patterns in products - everyday math · cards 0–10 and 1 each of number ... many of the patterns...

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576 Unit 7 Multiplication and Division

Advance Preparation

Teacher’s Reference Manual, Grades 1–3 pp. 204, 205

Patterns in ProductsObjective To review square-number facts, multiplication, and

division patterns.d

��

Key Concepts and Skills• Identify factors, products, square numbers,

and patterns in the Multiplication/Division

Facts Table.

[Operations and Computation Goal 3]

• Use the Multiplication/Division Facts

Table to generate fact families.


• Use arrays to find square products.


• Use the turn-around rule (Commutative

Property of Multiplication) to generate

multiplication facts.

[Patterns, Functions, and Algebra Goal 4]

Key ActivitiesChildren identify patterns in a sequence of

square numbers and in the Multiplication/

Division Facts Table.

Ongoing Assessment: Recognizing Student Achievement Use journal page 157. [Operations and Computation Goal 3]

Key Vocabularyproduct � square product � square number �

factor

MaterialsMath Journal 2, p. 157

Student Reference Book, p. 52

slate

Playing Name That Number Student Reference Book, pp. 299

and 300

per partnership: 4 each of number

cards 0–10 and 1 each of number

cards 11–20 (from the Everything Math

Deck, if available).

Children practice finding equivalent

names for a number.

Math Boxes 7�1Math Journal 2, p. 158

Children practice and maintain skills

through Math Box problems.

Home Link 7�1Math Masters, p. 206

Children practice and maintain skills

through Home Link activities.

READINESS

Building Square and Rectangular ArraysMath Masters, p. 207

cm cubes

Children build square and rectangular arrays

and look for patterns.

ENRICHMENTExploring a Pattern in a Sequence of ProductsMath Masters, p. 208

Student Reference Book, pp. 198 and 199

Children look for patterns in a sequence

of rectangular arrays and in the products

they represent.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 132

Children add the terms product and factor

to their Math Word Banks.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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Adjusting the Activity

Lesson 7�1 577

Array showing square products

2 × 2 = 4

3 × 3 = 9 (add 5 dots)

Part A

Product PatternsLESSON

7�1

Date Time

Math Message

Complete the facts.

1. 1 × 1 = 1

2. 2 × 2 = 4

3. 3 × 3 = 9

4. 4 × 4 = 16

5. 5 × 5 = 25

6. 6 × 6 = 36

7. 7 × 7 = 49

8. 8 × 8 = 64

9. 9 × 9 = 81

10. 10 × 10 = 100

A Two’s Product Pattern

Multiply. Look for patterns.

11. 2 × 2 = 4 12. 2 × 2 × 2 = 8

13. 2 × 2 × 2 × 2 = 16 14. 2 × 2 × 2 × 2 × 2 = 32

15. 2 × 2 × 2 × 2 × 2 × 2 = 64

Use the Two’s Product Pattern for Problems 11 through 15. Multiply.

16. 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

Part B

�

Try This

EM3MJ2_G3_U07_157-179.indd 157 1/18/11 3:34 PM

Math Journal 2, p. 157

Student Page

Interactive whiteboard-ready

ePresentations are available at

www.everydaymathonline.com to

help you teach the lesson.

1 Teaching the Lesson

� Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 2, p. 157)

Review answers to the Math Message problems. Ask children to share how they found answers to facts they have not yet memorized. Some children may suggest a count-by strategy: for example, for 5 × 5, count by 5s five times. 5, 10, 15, 20, 25

Ongoing Assessment: Journal

Page 157 �Recognizing Student Achievement

Use journal page 157, Part A to assess children’s progress toward learning

the multiplication facts. Children are making adequate progress if they use

strategies to correctly complete the facts in Problems 1 through 10. Some

children will demonstrate automaticity with the facts.


Point out how the array diagram highlights the number of dots that are added to an array to obtain the next array. For example, 5 dots are added to the 2-by-2 array to obtain the 3-by-3 array. This shows that 3 × 3 = 9 is 5 more than 2 × 2 = 4.

Draw square arrays for 4, 9, and 16 on the board without lines. Ask

children to imagine what a 5-by-5 array looks like. Have a volunteer draw the

5-by-5 array on the board.

4, 9, and 16 are square products

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

Getting Started

Math MessageTurn to page 157 in your new journal. Find the products in Problems 1 through 10.

Mental Math and ReflexesHave children practice quick recall of basic multiplication facts. Suggestions:

2 × 4 8 3 × 4 12 3 × 5 15 4 × 5 20

3 × 6 18 4 × 6 24 4 × 7 28 5 × 7 35

3 × 8 24 3 × 9 27 4 × 8 32 6 × 6 36

EM3cuG3TLG2_577-581_U07L01.indd 577EM3cuG3TLG2_577-581_U07L01.indd 577 1/23/11 12:25 PM1/23/11 12:25 PM


Adjusting the Activity

NOTE In Lesson 4-6, children used the

Multiplication/Division Facts Table to generate

fact families. They looked for patterns in the

table with teacher guidance. In this activity,

expect that children will be able to describe

many of the patterns on their own.

52 fifty-two

Basic Facts for Multiplication and DivisionSolving problems is easier when you know the basic number facts. Here are some examples of basic multiplication and division facts:

Basic multiplication facts:6 � 4 � 24, 10 � 7 � 70, 1 � 8 � 8, 3 � 9 � 27

Basic division facts:24 � 6 � 4, 70 � 10 � 7, 8 � 1 � 8, 27 � 3 � 9

The facts table shown below is a chart with rows and columns. It can be used to find all of the basic multiplication and division facts.

Operations and Computation

1 2 3

1 2 3 4 5 6 7 8 9 10

1 4 5 6 7 8 9 10

2 2 4 6 8

7 7

8 8

9 9

10 10

32 40 48 56

36 45 54 63 72

40 50 60 70 80 90

3 3 6 9

4 4 8 12 16

5 5 10 15 20 25

6 6 12 18 24 30 36

14 21 28 35 42 49

16 24

18 27

20 30

64

81

100

10 12 14 16 18 20

12 15 18 21 24 27 30

20 24 28 32 36 40

30 35 40 45 50

42 48 54 60

56 63 70

72 80

90

Multiplication/Division Facts Table

Student Reference Book, p. 52

Student Page

With the help of the children, list the number that is added to each square product to obtain the next square product. The numbers below the arrows name the number of dots that are added to each succeeding array. Point out this pattern on the dot array on journal page 157.

1 4 9 16 25 36 49 64 81 100

+3 +5 +7 +9 +11 +13 +15 +17 +19

Write a list of square products on the board. Remind children that these are called square products or square numbers. Draw arrays to illustrate the square numbers. Then ask why these numbers might be called square numbers. Starting with 2 × 2, each product can be represented by a square array.

Find 11 × 11 without using a calculator. Add 21 to continue the pattern

shown above. 11 × 11 = 10 × 10 + 21 = 100 + 21 = 121

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

� Finding Patterns in the SMALL-GROUP ACTIVITY

Multiplication/DivisionFacts Table(Student Reference Book, p. 52)

Algebraic Thinking Divide the class into small groups and ask children to turn to the Multiplication/Division Facts Table on page 52 in their Student Reference Book. Remind them that the shaded numbers across the top and down the left side of the table are called factors and that the rest of the numbers are the products of the factors. Ask each group to look for patterns in the table and record them on a sheet of paper. After a few minutes, bring the class together to share the patterns they found. To support English language learners, write the patterns on the board as the children describe them.

Examples:

● The products in the row for a factor are counts by that factor. For example, the products in the 3s row are counts by 3: 3, 6, 9, 12. The same is true of the products in the column for a factor.

● The numbers on the diagonal (from the upper-left to the lower-right corner) are square numbers.

● The square numbers divide the table into two parts that are mirror images of each other.

● All products in even-factor rows and columns are even numbers.

● Products in odd-factor rows and columns alternate between even and odd numbers.

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BBBBBBBBBBBBBBBBBBBB EEELEMMMMMMMMOOOOOOOOOBBBLBLBLBLBBLBROOOOROROROROROROROROROO LELELELEEEEEELEEMMMMMMMMMMMMLEMLLLLLLLLLLLLLLLLLLLLLRRRRRRRRRRRGGGLLLLLLLLLLLLLVINVINVINVINVINVINNNVINVINVINNVINVINVINVINV GGGGGGGGGGGOLOOOLOOOLOLOO VINVINVINVLLLLLLLLLVINVINVINVINVINVINVINVINVINVINVINVINVINVINNGGGGGGGGGGGOOOLOLOLOLOLOLOO VVVVLLLLLLLLLLVVVVVVVVVOSOSOOSOSOSOSOSOSOSOOSOSOSOSOOOOSOOSOSOSOSOSSOOSOSOSOSOSOSOSOSOSOSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS VVVVVVVVVVVVVVVVVVVVVVVLLLLLVVVVVVVVLLVVVVVVVLLLLLLLLVVVVVLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSSSSSSSSSSSSSSSSSSSSSSOOOOOOOOOOOOOOOOOOO GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGNNNNNNNNNNNNNNNNNNNNNNNNNNNIIIIIIIIIIIIIIIIIIIIISOLVING

ELL


5. Complete the Fact Triangle.

Write the fact family.

6 × 6 = 36

36 ÷ 6 = 6

3. Draw and label three parallel line

segments. Draw and label a line that

intersects all three line segments.

Date Time

2. Draw an array with 25 Xs arranged

in 5 rows.

How many Xs in each row? 5 Write a number model for the array.

5 × 5 = 25

6. Divide the rectangle into

4 equal parts.

1. This is a picture of a triangular

pyramid. This shape has

4 faces

6 edges

4 vertices

Math BoxesLESSON

7�1

116 64 65

55

N O

H

IK

J L

M

Sample answer:

99 100

4. Fill in the circle next to the correct

answer.

A 251 B 1,751

C 1,795 D 1,805

777

+ 1,028

57–59

6 6

36

×,÷

Sample answer:

EM3MJ2_G3_U07_157-179.indd 158 1/18/11 3:34 PM

Math Journal 2, p. 158

Student Page

Name Date Time

Which Way Out?HOME LINK

7�1

Today your child explored patterns in square products, such as 3 × 3 and 4 × 4. The activity below provides practice in identifying square products. Have your child start at the picture of the Minotaur and use a pencil so he or she can erase wrong turns. If it would be helpful, suggest that your child mark each square product before attempting to find a path.

Please return this Home Link to school tomorrow.

Family Note

199

According to Greek mythology, there was a monster called the Minotaur

that was half bull and half human. The king had a special mazelike

dwelling built, from which the Minotaur could not escape. The dwelling,

called a labyrinth (la buh rinth), had many rooms and passageways

that formed a puzzle. Whoever went in could not find their way out

without help. One day, a Greek hero, Theseus, decided to slay the

monster. To find his way out of the labyrinth, his friend Ariadne gave

him a very, very long ball of string to unwind as he walked through the

passageways. After Theseus slew the Minotaur, he followed the string

to escape.

Pretend you are

Theseus. To find your

way out, you may

go through only

those rooms

numbered with

square products.

Start at the Minotaur’s

chambers and draw

a path to the exit.

Possible paths:

EM3MM_G3_U07_206-236.indd 206 1/18/11 1:03 PM

Math Masters, p. 206

Home Link Master

Lesson 7�1 579

● The 1s products are consecutive counting numbers.

● The 2s products end in 2, 4, 6, 8, or 0.

● The 5s products end in 0 or 5.

● The 10s products end in 0.

● The sum of the digits in each of the 9s products is 9. For example, 4 × 9 = 36, and 3 + 6 = 9.

NOTE Zero can be divided by any nonzero number, but no number can be

divided by zero. Because this table is also used for division, the zero facts are

omitted from it.

� Exploring Multiplication Patterns INDEPENDENTACTIVITY


Algebraic Thinking Have children work for about 5 minutes to solve the problems in Part B on journal page 157. Bring the class together to share solution strategies. Mention that each product is twice as much as the product before it. To find the product in the Try This problem, children can double 64 (or add 64 + 64).

2 Ongoing Learning & Practice

� Playing Name That Number PARTNER ACTIVITY

(Student Reference Book, pp. 299 and 300)

Children practice finding equivalent names for a number as they play Name That Number. Encourage them to use as many operations as they can to name numbers. See Lesson 1-6 or pages 299 and 300 in the Student Reference Book for detailed instructions.

� Math Boxes 7�1 INDEPENDENTACTIVITY


Mixed Practice The Math Boxes in this lesson are paired with the Math Boxes in Lesson 7-3. The skill in Problem 6 previews Unit 8 content.

� Home Link 7�1 INDEPENDENTACTIVITY

(Math Masters, p. 206)

Home Connection Children read about the Greek myth of the Minotaur. They will then trace a path through a labyrinth, or maze, moving from one square product to another.



LESSON

7�1

Name Date Time

Product Patterns

Find each product. Then look for patterns.

1. 1 × 2 = 2

2.

2 × 3 = 6

3.

3 × 4 = 12

4.

4 × 5 = 20

5.

5 × 6 = 30

6.

6 × 7 = 42

7. What happens when you subtract each product

from the next larger product?

Each difference is two more than the

previous difference.

Read pages 198 and 199 in the Student Reference Book to learn more

about patterns in multiplication.

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Teaching Master

LESSON

7�1

Name Date Time

Square and Rectangular Arrays

Follow these steps:

1. Use centimeter cubes to build arrays for each fact.

2. Record the arrays on the grids.

3. Name the shapes of the arrays.

4. Write the number models that match the arrays.

1. 3 × 3 2. 3 × 4

3. 4 × 5 4. 4 × 4

The shape is a square .

Number model: 3 × 3 = 9

The shape is a rectangle .


The shape is a rectangle .


The shape is a square .


Compare the shapes and the number models for different arrays. What

patterns do you see?

The number models with two factors that are the same make

square arrays. When factors are different, rectangular arrays

are made.

Sample answers:

EM3MM_G3_U07_206-236.indd 207 1/18/11 1:03 PM


Teaching Master

3 Differentiation Options

READINESS

INDEPENDENTACTIVITY

� Building Square and 5–15 Min

Rectangular Arrays(Math Masters, p. 207)

To provide experience with square and rectangular arrays, have children use centimeter cubes to build arrays for given factors. They record their work on Math Masters, page 207. When the children have completed the page, have them share the patterns they found.

ENRICHMENT INDEPENDENTACTIVITY

� Exploring a Pattern in a 5–15 Min

Sequence of Products(Math Masters, p. 208; Student Reference Book, pp. 198 and 199)

To further explore multiplication patterns, have children look for patterns in a sequence of multiplication problems in which one factor is 1 more than the other factor (1 × 2, 2 × 3, 3 × 4, and so on).

Possible patterns:

● Each array has one more row and one more column than the preceding array.

● Each array has one more column than a square array. Therefore, each product can be expressed in the form n × n + n:

1 × 2 = 1 × 1 + 1 = 2

2 × 3 = 2 × 2 + 2 = 6

3 × 4 = 3 × 3 + 3 = 12

4 × 5 = 4 × 4 + 4 = 20

5 × 6 = 5 × 5 + 5 = 30

6 × 7 = 6 × 6 + 6 = 42

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Lesson 7�1 581

● If you subtract each product from the next larger product, each difference is 2 more than the preceding difference.

1 × 2 = 2

6 - 2 = 4

2 × 3 = 6

12 - 6 = 6

3 × 4 = 12

20 - 12 = 8

4 × 5 = 20

30 - 20 = 10

5 × 6 = 30

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 the children to write the terms product and factor, draw a picture representing each term, and write other related words. See the Differentiation Handbook for more information.

REMINDER Have children copy the sunrise/sunset data on journal page 125

to the graph on journal page 279.

Have children continue recording the sunrise, sunset, and length of day for your

location in their new journals on pages 279–281.

Have children copy their body measures from journal page 64

onto journal page 251. They will revisit their body measures in Lesson 10-7.

NOTE The data on journal page 43 will be graphed in Lesson 7-8. You may

want children to keep Journal 1 accessible until the data are used in Lesson 7-8,

or you might choose to make the data available by making copies of the class

record of temperature differences kept on Math Masters, page 48. Children will

continue to record the national high and low temperatures for the rest of the year

on journal page 175 as they did on journal page 43.


patterns in products - everyday math · cards 0–10 and 1 each of number ... many of the patterns...

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