Teaching the Lesson materials

Key ActivitiesStudents cut out Multiplication/Division Fact Triangles. They discuss the Multiplication/Division Facts Table, symbols, and vocabulary for multiplication; and they learn multiplication shortcuts and ways to solve unknown facts. They use Fact Triangles to practice multiplication facts.

Key Concepts and Skills• Find factors and multiples of numbers. [Number and Numeration Goal 3]• Identify square numbers. [Number and Numeration Goal 4]• Solve multiplication facts. [Operations and Computation Goal 3]• Identify and use patterns in the Multiplication/Division Facts Table.

[Patterns, Functions, and Algebra Goal 1]

Key Vocabulary multiplication facts • factors • products • factor pair • multiples •square numbers • turn-around facts

Ongoing Learning & Practice materials

Students play Name That Number to practice representing numbers in different ways.

Students practice and maintain skills through Math Boxes and Study Link activities.

Ongoing Assessment: Recognizing Student Achievement Use Math Masters,page 489. [Number and Numeration Goal 4]

Differentiation Options materials

Students make rectangular arrayswith centimetercubes.

Students use rectangular arrays to investigate prime numbers.

Students playBuzz and Bizz-Buzz to practicenaming multiples.

Students addsquare numbersto their Math Word Banks.

� Student Reference Book, p. 234� Teaching Masters (Math Masters,

pp. 76 and 77)� Teaching Aid Master (Math Masters,

p. 403)� Differentiation Handbook� 4 each of number cards 1–10� centimeter cubes

ELL SUPPORTEXTRA PRACTICEENRICHMENTREADINESS

3

� Math Journal 1, p. 57� Student Reference Book, p. 254� Study Link Master (Math Masters, p. 75)� Game Master (Math Masters, p. 489)� deck of number cards

2

� Math Journal 1, p. 56; Activity Sheets 1–4 � Student Reference Book, p. 16 � Study Link 3 �1� Transparencies (Math Masters,

pp. 408 and 409; optional)� paper clips � envelopes or resealable plastic bags� scissors� calculator� counters (optional)

See Advance Preparation

1

Objectives To review strategies for solving multiplication facts;

and to help students work toward instant recall of the

multiplication facts.

Additional InformationAdvance Preparation For Part 1, have plastic bags or envelopes available for storing the FactTriangles. Think about how to pair students for this activity so they will be at similar skill levels.

Lesson 3�2 163

Technology Assessment Management System

Math Masters, page 489See the iTLG.

164 Unit 3 Multiplication and Division; Number Sentences and Algebra

� Math Message Follow-UpAsk partners to explain to each other how they think the FactTriangles will be used.

Tell the class they will use Fact Triangles to practicemultiplication facts and another tool, the Multiplication/DivisionFacts Table, to discuss the terms factor, product, and multiple.

� Reviewing the Multiplication/Division Facts Table(Math Journal 1, p. 56; Math Masters, p. 408)

Introducing the Symbols º and /Ask students to open their journals to page 56. Display an overhead transparency of Math Masters, page 408. Point out the following:

� The table shows the multiplication facts.

� The 0s facts have been omitted to avoid confusion aboutdivision by 0 (which is not possible).

� The 10s facts are included; they are easy and useful.

Point out the � and / symbols in the upper left-hand corner of the table. Explain that in Grades 4, 5, and 6 of EverydayMathematics, an asterisk (�) is usually used to indicatemultiplication. A slash (/) is often used to indicate division, but the and the �� symbols are also used.

NOTE The symbols � and / are used for multiplication and division on computerkeyboards. The asterisk helps avoid confusion between the symbol and thevariable x in algebra. Using the slash for division clearly relates division to fractions. 1 / 2, �

12�, 1 2, and 2 ��1 all represent the same number.

WHOLE-CLASS

DISCUSSION

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

Getting Started

Math Message Cut apart the Multiplication/Division Fact Triangleson Activity Sheets 1–4 at the back of your journal. Write your initials on the back of each one.

Put the Fact Triangles and two paper clips into a plastic bag orenvelope. Think about how you will use the Fact Triangles.

Study Link 3�1 Follow-Up Have partners share the “What’s My Rule?” tablesand rules they created for Problem 6.

Mental Math and Reflexes In preparation for discussing multiples, pose calculatorskip-counting problems, starting with 0. Suggestions:

20s 0, 20, 40, 60, 80, 100, ...50s 0, 50, 100, 150, 200, 250, ...6s 0, 6, 12, 18, 24, 30, ...8s 0, 8, 16, 24, 32, 40, ...Back by 4s 0, �4, �8, �12, �16, �20, ...Back by 7s 0, �7, �14, �21, �28, �35, ...

1

1 2 3 4 5 6 7 8 9 10

1 10

2 4

7

8

9

10

3 9

4 16

5 25

6 36

49

64

81

908070605040302010

20

30

24 40

50

60

70

80

90

100

squarenumbers

factor

factor product

multiplesof 10

multiplesof 10

�, �

Multiplication/Division Facts Table from MathJournal 1, page 56 and Math Masters, page 408

Reviewing the Meaning of Factors, Products, and MultiplesRemind students that the numbers in the shaded row and shaded column in the table are called factors, and the rest of thenumbers are called products. For example, in 4 � 6 � 24, 4 and 6are factors of 24, and 24 is the product of 4 and 6. 4 and 6 are afactor pair of 24. A number may have more than one factor pair.For example, the factor pairs for 24 are 1 and 24, 2 and 12, 3 and8, and 4 and 6. To support English language learners, label and identify the factors and products on the transparency of MathMasters, page 408 as students do so on journal page 56.

Direct students to look across a row or down a column to find the multiples of a number. The multiples of 10, for example, are 10, 20, 30, 40, and so on.

Reviewing Square NumbersNext point out the shaded products on the diagonal. Remindstudents that these products are called square numbers. Eachsquare number is the product of a counting number multiplied byitself. For example, 9 is a square number, because 3 � 3 � 9.

Reviewing Turn-Around FactsPoint out that the products above the diagonal of square numbers are “mirror images” of the products below the diagonal.For example, 6 � 3 and 3 � 6 both equal 18. These are called turn-around facts. If you know a fact, you also know its turn-around fact. To support English language learners, ask: Why do you think these facts are called turn-around facts?

Reviewing Other Multiplication ShortcutsAsk students if they know any other multiplication shortcuts. For example:

� 0 times any number equals 0.

� 1 times any number equals the number.

� To find 2 times a number, double the number.

� To find 10 times a number, write 0 after the number.

Identifying the Facts to Be MemorizedExplain that although there are 100 basic multiplication facts,memorizing them is not so overwhelming. Point out the following:

� They already know the 1s facts. (Cross out the 1s row and 1scolumn on the Multiplication/Division Facts Table transparency.)

� The 2s and 10s facts are easy. (Cross out the correspondingrows and columns.)

� Because of the turn-around rule, all the facts below thediagonal are the same as those above. (Cross out the factsbelow the diagonal. Draw a ring around the 28 remainingfacts.)

Lesson 3�2 165

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

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

100

�, �

Adjustingthe Activity

Have students find common multiples of 2 and 3 or common multiples of 2, 3, and 5 andorganize them in Venn diagrams. For example:

ELL

Multiplesof 5

Multiplesof 3

Multiplesof 2

222

26

10 6 12

30

15

935

144 81628

20 18 24

272125

Multiplesof 2

Multiplesof 3

2 4

8 10

14 16

6 3 9

1512

18

AUDITORY � KINESTHETIC � TACTILE � VISUAL

Adjusting the Activity

166 Unit 3 Multiplication and Division; Number Sentences and Algebra

� Discussing Ways to Solve a Fact(Student Reference Book, p. 16)

Ask students to suggest ways to find the answer to a multiplication fact such as 8 � 7. Memorizing the answer is the most efficient method, but there are many ways to find theanswer if you cannot remember it. Several methods are described on Student Reference Book, page 16.

NOTE Students vary widely in their ability to memorize the multiplication facts.During fourth grade, most students will achieve automatic recall of almost allmultiplication facts, but a few may not. Keep working with those students, but donot equate knowing the facts with mathematical ability. Some students may nevermemorize all the facts, yet they may be successful in mathematics. The authorsknow a research mathematician who still finds 7 � 8 by thinking “7, 14, 28, 56.”Fact power is just one goal of Everyday Mathematics.

� Sorting Fact Triangles(Math Journal 1, p. 56; Math Masters, p. 409)

Ask students to count the Fact Triangles they stored—thereshould be 36.

Draw a large Multiplication/Division Fact Triangle on the board,or display a transparency of Math Masters, page 409, withnumbers inserted. Point out that the numbers in the bottomcorners of the triangle are factors and that the number under thedot is the product of the two factors. Demonstrate how to use theFact Triangle by covering the product.

Divide the class into partnerships. As students practice their facts,remind them to sort the Fact Triangles into two piles—“OK” and“Try Again”—as they did with the �, � Fact Triangles in Unit 1.

Ask each partner to color his or her OK facts and turn-aroundfacts in the table on journal page 56. At the end of the lesson,have students fasten each of the piles with paper clips.

ELL

Encourage the use of tools and strategies, such as:� counters or pictures to illustrate facts,� calculators for skip counting to find products, and� the Multiplication/Division Facts Table to find 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

PARTNER

ACTIVITY

WHOLE-CLASS

DISCUSSION

�, �

“6 times 7 equals what number?”

Basic Multiplication Facts

The symbols and * are both used to indicate multiplication.In this book, the symbol * is used most often.

A basic multiplication fact is a product of two one-digit factors.

8 * 5 � 40 is a basic fact. If you don't remember a basic fact, tryone of the following methods:

Use Counters or Draw a PictureTo find 8 * 5, make 8 groups of counters with 5 counters ineach group, or draw a simple picture to show 8 groups of 5objects. Then count all the objects.

Skip Count UpTo find 8 * 5, count up by 5s, 8 times: 5, 10, 15, 20, 25, 30, 35, 40. Use your fingers to keep track as you skip count.

Use Known FactsThe answer to a 4s fact can be found by doubling, then doublingagain. For example, to find 4 * 7, double 7 to get 14. Thendouble 14 to get 28.

The answer to an 8s fact can be found by doubling three times.For example, to find 8 * 6, double 6 to get 12. Double again toget 24. And then double a third time to get 48.

The answer to a 6s fact can be found by using a related 5s fact.For example, 6 * 8 is equal to 8 more than 5 * 8. 6 * 8 � 5 * 8 � 8 � 40 � 8, or 48.

There is a pattern to the 9s facts:

♦ The 10s digit in the product is 1 less than the digit that ismultiplying the 9.

For example, in 9 * 3 � 27, the 2 in 27 is 1 less than the 3in 9 * 3.

In 9 * 7 � 63, the 6 in 63 is 1 less than the 7 in 9 * 7.

♦ The sum of the digits in the product is 9.

For example, in 9 * 3 � 27, 2 � 7 � 9.

In 9 * 7 � 63, 6 � 3 � 9.

Whole Numbers

���

��

���

��

���

��

���

��

���

��

���

��

���

��

���

��

9s Facts

9 * 1 � 9

9 * 2 � 18

9 * 3 � 27

9 * 4 � 36

9 * 5 � 45

9 * 6 � 54

9 * 7 � 63

9 * 8 � 72

9 * 9 � 81

4s Facts

Double and thendouble again.

8s Facts

Double 3 times.

Student Reference Book, p. 16

Student Page

� Playing Name That Number(Student Reference Book, p. 254; Math Masters, p. 489)

Students play Name That Number to practice representingnumbers in different ways. See Lesson 2-2 for additional information.

Ongoing Assessment:Recognizing Student AchievementUse Math Masters, page 489 to assess students’ ability to use numerical expressions involving one or more of the basic four arithmetic

operations to give equivalent names for whole numbers. Students are makingadequate progress if they are able to create numerical expressions to name thetarget number for two rounds of play. Some students may write numericalexpressions that include parentheses or exponents.

[Number and Numeration Goal 4]

� Math Boxes 3�2(Math Journal 1, p. 57)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 3-4. The skill in Problem 5previews Unit 4 content.

� Study Link 3�2(Math Masters, p. 75)

Home Connection Students complete a Multiplication/Division Facts Table and name factors, products, and multiples of numbers.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

Math MastersPage 489 �

PARTNER

ACTIVITY

2 Ongoing Learning & Practice

57

Date Time

Math Boxes LESSON

3 �2

3. Earth is covered by a rocky outer layer called thecrust, which is made up of many elements.

a. Is there more aluminum or silicon in Earth’s crust?

b. What percentage of Earth’s crust is aluminum?

c. Which element makes up most of Earth’s crust?

oxygen

8%

siliconOxygen

47%Silicon28%

Iron, Calcium,Sodium,

Potassium,Others

17%

Aluminum8%

Elements Found in Earth’s Crust(percent by weight)

1. The numbers 28, 35, and 42 are allmultiples of __. Circle the best answer.

A 7

B 4

C 6

D 2

2. Complete the “What’s My Rule?” table and state the rule.

Rule: �95

9 162–166

in out

236 331

682 777

391 486

938

647 742

A B C D

5. Put these numbers in order from smallestto largest.

0.6 0.06 0.43 0.9

0.90.60.430.06

32 3390

4. Name as many line segments as you canin the figure below.

AB���, AC���, AD���, BC���, BD���, CD���

1,033

Math Journal 1, p. 57

Student Page

Lesson 3�2 167

STUDY LINK

3�2 Multiplication Facts 16 20

Name Date Time

1 6

2

7

8

9

3 3 9

4 8

5

6

14

1 2 3 4 5 7 8 9 10

10 10 40 50 60 70 80 9020 30 100

2 4 6 8 10 12 14 16 18 20

6 12 15 18 21 24 27 30

4 12 16 20 24 28 32 36 40

5 10 15 20 25 30 35 40 45 50

6 12 18 24 30 36 42 48 54 60

7 21 28 35 42 49 56 63 70

8 32 40 48 5616 24 64 72 80

9 36 45 54 63 7218 27 81 90

1 2 3 4 5 6 7 8 9 10º, /

1. Complete the Multiplication/Division Facts Table below.

2. List all the factors of 36.

3. List the factor pairs of 16. and , and , and

4. Name the product of 8 and 7.

5. Name four multiples of 4. , , , 242016856

44821611, 2, 3, 4, 6, 9, 12, 18, 36

Practice

Sample answers:

6. � 91 � 38 7. � 630 � 242 8. 1,462 � 697 � 76538853Math Masters, p. 75

Study Link Master

168 Unit 3 Multiplication and Division; Number Sentences and Algebra

LESSON

3�2

Name Date Time

Prime Numbers

7

A rectangular array shows two factors of a number.

1. On centimeter grid paper, draw as many 2. Write a number model arrays as you can for each of the for each array and its following numbers: turn-around fact.

2, 3, 4, 5, 6, 7, 11, 12, 15, 16

Example: Two different arrays can be made for 8.

1, 2, 4, and 8 are factors of 8.

2 � 4 � 8

4 � 2 � 88 � 1� 81� 8� 8

4. The numbers you listed in the left-handcolumn are called prime numbers. Usethe data you collected to write a definitionfor prime numbers.

factors—itself and 1.has exactly twoA prime numberSample answer:

Numbers with Numbers with Only 1 Array More than 1 Array

2 8

3 45 67 12

11 1516

5. There are 20 other prime numbers that are less than 100. List them below.

53, 59, 61, 67, 71, 73, 79, 83, 89, 9713, 17, 19, 23, 29, 31, 37, 41, 43, 47,

Try This

3. Record your results in the table.

Math Masters, p. 77

Teaching Master

Links to the Future

� Making Rectangular Arrays(Math Masters, p. 76)

To provide experience with multiplication facts using a concretemodel, have students make rectangular arrays with centimetercubes and record a number model for each array.

� Exploring Prime Numbers(Math Masters, pp. 77 and 403)

To extend students’ work with factors, have themconstruct as many arrays as possible for a set of givennumbers and use these arrays to develop a definition

of prime numbers. Ask students to write a list of primes less than 100.

Identifying prime and composite numbers is a Grade 5 Goal.

� Playing Buzz and Bizz-Buzz(Student Reference Book, p. 234)

To practice naming multiples, have students play Buzz and Bizz-Buzz.

� Building a Math Word Bank(Differentiation Handbook)

To provide language support for multiplication, have students usethe Word Bank Template found in the Differentiation Handbook.Ask students to write the term square numbers, draw picturesrelating to the term, and write other related words. See theDifferentiation Handbook for more information.

5–15 Min

SMALL-GROUP

ACTIVITYELL SUPPORT

5–15 Min

SMALL-GROUP

ACTIVITYEXTRA PRACTICE

5–15 Min

PARTNER

ACTIVITYENRICHMENT

5–15 Min

PARTNER

ACTIVITYREADINESS

3 Differentiation OptionsLESSON

3�2

Name Date Time

Rectangular Arrays

7

Rectangular arrays can be used to model multiplication facts. Follow these steps to build arrays with centimeter cubes.

1. Place a deck of number cards (1�10) facedown on the table. Turn over 2 cards.

◆ Use one card for the number of rows in the array.

◆ Use the other card for the number of cubes in each row.

2. Use centimeter cubes to build the array.

Example:

If and are turned over, you can make either array:

3. Record some of the arrays you made in the table below.

2

2

4

4

2 rows, 4 cubes per row

4 rows, 2 cubes per row

How many How many cubes How many Number rows? in each row? cubes in all? model

4 2 8 4 º 2 � 8

Math Masters, p. 76

Teaching Master