own MATHEMATICAL PRACTICES

MP2 Reason abstractly and quantitatively. Exercise 5 requires students to analyze a multistep problem involving combinations of coins.

SMARTER

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

DEEPER

Have students write an expression that could be used to solve the problem. Students might write (8 × 3) + 3. Remind students to complete the multiplication step first and then add.

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 might identify the 3 combinations that use only the same weights, but miss other combinations. Students should make a table to help them solve the problem. Meeting Individual Needs

Extend the Math Activity

Name

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Chapter 4 251

Solve.

1. Henry has a new album for his baseball cards. He uses pages thathold 6 cards and pages that hold 3 cards. If Henry has 36 cards, how many different ways can he put them in his album?

Henry can put the cards in his album _ ways.

2. Ms. Hernandez has 17 tomato plants that she wants to plant in rows. She will put 2 plants in some rows and 1 plant in the others. How many different ways can she plant the tomato plants? Make a table to solve.

Ms. Hernandez can plant the tomato plants _ ways.

3. WRITE Math Write a problem you can use the make a table strategy to solve. Then solve the problem.

Pages with 6 Cards 1 2 3 4 5

Pages with 3 Cards 10 8 6 4 2

Total Cards 36 36 36 36 36

5

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

Lesson 4.10Practice and Homework

Rows with 2 Plants

8 7 6 5 4 3 2 1

Rows with 1 Plant

1 3 5 7 9 11 13 15

Total Plants

17 17 17 17 17 17 17 17

Possible table is shown.

Check students’ work.

8

251 Chapter 4

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.

Additive and Multiplicative Comparison

Investigate In Grade 4, students will need to distinguish the difference between additive comparison and multiplicative comparison problems. The language used in this lesson provides a foundation for this skill.

• Write these problems on the board. Discuss how to write a number sentence to match each problem.1. A toy bird costs $5. A toy dog costs 2 times as much

as the toy bird. How much does the toy dog cost? 2 3 $5 5

2. A toy boat costs $12 more than a toy train. The train costs $15. How much does the boat cost? $15 1 $12 5

3. A toy truck costs $8. A toy car costs $2. How many times as much does the truck cost than the car?

3 $2 5 $8

• Explain to students that additive comparison problems show comparisons that involve addition. Read Exercise 2 aloud and help students identify the words “more than.” Then read Exercise 1 aloud and point out the words “times as much as,” and explain that this comparing sentence involves multiplication.

• Group students into pairs. Have each student write one type of comparison problem. Then students exchange problems and write equations using a box to represent the unknown. After solving the problems, the group discusses whether the equations and answers are correct.

Summarize Give students an opportunity to share their problems and solutions with the class. They should be able to match the language in the problem to the corresponding known and unknown numbers.

Personal Math Trainer

FOR MORE PRACTICE GO TO THE

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Lesson Check (3.OA.D.8)

1. The table shows different ways that Cameron can display his 12 model cars on shelves. How many shelves will display 2 cars if 8 of the shelves each display 1 car?

2 4 6 8 10

5

12

4

12

3

12 12 12Total cars

Shelves with 2 Cars

Shelves with 1 Car

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

2. Find the sum. 3. The school cafeteria has an order for 238 hot lunches. What is 238 rounded to the nearest ten?

4. Tyler made a picture graph to show students’ favorite colors. This is the key for his graph.

Each = 3 votes.

If 12 students voted for green, how many should there be in the green row of the graph?

5. There are 5 bikes in each bike rack at the school. There are 6 bike racks. How many bikes are in the bike racks?

317 + 151

__

2 shelves

240

30 bikes4

468

Lesson 4.10 252

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.