6
For the student For the teacher Teacher Edition Math on the Spot Video Tutor Online Assessment System Soar to Success Math Online Intervention i Tools Virtual Manipulatives Digital Management Center organizes program resources by TEKS! Interactive Student Edition provides students with an interactive learning environment! Resources e Texas Essential Knowledge and Skills Number and Operations—3.4.D Determine the total number of objects when equally-sized groups of objects are combined or arranged in arrays up to 10 by 10 3.4.E Represent multiplication facts by using a variety of approaches such as equal-sized groups and arrays Algebraic Reasoning—3.5.B Represent and solve one- and two-step multiplication problems within 100 using arrays and equations Also 3.4.K MATHEMATICAL PROCESSES 3.1.C Select tools, technology, and techniques 3.1.D Communicate mathematical ideas and reasoning Are You Ready? Access Prior Knowledge Use the Are You Ready? 6.4 in the Assessment Guide to assess students’ understanding of the prerequisite skills for this lesson. Vocabulary array Go to Multimedia eGlossary at thinkcentral.com Materials square tiles, MathBoard 6.4 Model with Arrays How can you use arrays to model multiplication and find factors? Essential Question ? Lesson Opener Making Connections Engage students in a discussion about multiplication. Ask the following questions. What multiplication sentence can you write for 9 + 9? (2 × 9) What addition sentence can you write for 4 × 3? (3 + 3 + 3 + 3) How is multiplication like addition? (Possible answer: You can use the Commutative Property and the Associative Property to find answers to multiplication and addition problems.) Using the Digital Lesson If you want, ask students to look at the image of the leaves lined up in rows and columns. Ask students to discuss how this model makes it easier to solve the problem. Learning Task What is the problem the students are trying to solve? Connect the story to the problem. What question are you asked to answer? (Possible answer: how many leaves blew off the tree) What is one way you could find out how many leaves were blown from the tree? (Possible answer: by finding the sum of all the leaves) Suppose the wind blew 15 leaves from the tree. What addition sentence could you write showing 3 equal groups of leaves? (5 + 5 + 5 = 15) What multiplication sentence could you write for this addition sentence? (3 × 5 = 15) Literacy and Mathematics Have students work in pairs and create a story problem that involves multiplying 2 whole numbers to find the solution. Have them write the multiplication sentence as a repeated addition sentence and explain why the 2 equations are equivalent. Have students discuss the 4 seasons of a year and name the season when trees typically drop their leaves. Ask students if they have ever collected fallen leaves. How can you use arrays to model multiplication and find factors? Lesson 6.4 191A

6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

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Page 1: 6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

For the student For the teacher

Teacher Edition

Math on the Spot Video Tutor Online Assessment

System

Soar to Success Math Online Intervention

iTools Virtual Manipulatives

Digital ManagementCenter organizes program resources by TEKS!

Interactive Student Edition provides students

with an interactive learning environment!

Resources

e

Texas Essential Knowledge and Skills

Number and Operations—3.4.D Determine the total number of objects when equally-sized groups of objects are combined or arranged in arrays up to 10 by 10

3.4.E Represent multiplication facts by using a variety of approaches such as equal-sized groups and arrays

Algebraic Reasoning—3.5.B Represent and solve one- and two-step multiplication problems within 100 using arrays and equations

Also 3.4.K

MATHEMATICAL PROCESSES3.1.C Select tools, technology, and techniques3.1.D Communicate mathematical ideas and reasoning

Are You Ready?Access Prior KnowledgeUse the Are You Ready? 6.4 in the Assessment Guide to assess students’ understanding of the prerequisite skills for this lesson.

Vocabularyarray

Go to Multimedia eGlossary at thinkcentral.com

Materialssquare tiles, MathBoard

6.4 Model with Arrays How can you use arrays to model multiplication and find factors?

Essential Question?

Lesson OpenerMaking ConnectionsEngage students in a discussion about multiplication. Ask the following questions.

What multiplication sentence can you write for 9 + 9? (2 × 9) What addition sentence can you write for 4 × 3? (3 + 3 + 3 + 3) How is multiplication like addition? (Possible answer: You can use the Commutative Property and the Associative Property to find answers to multiplication and addition problems.)

Using the Digital LessonIf you want, ask students to look at the image of the leaves lined up in rows and columns. Ask students to discuss how this model makes it easier to solve the problem.

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

• What question are you asked to answer? (Possible answer: how many leaves blew off the tree)

• What is one way you could find out how many leaves were blown from the tree? (Possible answer: by finding the sum of all the leaves)

• Suppose the wind blew 15 leaves from the tree. What addition sentence could you write showing 3 equal groups of leaves? (5 + 5 + 5 = 15) What multiplication sentence could you write for this addition sentence? (3 × 5 = 15)

Literacy and Mathematics• Have students work in pairs and create a story problem that involves multiplying

2 whole numbers to find the solution. Have them write the multiplication sentence as a repeated addition sentence and explain why the 2 equations are equivalent.

• Have students discuss the 4 seasons of a year and name the season when trees typically drop their leaves. Ask students if they have ever collected fallen leaves.

How can you use arrays to model multiplication and

f ind factors?

Lesson 6.4 191A

Page 2: 6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

HandsOn

Unlock the ProblemUnlock the Problem

Essential Question? How can you use arrays to model multiplication and find factors?

Model with Arrays6.4 Number and Operations—3.4.D, 3.4.E

Algebraic Reasoning—3.5.B Also 3.4.KMATHEMATICAL PROCESSES3.1.C, 3.1.D

Many people grow tomatoes in their gardens. Lee plants 3 rows

of tomato plants with 6 plants in each row. How many tomato

plants are there?

Activity 1Materials ■ square tiles ■ MathBoard

• You make an array by placing the same number of tiles in

each row. Make an array with 3 rows of 6 tiles to show the

tomato plants.

• Now draw the array you made.

• Find the total number of tiles.

Multiply. 3 × 6 = _

↑ ↑ number number of rows in each row

So, there are _ tomato plants.

Mathematical ProcessesMath Talk

Does the number of tiles change if you turn the array to show 6 rows

of 3? Explain.

Name

Check students’ drawings.

18

18

No; possible explanation: if you only turn the tiles, the number stays the same because you do not add or take away any of the tiles.

Module 6 191

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English Language Learners Language SupportELL

1 2 3 48 9 10 11 12

13 14 15 16 17 18 1920 21 22 23 24 25 2627 28 29 30 31

56 7

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

July

ELPS 1.A.1, 3.B.3, 3.D.2Leveled Activities ELPS

Beginning: Activity 8 2.I.4, 3.G.1, 3.H.3

Intermediate: Activity 3 2.D.2, 2.E.3, 3.F.2

Advanced: Activity 57 2.C.4, 3.D.2, 3.E

Advanced High: Activity 59 1.F, 3.E, 3.H.3, 4.C.3

Go to thinkcentral.com for the ELL Activity Guide containing these leveled activities.

Strategy: Identify Relationships• Students can learn the concept of an array by relating common objects

to arrays.• Review the definition of array.• Ask students to identify arrays in the

classroom. Possible answer: desks in a row, papers on a bulletin board, or the calendar

• Choose one array in the classroom, and have students describe it using this model: The calendar has 5 rows of squares with 7 squares in each row.

VisualSmall Group

Unlock the ProblemWhat do many people grow in gardens? Read the problem to find out.

Be sure students understand that rows go across.

Activity 1This activity introduces students to an array as another way to represent multiplication.

Work through the activity with students.

• How many tiles will you put in each row? 6 tiles How many rows do you have? 3 rows

• What method did you use to find the product? Possible answer: I skip counted 3 times by the number in each row.

• What if Lee planted 7 tomato plants in each row, instead of 6? How would your array be different? Each row of the array would have 7 tiles. I’d have to add 1 tile to each row.

• How could you find the new total number of tiles? Possible answers: I know 3 × 6 = 18. I added 1 more tile to each of 3 rows, so I can add 3 to 18 to get 21; I can multiply 3 × 7 = 21.

HandsOn

Math Talk Use Math Talk to focus on students’ understanding of the proper format of rows and columns. The first factor tells the number of rows and the second factor tells the number in each row. So, turning the array helps introduce the concept of the Commutative Property of Multiplication.

Mathematical Processes

191 Module 6

Page 3: 6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

HandsOn

Share and ShowShare and Show

8 rows of _

8 × _ = 8

_ rows of _

_ × _ = 8

_ rows of _

_ × _ = 8

1 row of 8

1 × 8 = 8

Activity 2

Materials ■ square tiles ■ MathBoard

Use 8 tiles. Make as many different arrays as you can,

using all 8 tiles. Draw the arrays. The first one is done for you.

A B

C D

You can make _ different arrays using 8 tiles.

1. Complete. Use the array.

_ rows of _ = _

_ × _ = _

Write a multiplication sentence for the array.

2.

3.

1

1

2 4 4

4 2

2

4

4

2

2

2

10

10

20

20

4 × 6 = 24 3 × 8 = 24

192

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Go to Go to thinkcentral.com for additional enrichmentactivities in the Enrich Activity Guide.

Enrich

1

2

3

a student misses the checked exercises

Quick Check

IF

THENDifferentiate Instruction withRtI Tier 1 Lesson 26

Activity 2In this activity, students make arrays to show all the factors of 8.

• What are all the factors of 8? 1, 2, 4, and 8

• Which number in the multiplication sentences represents the number of rows? the first number

Share and ShowThe 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.

HandsOn

Materials: square tiles, 1-Centimeter Grid Paper (see eTeacher Resources)

• Write the following numbers on the board

8 12 16 20

• Have students choose one of the numbers and make as many different arrays as they can with that number of squares.

• Then have students record the arrays on the grid paper and write the corresponding multiplication sentence for each array.

SpatialIndividual

COMMON ERRORSError Students may incorrectly make an array to solve a problem.

Example 3 × 7 =

There are 24 tiles in the array.

Springboard to Learning Have students circle the second factor in a multiplication sentence to remind them that this number represents the number in each row. Then have them draw this number of tiles in the first row and use the first number to determine how many rows to draw.

CE

Lesson 6.4 192

Page 4: 6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

Problem SolvingProblem Solving

Write MathWrite Math Show Your Work

4 rows of 6

2 rows of 8

4 rows of 7

5 rows of 9

Mr. Bloom’s Garden

Beans

Carrots

Beets

Corn

Vegetable Planted In

Name

Use the table to solve 4–6.

4. Representations Mr. Bloom grows

vegetables in his garden. Draw an array

and write the multiplication sentence to

show how many corn plants Mr. Bloom

has in his garden.

5. Write MathWrite Math Could

Mr. Bloom have planted his carrots

in equal rows of 4? If so, how many

rows could he have planted? Explain.

6. Multi-Step List the vegetables in order

by the number of plants. Put the names

in order from greatest to least.

7. Mr. Bloom has 12 strawberry

plants. Describe all of the different

arrays that Mr. Bloom could make using

all of his strawberry plants.

5 × 9 = 45; 45 corn plants

corn 45; beets 28; beans 24; carrots 16

Yes; possible explanation: he has 16 carrot

plants because 2 × 8 = 16. Skip count

by 4s to see if you get 16: 4, 8, 12, 16. You

count 4 times, so he could plant 4 rows.

2 rows of 6; 6 rows of 2; 1 row of 12;

12 rows of 1; 3 rows of 4; 4 rows of 3

Check students’ drawings.

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Module 6 • Lesson 4 193

1

column

row

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3.4.D, 3.4.E

Name

An array is a set of objects arranged in rows and columns.

Write a multiplication sentence for each array.

This array has 2 rows and 5 columns.

Count by 5s.

2 rows of 5 are 10.

The multiplication sentence is 2 × 5 = 10.

This array has 5 rows and 2 columns.

Count by 2s.

5 rows of 2 are 10.

The multiplication sentence is 5 × 2 = 10.

Write a multiplication sentence for the array.

1.

× =

2.

× =

3.

× =

4.

× =

LESSON 26

Model with ArraysOBJECTIVE Use arrays to model products and factors.

Possible sentences are shown.

3

35

4 4

3 6

4 16

18

12

15

Number and Operations 51

marigolds

orange

tulips

blue

petunias

white

pansies

purple

begonias

pink

mums

yellow

CLUES

Enrich© Houghton Mifflin Harcourt Publishing Company

Name Enrich 29

E29

Garden Arrays

Ed’s Garden Nursery is displaying new fl owers. Each type of

fl ower is arranged in an array. Use the clues to label each part of

the fl ower display. Then fi nd the number of each type of fl ower.

Color the display.

1. Yellow mums 2 3 8 array = mums

2. Purple pansies 6 3 4 array = pansies

3. Pink begonias 5 3 5 array = begonias

4. Orange marigolds 3 3 7 array = marigolds

5. White petunias 3 3 4 array = petunias

6. Blue tulips 3 3 8 array = tulips

7. Add a 2 3 5 array for a red rose garden. Label it and color it red.

8. Explain how you decided where each type of flower belonged.

Check students’ array.

Possible answer: I looked at the rows and how many were

in each row and matched it with an array for a fl ower.

16

24

25

21

12

24

Enrich 29RtI Tier 1 Lesson 26

Problem SolvingHave students read Problem 4. Ask them to describe how they will solve the problem.

For Problem 6, students must analyze the data in the table and list the vegetables in Mr. Bloom’s garden in order from greatest number of plants to least number of plants.

ProblemsProblems 5 and 7 require students to extend their thinking to find different ways to arrange the plants in Mr. Bloom’s garden.

Go DeeperBe sure students understand that arrays are made of rows of objects with the same number of objects in each row.

The first factor tells the number of rows, and the second factor tells the number of objects in each row.

• Can you think of any objects around your house that look like arrays? Possible answers: egg cartons, muffin tins, 6-pack of juice, 8-pack of cereal, shoe rack

Have students sketch a picture and write a multiplication sentence for each real-world example.

Math on the Spot Video Tutor

Through the Math on the Spot Video Tutor, students will be guided through an interactive solving of this type of H.O.T. problem. Use this video to also help students solve the H.O.T. problem in the Interactive Student Edition. With these videos and the H.O.T. problems, students will build skills needed in the TEXAS assessment.

MV

Math on the Spot videos are in theInteractive Student Edition and atthinkcentral.com.

1

2

3

193 Module 6

Page 5: 6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

Mathematical Processes

Daily Assessment TaskDaily Assessment Task

TEXAS Test Prep11. What multiplication sentence does this array show?

A 2 × 3 = 6 C 2 × 5 = 10

B 4 × 1 = 4 D 2 × 4 = 8

Fill in the bubble for the correct answer choice.

8. Jacob made this quilt. It has 5 rows of blocks with

6 blocks in each row. Which multiplication sentence

shows the number of blocks in the quilt?

A 4 × 5 = 20 C 5 × 6 = 30

B 6 × 6 = 36 D 5 × 5 = 25

9. Apply Karen has 6 animal posters. Which describes

one array Karen could make using all of her posters?

A 2 rows of 4 C 1 row of 5

B 2 rows of 3 D 3 rows of 3

10. Multi-Step Ashley and

Daniel each have a garden

with flowers planted in

rows. Which multiplication

sentence shows the number

of flowers in both gardens?

A 3 × 7 = 21

B 4 × 3 = 12

C 4 × 4 = 16

D 4 × 7 = 28

Daniel’s GardenAshley’s Garden

194

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Games

THENIF

YES

NO

Daily Assessment Task 1

2

3

Differentiated Centers Kit

ActivitiesHurray for Arrays!Students complete blue Activity Card 15 by using arrays to model multiplication facts.

LiteratureCollections Times FourStudents read the book and determine how to use multiplication to find the total number of objects in each collection.

GamesMultiplication BingoStudents practice multiplication facts through 10.

TEXAS Test Prep CoachTest Prep Coach helps teachers to identify common errors that students can make.

In the Test Prep exercise, if students selected:

A or C They miscounted the number in each row.

B They counted only 1 row.

Essential Question? WriteMathWriteMath

How can you use arrays to model multiplication and find factors? Possible answer: the number of rows is the first factor, and the number in each row is the second factor.

• Soar to Success MathWarm-Up 12.35

• Enrich 29

• Homework and Practice Lesson 6.4

Can students use arrays to model multiplication and find factors?

Lesson 6.4 194

Page 6: 6.4 Model with Arrays€¦ · 4 rows of 6 2 rows of 8 4 rows of 7 5 rows of 9 Mr. Bloom’s Garden Beans Carrots Beets Corn Vegetable Planted In Name Use the table to solve 4–6

TEXAS Test PrepLesson CheckLesson Check

8. Ryan makes this array. Which

equation does Ryan’s array show?

A 3 × 7 = 21

B 3 + 7 = 10

C 3 × 6 = 18

D 3 × 7 = 20

9. Ahmad makes this array. Which

equation does Ahmad’s array show?

A 2 + 6 = 8

B 6 + 2 = 8

C 2 × 6 = 12

D 2 × 6 = 26

10. Becky has 24 stamps. Which

describes one array Becky can

make using all of the stamps?

A 3 rows of 6

B 2 rows of 9

C 4 rows of 6

D 5 rows of 3

11. Keesha makes an array of 5 rows of

7 stickers. How many stickers does

Keesha have?

A 28

B 12

C 57

D 35

12. Multi-Step DeShawn makes an

array using 3 rows of 5 tiles. How

many tiles does Deshawn have if he

adds 2 more rows to the array?

A 15

B 9

C 20

D 25

13. Multi-Step Ming makes an array

using 2 rows of 7 tiles. She adds

3 more rows to the array. Which

multiplication sentence does

Ming’s array show?

A 2 × 7 = 14

B 3 × 7 = 21

C 5 × 7 = 35

D 9 × 3 = 27

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Problem SolvingProblem Solving

Science 3 rows of 4

2 rows of 8

4 rows of 5

Art

Reading

Kind ofBook

Number ofRows

Anna’s Bookshelves

Homeworkand Practice

Name

Model with Arrays6.4

Number and Operations—3.4.D, 3.4.EAlgebraic Reasoning—3.5.B Also 3.4.KMATHEMATICAL PROCESSES 3.1.C, 3.1.D

Write a multiplication sentence for the array.

Use the table to solve 3–5.

1. 2.

6. Sasha makes an array of tiles. She has

3 rows of 9 tiles. Write a multiplication

sentence to show Sasha’s array.

7. Dexter makes an array of tiles. He has

5 rows of 6 tiles. Write a multiplication

sentence to show Dexter’s array.

3. Draw an array and write a

multiplication sentence to show

how many science books are on

Anna’s bookshelves.

4. Write a number sentence to show how

many reading books are on Anna’s

bookshelves.

5. Describe three different arrays that

Anna can make to show how many art

books she has.

3 × 5 = 15

3 × 4 = 12; 12 books

8 rows of 2; 4 rows of 4; 1 row of 16

Possible answers are given.

4 × 5 = 20; 20 books

Check students’ drawings.

4 × 7 = 28

5 × 6 = 303 × 9 = 27

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Module 6 • Lesson 4 195

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Homework and PracticeUse the Homework and Practice pages to provide students with more practice on the concepts and skills of this lesson.

195-196 Module 6