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248 Unit 4 Multiplication and Division

Advance PreparationGather a large supply of pennies or other counters. Choose a place for an Arrays Exhibit. See Home

Link 4�2 for details. Display the Guide to Solving Number Stories on Math Masters, page 406 and a

multiplication/division diagram.

Key Concepts and Skills• Use basic facts to solve multiplication

number stories.  

[Operations and Computation Goal 3]

• Use arrays to model multiplication.  

[Operations and Computation Goal 6]

• Write number sentences to model

number stories.  

[Patterns, Functions, and Algebra Goal 2]

• Model the turn-around rule for multiplication

(Commutative Property of Multiplication)

using an array model.  

[Patterns, Functions, and Algebra Goal 4]

Key ActivitiesChildren use arrays to represent multiples of

equal groups. Children solve multiplication

number stories using arrays, multiplication/

division diagrams, and number models.

Ongoing Assessment: Recognizing Student Achievement Use journal page 81. [Operations and Computation Goal 6]

Key Vocabularyarray � factor � product

MaterialsMath Journal 1, p. 81

Home Link 4�1

transparency of Math Masters, pp. 406 and

419 (optional) � per person: 40 counters �

slate � number grid � calendar � number line

(optional) � calculator (optional)

Measuring Perimeters of PolygonsMath Journal 1, p. 82

ruler

Children measure perimeters

of polygons and draw figures

with given perimeters.

Math Boxes 4�2Math Journal 1, p. 83

Children practice and maintain skills

through Math Box problems.

Home Link 4�2Math Masters, p. 87

Children practice and maintain skills

through Home Link activities.

READINESS

Building ArraysMath Masters, pp. 88 and 416

centimeter cubes or other counters � 

1 six-sided die

Children build arrays.

ENRICHMENTExploring Square NumbersMath Masters, pp. 89 and 416

per group: centimeter cubes or pennies

(or both), tape

Children explore square arrays.

ELL SUPPORT

Building a Math Word Bank Differentiation Handbook, p. 132

Children add the term array to their Math

Word Banks.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

������� Multiplication ArraysObjective To provide opportunities to use arrays, multiplication/

division diagrams, and number models to represent multiplication

number stories.

d

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Lesson 4�2 249

rows

6

trombones per row

4

trombonesin all

24

24 = 6 × 4

6 × 4 = 24 6× 4

244

6

rows

4

trombones per row

6

trombones in all

24

4

6

24 = 4 × 64 × 6 = 24

4× 6

24

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASS ACTIVITY

Have a volunteer draw an array on the board showing how he or she arranged the counters. Use circles or Xs to represent trombone players.

Ask others to draw different arrangements on the board. Explain that the pictures show the number of trombone players in arrays. An array is a group of objects arranged in rows and columns. Each row has the same number of objects, and each column has the same number of objects. Arrays provide a way to model multiplication. Possible arrays: 4-by-6, 6-by-4, 3-by-8, 8-by-3, 2-by-12, 12-by-2, 1-by-24, and 24-by-1

1. Draw an array on the board representing 4 rows of trombone players with 6 players in each row.

2. Display a multiplication/division diagram, and have a volunteer fill in the known quantities.

3. Write 24 in the bottom right box.

4. Call attention to the fact that the numbers do not have the same units or labels, as they do in addition and subtraction stories.

5. Ask a volunteer to write a number model for this array: 4 × 6 = 24.

6. Next, draw an array to represent 6 rows of trombone players with 4 players in each row.

Getting Started

Home Link 4�1 Follow-Up Ask several children to share the multiples of equal groups they found. They will share some of their stories later in this lesson.

Math Message There are 24 trombone players in a big parade.

Use counters to represent trombone players. Arrange the counters to show them in equal rows.

Mental Math and ReflexesHave children count chorally. They may use number grids, number lines, calculators, or calendars as needed.

Skip count forward and backward by 2s, 5s, and 10s.

Skip count forward and backward by 6s.

Skip count forward and backward by 8s.

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250 Unit 4 Multiplication and Division

NOTE A factor is a number that is multiplied

by another number. A product is the result of

multiplying factors. It is beneficial to use these

words when discussing multiplication number

stories, but do not insist that children use them.

Repeat Steps 1 through 5: Fill in a multiplication/division diagram and write a number model. Compare the two arrays, emphasizing that although the total number of players is the same in each array, the number of rows and the number of players per row are reversed. Remind children of the turn-around rule for multiplication (Commutative Property of Multiplication), which states that two numbers can be multiplied in either order without changing the product: 4 × 6 = 6 × 4

� Solving Multiplication WHOLE-CLASS ACTIVITY

Number Stories Using Arrays(Math Masters, p. 419)

Children draw arrays on their slates to solve simple multiplication number stories. Pose multiplication number stories in which arrays are obvious, such as the following.

� 4 boxes of crayons, 8 crayons per box

� 6 shoe boxes, 1 pair of shoes per box

� 5 rows of chairs, 3 chairs per row

� 5 shelves of books, 8 books per shelf

� 2 cartons of eggs, 12 eggs per carton

For each story:

1. Write a brief summary of the story on the board.

2. Fill in a multiplication/division diagram, going over the number of rows and the number of items per row (the factors) and the total number of things (the product) as children tell you what information to enter into the diagram. To support English language learners, distinguish between the common meaning and mathematical meaning of product.

3. Ask a volunteer to write a number model for the story on the board while the other children write number models on their slates. Children may use either a ? or a letter variable to represent the unknown quantity.

4. Have children draw arrays on their slates or arrange counters on their desks to model the story. As they do this, have a volunteer draw an array on the board or arrange counters on the overhead projector.

5. Have children solve the problem and share their solution strategies. Remind them to include the unit in the answer.

6. Ask children to discuss the reasonableness of their answers and whether their answers make the number model true. Finally, have children write summary number models on their slates for the story.

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BBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMOOOOOOOOOBBBLBLBLBLBBLBOOROROROOROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVVINVINVINVINNNNVINVINVINVINVINVINVINGGGGGGGGGGOLOOOLOLOLOOLOO VINVINVLLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLLOOO VVVLLLLLLLLLLVVVVVVVVVVOSOSOOSOSOSOSOSOSOSOOSOSOSOOSOOOOOSOSOSOSOSOSOSOOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLLVVVVVVVVVLLVVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIISOLVING

Adjusting the Activity

Hold up two egg cartons to provide a visual

for 24 eggs in 2 cartons or 2 dozen eggs.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

ELL

Name Date Time

Multiplication/Division Diagram

per in all

per in all

per in all

Math Masters, p. 419

Teaching Aid Master

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Lesson 4�2 251

More Multiplication Number StoriesLESSON

4�2

Date Time

� Fill in the multiplication/division diagram. Write a number model.

� Make an array with counters. Mark the dots to show the array.

� Find the answer. Write the unit with your answer.

1. Mrs. Kwan has 3 boxes of scented markers. Each box has 8 markers.

How many markers does she have?

markers markers boxes

per box in all

3 8 ?

Number model: 3 × 8 = ? Answer: 24 markers

(unit)

2. Monica keeps her doll collection in a case with 5 shelves. On each

shelf there are 6 dolls. How many dolls are in Monica’s collection?

dolls dolls

shelves per shelf in all

5 6 ?

Number model: 5 × 6 = ? Answer:

(unit)

3. During the summer Jack mows lawns. He can mow 4 lawns per day.

How many lawns can he mow in 7 days?

lawns lawns

days per day in all

7 4 ?

Number model: 7 × 4 = ? Answer: 28 lawns

(unit)

30 dolls

�

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Math Journal 1, p. 81

Student Page

Measuring PerimeterLESSON

4�2

Date Time

82 i ht t

Measure the perimeter of each shape in inches.

Perimeter: 6 inches

Perimeter: 3 3

_ 4 inches

Perimeter: 7 inches

5. Draw any shape with

a perimeter of

20 units.

What is the area of

your shape?

square units

in.

2 in.

2

in.1in.1

in.2

in.1in.1

in.1in.

12

in.11 2

in.1

in.1in.1

in.1

in.3

in.1

in.1in.1

Sample answer:

in.11 4

in.114 in.11

4

Perimeter: 10 inches

4.

1. 2.

3.

150 151

Try This

Sample answer: 16

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Math Journal 1, p. 82

Student Page

� Representing Multiplication WHOLE-CLASS ACTIVITY

Situations with Arrays(Math Masters, p. 419)

Have children share stories from Home Link 4-1 about multiples of equal groups. Follow steps 1–5 from the previous activity. Make the point that arrays can be used to represent multiplication situations, even when the real-life groups do not look like arrays. Arrays are easy to draw, and they simplify problem solving.

� Solving More PARTNER ACTIVITY

MultiplicationNumber Stories(Math Journal 1, p. 81)

Partners or individuals make arrays and use diagrams to solve multiplication number stories. Children may use counters if they wish. Circulate and ask guiding questions such as those on the Guide to Solving Number Stories.

When reviewing the answers to the journal page, ask follow-up questions so children can practice more difficult multiplication facts. For example:

● Suppose Mrs. Kwan has 7 boxes of markers. How many markers would she have now? 56 markers

● If Monica had 60 dolls, how many shelves would she need? 10 shelves

Ongoing Assessment: Journal

Page 81 �Recognizing Student Achievement

Use journal page 81 to assess whether children are making progress toward

using arrays and multiples of equal groups to demonstrate the meaning of

multiplication. Children are making adequate progress if they are able to draw

arrays and find the answer for Problems 1, 2, and 3. Some children may be able

to fill in the diagrams and write the number models.

[Operations and Computation Goal 6]

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BBBBBBBBBBBBBBBBBB ELELEELEMMMMMMMMOOOOOOOOOBLBLBLBLBLBLBLBLBLROOOROROROROROROROROROROO LELELELEEEEEELEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGGLLLLLLLLLLLLLVINVINVINVINNNVINVINVINVINNVINVINVINVINVV GGGGGGGGGGGOLOOOOLOLOOLOO VINVINVVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOLOLOLOLOLOLOLOOOLO VVVVVLLLLLLLLLLVVVVVVVVOOSOSOSOSOSOSOSOSOSOOSOSOSOOOOOOSOSOSOSOSOSOSOSOOOSOOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVVLLLLVVVVVVVVLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSS GGGGGGGGGGGGGGGGGGGOOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

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252 Unit 4 Multiplication and Division

Unit3. Solve. Make a ballpark

estimate to checkthat the answermakes sense.

� 648 � 209

estimate: Sample answer:650 � 200 � 850

857

Date Time

2. 10 packs of gum on the shelf in thecandy store. 8 sticks of gum perpack. How many sticks of gum inall?

Answer: 80 sticks

4. Solve.

3 � 5 �

3 nickels � ¢

� 4 � 5

¢ � 4 nickels2020

1515

1. Complete the bar graph.

86 87 259

10

8

6

4

2

0A B C D

Players

Poin

ts

Point Totals Player A scores4 points.Player B scores8 points.Player C scores3 points.Player D scores9 points.

192

sticks of stickspacks gum per of gum

pack in all

10 8 ?

6. 1,798

Which digit is in the tens place? Which digit is in the hundreds place?

Which digit is in the ones place?

Which digit is in the thousands place? 1

8

79

200 201 18 19

5. Fill in the empty frames.

�1

�31 23

6515

52 5356

Math BoxesLESSON

4�2

Math Journal 1, p. 83

Student Page

NOTE Some teachers include a small

section in their display titled Almost an Array.

This section may include children’s

contributions that are close but not quite

correct. It is important to discuss why such

contributions are considered almost an array

but not a true array.

Write these problems on the back of this page. Solve. Show your work.

4. 331 � 27 � 5. � 187 � 17

6. 907 � 479 � 428

204358

Name Date Time

ArraysHOME LINK

4�2

Your child is learning how to represent multiplication problems using pictures called arrays.An array is a group of items arranged in equal rows and equal columns. Help your child usecounters, such as pennies or macaroni, to build the array in each problem. Yourchild should record each solution on the dots next to the problem.

Please return this Home Link to school tomorrow.

FamilyNote

64 65

For the next few weeks, look for pictures

of items arranged in equal rows and

columns, or arrays. Look in newspapers or

magazines. Have people in your family help

you. Explain that your class is making an

Arrays Exhibit.

Make an array with counters. Mark the dots to show the array.

1. 4 rows with 6 counters per row

a 4-by-6 array

counters

2. 3 rows with 8 counters per row

a 3 � 8 array

counters

3. 2 rows with 12 counters per row

a 2 � 12 array

counters24

24

24

This is a 5-by-6 array.

There are 5 rows. There

are 6 dots in each row.

There are 30 dots in all,

since 5 � 6 � 30.

Unit

Practice

Math Masters, p. 87

Home Link Master

2 Ongoing Learning & Practice

� Measuring Perimeters INDEPENDENTACTIVITY

of Polygons(Math Journal 1, p. 82)

This journal page reviews content covered in Unit 3. Problems 1–4 provide children with practice measuring line segments. Children measure the perimeters of polygons to the nearest 1 _ 4 inch. Problem 5 involves children drawing a figure with a given perimeter and then counting unit squares to find the area.

� Math Boxes 4�2 INDEPENDENTACTIVITY

(Math Journal 1, p. 83)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 4-4. The skill in Problem 6 previews Unit 5 content.

Writing/Reasoning Have children write an answer to the following: Why are estimates helpful in Problem 3? Sample answer: Estimates help you know whether your answer makes sense.

� Home Link 4�2 INDEPENDENTACTIVITY

(Math Masters, p. 87)

Home Connection Children use arrays to solve multiplication stories. They find pictures from newspapers, magazines, and other sources that show

arrays of items. You may want to use these pictures to build an Arrays Exhibit.

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LESSON

4�2

Name Date Time

Building Arrays

Materials � cm cubes or other counters

� 1 six-sided die

� centimeter grid paper (Math Masters, p. 416)

1. Roll the die 2 times. The first number tells how many rows to put in

the array. The second number tells how many counters to put in

each row.

Example

You roll a 1 first and then a 5. You make this figure.

2. Draw the arrays you make on centimeter grid paper.

3. Then, fill in the Array Table below.

How manyHow many

How manyrows?

cubes in eachcubes in all?row?

1

2

3

4

5

Math Masters, p. 88

Teaching Master

LESSON

4�2

Name Date Time

Square Products

Work in a small group.

Materials � centimeter grid paper (Math Masters, p. 416)

� centimeter cubes or pennies (or both)

� tape

Directions

1. Each person chooses a different number from 2 to 10.

2. Build an array that shows your number multiplied by itself. Usecentimeter cubes or pennies.

3. Draw each array on centimeter grid paper. Write each numbermodel under each array.

4. Continue to build arrays with other numbers. Draw the arrays ongrid paper. You may need to tape pieces of grid paper together forthe larger arrays.

5. Look at the arrays you made. Why do you think the products of thenumber models you wrote are called square products?

Sample answer: All of the arrays are squares.

2 � 2 � 4

4 � 4 � 16

3 � 3 � 9

Math Masters, p. 89

Teaching Master

Lesson 4�2 253

3 Differentiation Options

READINESS INDEPENDENTACTIVITY

� Building Arrays 5–15 Min

(Math Masters, pp. 88 and 416)

To provide children with experience forming arrays, children build arrays with counters, following directions on Math Masters, page 88. They record the arrays on centimeter grid paper, Math Masters, page 416, and fill in the Array Table on page 88.

ENRICHMENT SMALL-GROUP ACTIVITY

� Exploring Square Numbers 15–30 Min

(Math Masters, pp. 89 and 416)

Algebraic Thinking To further explore arrays, have children build square arrays with counters for as many numbers as possible, beginning with 2. They record their arrays on centimeter grid paper (Math Masters, page 416) and write a number model under each array. As children build arrays for larger products, they might need to tape sheets of grid paper together. Have children arrange the number models and products for each array in order from smallest to largest. Ask: What patterns do you see? Sample answer: The differences between consecutive products are odd numbers, and they increase by 2.

ELL SUPPORT SMALL-GROUP ACTIVITY

� Building a Math Word Bank 5–15 Min

(Differentiation Handbook, p. 132)

To provide language support for multiplication, have children use the Word Bank template found on Differentiation Handbook, page 132. Ask children to write the word array, draw a picture representing the word, and write other related words. See the Differentiation Handbook for more information.

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