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Lesson 1�2 21

Advance PreparationPost the “Working with a Partner” principles. See Teacher’s Reference Manual, pages 46–48. Prepare a

display area for the Class Data Pad and Arrays Museum. Collect several arrays as examples. Refer to

Teacher’s Reference Manual, page 13.

Teacher’s Reference Manual, Grades 4–6 pp. 16, 79–83, 107–111, 267–271

Key Concepts and Skills• Find factors of a number.

[Number and Numeration Goal 3]

• Write number sentences for rectangular

arrays. [Operations and Computation Goal 7]

• Use the turn-around rule for multiplication.

[Patterns, Functions, and Algebra Goal 4]

Key ActivitiesStudents discuss rectangular arrays using

examples in the Arrays Museum and ones

they draw or make with counters. They write

multiplication number models to represent

rectangular arrays.

Ongoing Assessment: Recognizing Student Achievement Use journal page 5. [Operations and Computation Goal 7]

Key Vocabularyrectangular array � number model �

Commutative Property of Multiplication �

turn-around rule (for multiplication)

MaterialsMath Journal 1, p. 5

Student Reference Book, p. 10

Study Link 1�1

Math Masters, p. 413 (optional)

18 counters � Class Data Pad � slate

Recognizing Patterns in Extended FactsMath Journal 1, pp. 6 and 7

Students practice solving extended

multiplication and division fact

problems using Fact Triangles.

Math Boxes 1�2Math Journal 1, p. 8

Students practice and maintain skills

through Math Box problems.

Study Link 1�2Math Masters, p. 8

Students practice and maintain skills

through Study Link activities.

READINESS

Defining Rows and ColumnsMath Masters, p. 9

per partnership: 40 centimeter cubes � 2 dice

Students practice building arrays.

ENRICHMENTExploring Magic Square and Heterosquare ArraysMath Masters, p. 10

Students explore rectangular arrays by

solving magic square and heterosquare

array problems.

ELL SUPPORT

Describing Exhibits in the Arrays Museum Students practice new vocabulary by

describing items in the Arrays Museum.

Teaching the Lesson Ongoing Learning & Practice Differentiation Options

Rectangular ArraysObjectives To review rectangular arrays and the use of

multiplication number models to represent such arrays.

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Factors of a Counting Number

A rectangular array is an arrangement of objects into rowsand columns that form a rectangle. All rows and columns mustbe filled. Each row has the same number of objects. Eachcolumn has the same number of objects. A multiplicationnumber model can represent a rectangular array.

Counting numbers can have more than one factor pair. 1 and15 are another factor pair for 15 because 1 * 15 = 15.

To test whether a counting number a is a factor of anothercounting number b, divide b by a. If the result is a countingnumber and the remainder is 0, then a is a factor of b.

One way to find all the factors of a counting number is to findall the factor pairs for that number.

Whole Numbers

The counting numbersare 1, 2, 3, and so on.

Whenever you are askedto find the factors of acounting number:(1) each factor must be a

counting number, and(2) the other number in

its factor pair must also be a counting number.

Check your answers on page 433.

List all the factors of each number.1. 15 2. 8 3. 28 4. 36 5. 11 6. 100

This rectangular array has 15 red dots.It has 3 rows with 5 dots in each row.3 * 5 � 15 is a number model for this array.3 and 5 are counting-number factors of 15.15 is the product of 3 and 5.3 and 5 are a factor pair for 15.

4 is a factor of 12 because 12 / 4 gives 3 with a remainder of 0.

6 is not a factor of 14 because 14 / 6 gives 2 with a remainder of 2.

Find all the factors of the number 24.

The factors of 24 are1, 2, 3, 4, 6, 8, 12, and 24.

Number Models Factor Pairs

24 � 1 * 24 1, 24

24 � 2 * 12 2, 12

24 � 3 * 8 3, 8

24 � 4 * 6 4, 6

Student Reference Book, p. 10

Student Page

Adjusting the Activity

22 Unit 1 Number Theory

Getting Started

Mental Math and Reflexes Pose basic and extended division facts. Have students write the answers for each set of problems. At the end of each set, ask students to describe the patterns they see among the dividends, divisors, and quotients. Suggestions:

Math MessageArrange 12 counters into as many different rectangular arrays as you can. Then choose and draw one of the arrays.

Study Link 1�1 Follow-UpDiscuss student’s responses and their number pattern poems. Ask: If you were writing a poem about arithmetic, how would you finish this sentence: Arithmetic is…? List the mathematics vocabulary that students use on the Class Data Pad. Emphasize how using mathematics vocabulary makes communicating their ideas to others easier and more efficient.

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

Ask students to share the rectangular arrays they drew. Have one student describe the array and another draw the array from the description. To support English language learners, clarify the noun/adjective relationship between rectangle and rectangular. Mentally note students’ use and understanding of appropriate vocabulary (rows, columns, in each row, in each column). Array possibilities for 12: 1-by-12, 12-by-1, 2-by-6, 6-by-2, 3-by-4, and 4-by-3.

Draw the following visual R O W C

reference on the board: O

L

U

M

N

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

▶ Reviewing Arrays

WHOLE-CLASS ACTIVITY

(Student Reference Book, p. 10; Math Masters, p. 413)

Algebraic Thinking Display examples of rectangular arrays from the Arrays Museum. Stress these key elements:

� Each row has the same number of objects.

� Each column also has the same number of objects.

� Each array has a rectangular shape.

ELL

ELL

NOTE Some students may benefit from

doing the Readiness activity before you begin

Part 1 of each lesson. See the Readiness

activity in Part 3 for details.

Interactive whiteboard-ready

ePresentations are available at

www.everydaymathonline.com to

help you teach the lesson.

7 ÷ 1 7

70 ÷ 10 7

700 ÷ 100 7

7,000 ÷ 1,000 7

28 ÷ 4 7

280 ÷ 40 7

2,800 ÷ 400 7

28,000 ÷ 4,000 7

56 ÷ 7 = 8

560 ÷ 70 = 8

560,000 ÷ 70,000 = 8

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Lesson 1�2 23

Ask students to name the number of rows and columns in each example.

During this unit, students should collect other examples of arrays to add to the Arrays Museum.

Consider reserving a section of the Arrays Museum for arrangements

that are “almost arrays” but do not satisfy all the conditions for rectangular

arrays. Examples include some calculator keypads; certain playing cards, such

as the nine of diamonds; a double-3 domino; and a calendar month, such as the

month of August.

AugustS M T W T F S

7142128

6132027

51219

18152229

29162330

310172431

4111825 26

This is not an array:

The fifth row has only 3 days.


Arrays were first introduced in Second Grade Everyday Mathematics. Have students focus on labeling arrays in terms of rows and columns and representing arrays with number models. Assign student groups to:

1. Read page 10 in the Student Reference Book.

2. Complete one of the Check Your Understanding problems.

3. Draw a rectangular array from one of their factor pairs and write the number model that represents the array.

Circulate and assist.

Review multiplication number models as a way of representing rectangular arrays. Have groups present their arrays and number models.

Rectangular arrays can help students visualize factors and the Commutative Property of Multiplication. Ask students to take out and arrange 6 counters into an array. Show students’ responses on the board or transparency of Math Masters, page 413 until all four possibilities have been displayed and discussed. (See margin.)

To avoid confusion when naming an r-by-c array, let r represent the number of rows and c the number of objects in each row (the number of columns).

Point out that both the 3-row-by-2-column and the 2-row-by-3-column arrays have the same number of dots, but not the same number of rows and columns. Tell students that this models a property of multiplication. The order in which two numbers are multiplied makes no difference in their product: 2 ∗ 3 = 6 and 3 ∗ 2 = 6.

3-row-by-2-column array

Number model: 3 ∗ 2 = 6








Date Time

5

A rectangular array is an arrangement of objects into rows and columns. Each row

has the same number of objects, and each column has the same number of objects.

A multiplication number model can be written to describe a rectangular array. The first

factor is the number of rows in the array. The second factor is the number of columns.

The product is the total number of objects.

This is an array of 8 dots.

It has 4 rows with 2 dots in each row.

It has 2 columns with 4 dots in each column.

The number model is next to the array.

This is another array of 8 dots.

It has 2 rows with 4 dots in each row.

It has 4 columns with 2 dots in each column.

Label this array by writing the number model next to it.

1. a. Take 10 counters. Make as many

different rectangular arrays as you

can using all 10 counters.

b. Draw each array on the grid at the

right by marking dots.

c. Write the number model next to each array.

2. a. How many dots are in the array at the right?

18 dots

b. Write a number model for the array.

3 * 6 = 18

c. Make as many other arrays as you

can with the same number of dots

that were used for the array in Part 2a.

Draw each array on the grid at the right.

Write a number model for each array.

2 ∗ 4 = 8

4 ∗ 2 = 8

10 ∗ 1 = 10 1 ∗ 10 = 10

2 ∗ 5 = 105 ∗ 2 = 10

18 ∗ 1 = 18

3 ∗ 6 = 18

6 ∗ 3 = 18

1 ∗ 18 = 18

2 ∗ 9 = 18

9 ∗ 2 = 18

�Arrays

LESSON

1�2

EM3cuG5MJ1_U01_001-028.indd 5 1/11/11 11:29 AM

Math Journal, p. 5

Student Page

Date Time

Read the information about extended multiplication and division facts on Student

Reference Book, pages 18 and 21. If you know the basic multiplication and

division facts, then you can solve extended fact problems such as 30 ∗ 20 and

1,800 � 30 mentally. Just as there are four related facts for each basic fact, there

are also four related facts in an extended fact family.

2 ∗ 3 = 6

3 ∗ 2 = 6

6 � 2 = 3

6 � 3 = 2

20 ∗ 30 = 600

30 ∗ 20 = 600

600 � 20 = 30

600 � 30 = 20

20 ∗ 30 = ? Think: 2 [3s] = 6. Then 20 [30s] is 100 times as much. 20 ∗ 30 = 600

1. Write the extended fact family represented by each of these Fact Triangles.

a. 30 ∗

70 = 2,100

70 ∗

30 =

2,100

2,100 �

70 =

30

2,100 �

30 =

70

b. 60 ∗

20 = 1,200

20 ∗

60 =

1,200

1,200 �

20 =

60

1,200 �

60 =

20

•

30 70

2,100

∗, �

•

60 20

1,200

∗, �

•

2

6

3

∗, �

Multiplication and Division Extended FactsLESSON

1�2

•

20

600

30

∗, �

EM3cuG5MJ1_U01_001-028.indd 6 1/22/11 9:50 AM

Math Journal 1, p. 6

Student Page



Students have used this property of multiplication in turn-around facts as shortcuts to learning new facts. Ask if students know what this property is called. Some students will respond that the property is the turn-around rule for multiplication. Some students might know to use the term Commutative Property of Multiplication, but do not insist that students use this term.

Teach students a physical representation of the Commutative Property

of Multiplication to indicate “turn-around” facts. This gesture demonstrates the

idea of switching the numbers and can be used to remind students when the

turn-around rule is being applied.


▶ Finding All Possible Rectangular PARTNER ACTIVITY

Arrays for a Number(Math Journal 1, p. 5)

Review the “Working with a Partner” principles. Ask students for additional suggestions to help make the classroom more pleasant when students are working with partners or in small groups.

Ask partners to make all possible rectangular arrays using 8 counters. 1-by-8, 8-by-1, 2-by-4, 4-by-2 Partners then work on journal page 5. Circulate and assist.

NOTE Some students might find it easier to work on a full sheet of dot paper for

Problem 2. (Math Masters, p. 413)

Ongoing Assessment: Journal

Page 5 �Recognizing Student Achievement

Use journal page 5 to assess students’ ability to build arrays and identify factors

that describe arrays. Students are making adequate progress if they correctly

arrange and label the arrays for both 10 and 18 by using counters and/or

drawing on the journal page.

[Operations and Computation Goal 7]

ELL

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Date Time

8

5. Draw a line from each spinner to the number that represents the shaded parts.

3. Make an array for each of these

number sentences.

a. 3 ∗ 9 = 27

b. 6 ∗ 7 = 42

10 4

129

1. Marcus drew 8 cards from a pile:

10, 8, 4, 5, 8, 6, 12, and 1.

Find the following landmarks:

a. Maximum

b. Minimum

c. Range

d. Median

12

1

11

7

2. Name five numbers between 0 and 1.

Answers vary.

4. a. Write the largest number you can

make using each of the digits

7, 1, 0, 2, and 9 just once.

97,210

b. Write the smallest number. (Do not

start with 0.)

10,279

26 56119

Math BoxesLESSON

1�2

50%0.754

1

31

EM3cuG5MJ1_U01_001-028.indd 8 1/11/11 11:29 AM


Student Page

Math Masters, p. 8

Study Link Master

STUDY LINK

1�2 More Array Play

Name Date Time

10

A rectangular array is an arrangement of objects in rows and

columns. Each row has the same number of objects, and each

column has the same number of objects. We can write a

multiplication number model to describe a rectangular array.

For each number below, use pennies or counters to make as

many different arrays as possible. Draw each array on the grid

with dots. Write the number model next to each array.

1. 5 2. 14

3. 18

4 ∗ 3 = 12

4. 487 + 308 = 795 5. 679 - 408 = 271 6. 14 ∗ 7 = 98

7. 164 ∗ 6 = 984 8. 45 ÷ 9 = 5

Practice

5 º 1 � 5

1 º 5 � 5

14 º 1 � 14

1 º 14 � 14

2 º 7 � 147 º 2 � 14

2 º 9 � 18

6 º 3 � 18

3 º 6 � 181 º 18 � 18

18 º 1 � 18

9 º 2 � 18

EM3cuG5MM_U01_002-032.indd 8 1/13/11 3:17 PM

Date Time

c. 3,200 =

80 ∗ 40

3,200 =

40 ∗

80

80 =

3,200 �

40

40 =

3,200 �

80

2. Complete your own Fact Triangle with extended multiplication and division facts.

∗ =

∗ =

= �

= �

3. Look at the four sets of facts you wrote.

a. Describe a pattern for finding the product when you multiply with extended facts.

Sample answer: First find the basic fact. Then

count the number of zeros in each factor,

and attach that many zeros to the product.

b. Describe a pattern for finding the quotient when you divide with extended facts.

Sample answer: First find the basic fact. Then

subtract the number of zeros in the divisor

from the remaining zeros in the dividend.

Attach that many zeros to the quotient.

4. Do your patterns in Problem 3 work for 400 ∗ 50 and

for 2,000 � 40? If not, adjust your patterns as necessary. Answers vary.

•

80 40

3,200∗, �

Answers vary.

Multiplication and Division Extended Facts cont.LESSON

1�2

•

∗, �

EM3cuG5MJ1_U01_001-028.indd 7 1/22/11 9:50 AM


Student Page

Lesson 1�2 25

2 Ongoing Learning & Practice

▶ Recognizing Patterns in PARTNER ACTIVITY

Extended Facts(Math Journal 1, pp. 6 and 7)

Students write the extended fact families represented by the numbers on multiplication and division Fact Triangles. They describe patterns in the number of zeros in the factors and products.

▶ Math Boxes 1�2

INDEPENDENT ACTIVITY

(Math Journal 1, p. 8)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1-4. The skill in Problem 5 previews Unit 2 content.

▶ Study Link 1�2

INDEPENDENT ACTIVITY

(Math Masters, p. 8)

Home Connection Students build and draw rectangular arrays to represent numbers and write the associated number models.

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LESSON

1�2

Name Date Time

Rows and Columns

A rectangular array is an arrangement of objects in rows and columns. Each row has the

same number of objects, and each column has the same number of objects.

Work with a partner to build arrays. For each array, take turns rolling dice. The first die is the

number of rows. Write this number in the table under Rows. The second die is the number of

cubes in each row. Write this number under Columns. Then use centimeter cubes to build

the array on the dot grid. How many cubes are in the array? Write this number under Array

Total on the dot grid table.

Rows Columns Array Total Rows Columns Array Total

Math Masters, p. 9

Teaching Master

Name Date Time

A magic square is an array of positive whole

numbers. The sum of the numbers in each

row, column, and diagonal will be the same.

1. Complete this magic square.

A heterosquare is like a magic square,

except that the sum of the numbers in each

row, column, and diagonal are different. A

3-by-3 array for a heterosquare will have an

arrangement of the numbers 1–9.

2. Complete this heterosquare, and write

the sum for each row, column, and the

two diagonals.

LESSON

1�2 Magic Square and Heterosquare Arrays

A rectangular array is an arrangement of objects in rows and columns. The objects in an

array can be numbers or numerical expressions. The Multiplication/Division Facts Table on

the inside front cover of your journal is an example of numbers arranged in an array. The

objects can also be words or symbols that represent elements of a given situation. For

example, a plan for after-school snacks could be arranged in a 1-by-5 array, using A for

apple, B for banana, and so on.

16 3 13

8115

6

4 15 1

7

2

1010

9

10

9 12

14

34

34

34

7 6

4

29

13

14

1124 9 12

18

15

18

5

3

18

5

3

3. Create a magic square or heterosquare for your partner to solve.

Answers vary.

Math Masters, p. 10

Teaching Master


3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Defining Rows and Columns 15–30 Min

(Math Masters, p. 9)

To explore factoring numbers using a concrete model, have students build arrays and find the total number of counters for each array. Have students describe their arrays using the words row and column.

ENRICHMENT PARTNER ACTIVITY

▶ Exploring Magic Square 15–30 Min

and Heterosquare Arrays(Math Masters, p. 10)

To further explore rectangular arrays, have students solve magic square and heterosquare array problems. Arrays are also used to organize numbers, numerical expressions,

and symbols to represent rules. In a magic square, the rule is that the sum of each row, column, and diagonal is the same. In a heterosquare, these sums will be different. Partners complete these two types of arrays and make an array of either type using their own numbers. Have students display their arrays in the Arrays Museum.

This activity also provides practice with adding, subtracting, and comparing whole numbers.

ELL SUPPORT

SMALL-GROUP ACTIVITY

▶ Describing Exhibits in 5–15 Min

the Arrays MuseumTo provide language support for multiplication, have students look at the Arrays Museum. Ask them to describe the arrays in the museum using language from the lesson. They might describe the rows, columns, shape, and the contents of the arrays.

Planning Ahead

Remind students to collect examples of arrays for the Arrays Museum. The Arrays Museum will be used again in Lesson 1-3 and in subsequent lessons.


rectangular arrays - everyday math - login · pdf filenumber modelcan represent a rectangular...

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