180 Unit 3 Multiplication and Division; Number Sentences and Algebra

Teaching the Lesson materials

Key ActivitiesStudents explore the relationships between multiplication and division. They use theMultiplication/Division Facts Table to solve division facts, and they use Multiplication/DivisionFact Triangles to generate fact families and to practice division facts.

Key Concepts and Skills• Solve multiplication facts. [Operations and Computation Goal 3]• Use multiplication facts to generate related division facts. [Operations and Computation Goal 3]• Apply multiplication and division facts and extended facts to solve problems.

[Operations and Computation Goal 3]• Write multiplication and division number sentences. [Patterns, Functions, and Algebra Goal 2]

Key Vocabulary dividend • divisor • quotient • remainder • fact family

Ongoing Assessment: Recognizing Student Achievement Use journal page 61. [Patterns, Functions, and Algebra Goal 2]

Ongoing Learning & Practice materialsStudents play Beat the Calculator to practice multiplication facts.

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

Differentiation Options materials

Students playDivision Arrays.

Students explore the relationshipbetween divisionand fractions.

Students completeFact Triangles andwrite the related factfamilies.

Students create aposter for dividend,divisor, quotient, andremainder.

� Student Reference Book, p. 240� Teaching Master (Math Masters, p. 86)� Teaching Aid Master (Math Masters,

p. 418)� Game Master (Math Masters, p. 470)� number cards 6–18; 1 six-sided die;

18 counters; grid paper (optional)

ELL SUPPORTEXTRA PRACTICEENRICHMENTREADINESS

3

� Math Journal 1, p. 62� Student Reference Book, p. 233� Study Link Master (Math Masters,

p. 85)� Game Master (Math Masters, p. 461)� number cards 1–10 (4 of each);

calculator

2

� Math Journal 1, p. 61� Study Link 3�4� Transparencies (Math Masters,

pp. 408 and 409; optional)� º, / Fact Triangles� slate

1

Objectives To guide exploration of the relationship between

multiplication and division; and to provide practice with

division facts.

Technology Assessment Management System

Journal page 61, Problem 1See the iTLG.

10, 12, 14, , , 12, 16, 20, , , 30, 25, 20, , , 18, 15, 12, , ,

18, 27, 36, , , 24, 30, 36, , , 49, 42, 35, , , 40, 32, 24, , ,

24, 28, 32, , , 24, 36, 48, , , 63, 56, 49, , , 72, 64, 56, , ,

Lesson 3�5 181

Getting Started

Math MessageHow many bags of oranges, with 5 oranges ineach bag, can be made with 35 oranges? Howmany can be made with 42 oranges?

Study Link 3�4 Follow-Up In small groups, have students compare answersand pose their mystery-number problems.

Mental Math and Reflexes Have students name the next three numbers in each pattern. Suggestions:

Adjusting the Activity

� Math Message Follow-UpAs you and the students discuss the first problem, use alternativestatements such as these:● How many 5s in 35?● What number multiplied by 5 gives 35?● 5 times what number is 35?● How much is 35 divided by 5?

Point out that in the division fact 35 / 5 � 7, the number 35 is thedividend, 5 is the divisor, and 7 is the quotient.

In the second problem, there will still be 2 oranges left over aftermaking 8 bags with 5 oranges in each bag. Remind students that2 is called the remainder.

ELL

To help students keep the terms dividend and divisor straight, point out that the suffix -or indicates someone or something doing the action—doctor,senator, tailor, generator, and so on. So the divisor is the number that is doingthe dividing. The dividend is the number that is being divided.

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

WHOLE-CLASS

DISCUSSION

1 Teaching the Lesson

24 28 3215 10 59 6 3

16 18 2042 48 5428 21 1416 8 0

45 54 6360 72 8442 35 2848 40 32

36 40 44

182 Unit 3 Multiplication and Division; Number Sentences and Algebra

61

Date Time

Multiplication and DivisionLESSON

3� 5

1. Choose 3 Fact Triangles. Write the fact family for each.

� � � � �

� � � � �

/ � / � �

/ � / � �

2. Solve each division fact.

a. 27 / 3 � Think: How many 3s in 27?

b. � 45 / 5 Think: How many 5s in 45?

c. 36 6 � Think: 6 times what number equals 36?

d. 24 / 8 � Think: 8 times what number equals 24?

3. A cashier has 5 rolls of quarters and 6 rolls of dimes in the cash register.Each roll of quarters is worth $10, and each roll of dimes is worth $5.

a. How much are the rolls of quarters and dimes worth in all? $

b. How many quarters are in 1 roll? quarters

c. How many quarters are in the 5 rolls? quarters

d. How many dimes are in 1 roll? dimes

e. How many dimes are in the 6 rolls? dimes

f. There is also $7.50 worth of half-dollars in the cash register. How many half-dollars is that? half-dollars15

30050

2004080

36

99

16 20

Answers vary.

Equivalents

3 � 4 12 / 3 12 3 3 � 5(� means “is less than”)

3 4 �132� 3�1�2� 5 � 3

(� means “is greater than”)

�

Try This

Math Journal 1, p. 61

Student Page

Adjusting the ActivityUse a piece of paper to track

across and up. For example, to solve 56 / 7,line up the bottom of a piece of paper alongthe top of the 7s row. Scan across the 7srow to find 56. Line up the right edge of thepaper with the left side of the column containing 56. Then scan up that column tofind the answer, 8, at the top.

1

1 2 3 4 5 6 7 8 9

1

2 4

7

8

9

10

3 9

4 16

5 25

6 36

4921147

8

9

423528

2416 48 564032

2718 5445 726336

302010 605040 908070

64

81

10

10

20

30

40

50

60

70

80

9

18

27

36

45

54

63

8

16

24

32

40

48

56

72

90

100

� Exploring the Relationshipbetween Multiplication and Division(Math Journal 1, p. 61; Math Masters, p. 409)

On the board, draw the �, / Fact Triangle shown below, or use atransparency of Math Masters, page 409.

Remind students of the following symbols: � or for multiplication, and or / for division. Then show them how to use the Fact Triangle to generate a multiplication/division fact family:

6 � 7 � 42

7 � 6 � 42

42 / 6 � 7

42 / 7 � 6

Do several more examples with the class. Then have studentschoose three of their Fact Triangles and record the fact families for those triangles in Problem 1 on journal page 61.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 61, Problem 1 to assess students’ ability to use conventionalnotation to write multiplication and division number sentences. Students aremaking adequate progress if they are able to record the appropriate fact families.Some students may use factors greater than 10.

[Patterns, Functions, and Algebra Goal 2]

� Using the Multiplication/Division Facts Table for Division(Math Journal 1, p. 61; Math Masters, p. 408)

You may want to use an overhead transparency of theMultiplication/Division Facts Table (Math Masters, page 408) toshow students how to use it for division. For example, to find theanswer to 24 / 3, move across the 3s row to the cell containing 24.Then move up the column containing 24 to find the answer, 8, atthe top. Tell students that the arrow (see below) from the 3 to the24 can represent the question “How many 3s in 24?”

Have students complete Problems 2 and 3 on journal page 61 ontheir own or with a partner.

1

1 2 3 4 5 6 7 8 9 10

1 10

2 4

3 9

4 16

20

24 30

40

�, �

WHOLE-CLASS

ACTIVITY

Journal

page 61

Problem 1 �

WHOLE-CLASSDISCUSSION

•

6 7

42

�, �

AUDITORY � KINESTHETIC � TACTILE � VISUAL

Lesson 3�5 183

Adjusting the Activity

� Practicing Division Facts with Fact TrianglesDemonstrate the following procedure:

1. Student A takes the top triangle from his or her OK pile and asks Student B a division fact from the triangle, such as“42 divided by 6 is equal to what number?”

2. Partners help each other. If Student B is not sure of theanswer, Student A gives a clue. For example, if the fact is 42divided by 6, Student A can help Student B by asking, “Whatnumber times 6 is equal to 42?” or “How many 6s in 42?” IfStudent B still is not sure, Student A reads the answer.

3. Partners take turns asking and answering until they havegone through the triangles in both their OK and Try Againpiles, or as long as time permits.

� Playing Beat the Calculator(Student Reference Book, p. 233; Math Masters, p. 461)

Students play Beat the Calculator to develop automaticity withmultiplication facts.

Have students play the extended-facts version of the game on StudentReference Book, page 233.

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

� Math Boxes 3�5(Math Journal 1, p. 62)

Mixed Practice Math Boxes in this lesson are linked withMath Boxes in Lessons 3-1 and 3-3. The skill in Problem 6previews Unit 4 content.

� Study Link 3�5(Math Masters, p. 85)

Home Connection Students fill in the missing numbersin multiplication and division facts.

INDEPENDENT

ACTIVITY

INDEPENDENT

ACTIVITY

SMALL-GROUP

ACTIVITY

2 Ongoing Learning & Practice

PARTNER

ACTIVITY Math Boxes LESSON

3� 5

5. Complete.

a. 7, 11, 15, , ,

Rule:

b. , , , 22, 25, 28

Rule:

c. , 14, , 28, , 42

Rule: �735217

�3191613

�4272319

1. Write �, �, or � to make each numbersentence true.

a. 5,389 3,389

b. 70,642 70,699

c. 6 million 6,000,000

d. 8,000,032 8 million, 32 thousand

e. 400 � 30 � 5 4,000 � 30 � 5��

��

�

3. Make a ballpark estimate. Write a numbermodel to show your strategy.

a. 13,685 � 8,379

� �

b. 7,602 � 3,213

� � 4,4003,2007,600

6,0008,00014,000

Sample answers:

4. Complete.

a. 31 in. � ft in.

b. 17 ft � yd ft

c. ft � 14 yd

d. in. � 2 yd 1 ft

e. 2�14� miles � ft11,8808442

2572

6. Solve mentally or with a paper-and-pencilalgorithm.

a. $2.27 b. $5.00� $4.96 � $3.64

73

181 129

34–37160 161

2. The number of glasses of milk drunk by 10 students in a week:

16, 13, 15, 20, 8, 10, 15, 12, 10, 18

What is the range? Circle the best answer.

A 8

B 20

C 12

D 14

62

Date Time

6 149

$7.23 $1.36

Math Journal 1, p. 62

Student Page

STUDY LINK

3�5 Missing Numbers

16 20

Name Date Time

Complete each fact by filling in the missing numbers. Use the Multiplication/Division Facts Table to help you.

1. 30 / 6 � 2. 21 / � 3

3. 9 � 8 4. 100 / � 10

5. / 4 � 8 6. 25 �

7. � 42 / 8. 8 / �

9. 4 � / 10. � 1

11. / 2 � 12. 10 º �

32107275

Try This

13. 5 º º � 30

14. 54 � º º 63323

1 2 3

, 1 2 3 4 5 6 7 8 9 10

4 5 6 7 8 9 10

2 4 6 8

7

8

9

10 10

32 40 48 56

36 45 54 63 72

40 50 60 70 80 90

3 6 9

4 8 12 16

5 10 15 20 25

1

2

7

8

9

3

4

5

6 6 12 18 24 30 36

14 21 28 35 42 49

16 24

18 27

20 30

64

81

100

º /

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

15. � 989 � 657 16. 314 � 4,719 �

17. 887 � 598 � 18. � 2,004 � 7161,2882895,0331,646

Practice

Answersvary.

Sample answers.

Math Masters, p. 85

Study Link Master

184 Unit 3 Multiplication and Division; Number Sentences and Algebra

� Playing Division Arrays(Student Reference Book, p. 240; Math Masters, p. 470)

To explore the connections between multiplication and divisionusing a concrete model, have students play Division Arrays andrecord their work on Math Masters, page 470. Some students mayfind it helpful to organize the counters on grid paper.

� Exploring the Relationshipbetween Division and Fractions(Math Masters, p. 86)

To apply students’ understanding of division, have them usedivision facts to solve a fraction problem. Both fractions anddivision involve sharing something equally or forming equalgroups. Fraction notation is one way to write a division problem.

Some students may solve the pizza problem by dividing each pizzainto 4 equal pieces. Each person gets 1 piece, or �

14�, of each pizza.

So one person’s share of the 3 pizzas is three one-fourths, or �34� of

one pizza.

5–15 Min

PARTNER

ACTIVITYENRICHMENT

15–30 Min

PARTNER

ACTIVITYREADINESS

3 Differentiation Options

Name Date Time

Division Arrays Record Sheet 132

4

Your score is the number of counters per row. If there are 0 leftover counters, your score is double the number of counters per row.

Round Rows Counters Counters Leftover Scoreper Row in All Counters

Sample 3 5 15 0 10

1

2

3

4

5

Total Score

Name Date Time

Division Arrays Record Sheet 132

4

Your score is the number of counters per row. If there are 0 leftover counters, your score is double the number of counters per row.

Round Rows Counters Counters Leftover Scoreper Row in All Counters

Sample 3 5 15 0 10

1

2

3

4

5

Total Score

240

240

Math Masters, p. 470

Game MasterLESSON

3�5

Name Date Time

Fractions and Division

Four friends want to share 3 pizzas evenly. Each person cannot have a wholepizza, so the pizzas need to be divided.

1. How many slices of pizza should each person get? slices.Use a drawing to explain your answer.

2. Marisa says these expressions all mean the same thing:

�14� of 3 �

34� 3 4 4�3�

Do you agree or disagree? Explain your answer.

3

46

Sample answer: Agree. They are all different names for the same thing. �

14� of 3 pizzas is �

34� of a pizza, so �

14� of 3 � �

34�. The

fraction bar, , and �� all mean division.

Math Masters, page 86

ExampleExample Dave draws a 14-card and takes 14 counters. He rolls a 3 and makes an array with 3 rows by putting 4 counters in eachrow. Two counters are left over.

Dave scores 4 because there are 4 counters in each row.

ExampleExample Marsha draws a 15-card and takes 15 counters. She rolls a 3 and makes an array with 3 rows by putting 5 countersin each row.

Her score is 5 * 2 � 10 because there are 5 counters in each row, with none left over.

Games

Division Arrays

Materials � number cards 6–18 (1 of each)� 1 six-sided die� 18 counters

Players 2 to 4Skill Division and equal sharesObject of the game To have the highest total score. Directions

1. Shuffle the cards. Place the deck number-side down on the table.

2. Players take turns. When it is your turn, draw a card and take the number of counters shown on the card. You will use the counters to make an array.

♦ Roll the die. The number on the die is the number of equal rows you must have in your array.

♦ Make an array with the counters. ♦ Your score is the number of counters in 1 row. If there

are no leftover counters, your score is double the number of counters in 1 row.

3. Keep track of your scores. The player with the highest totalscore at the end of 5 rounds wins.

Student Reference Book, p. 240

Student Page

� Practicing with Fact Families(Math Masters, p. 418)

To practice the connection between multiplication and division,have students complete Fact Triangles and related fact families.

� Creating a PosterTo provide language support for division, have students create aposter representing the terms dividend, divisor, quotient, andremainder. Ask them to write a division number sentence andlabel all of the parts. Students can color code the number sentenceso that each part is drawn in the same color as the label.

5–15 Min

SMALL-GROUP

ACTIVITYELL SUPPORT

5–15 Min

INDEPENDENT

ACTIVITYEXTRA PRACTICE

Lesson 3�5 185

Planning Ahead

Lesson 3-6 provides a detailed introduction to the World Tour.Read through the lesson well in advance, along with the relatedpages in the World Tour section of the Student Reference Book. Inaddition to a globe and a large map of the world, try to collect asmany reference materials as possible for a World Tour Corner.The authors recommend several copies of an almanac, such as theWorld Almanac, a full-sized atlas, and several student atlasessimilar to the National Geographic World Atlas for YoungExplorers (National Geographic Society, 2003).

Ask students to bring country guide books to school. Begin acollection of travel information, including airline schedules andfares, travel articles, weather reports, and currency exchangerates from the local newspaper or the Internet.

Fact Families

Name Date Time

1.

Complete each Fact Triangle. Write the fact family for each Fact Triangle.

•

�, �

2.

•

�, �

3.

•

�, �

4.

•

�, �

Math Masters, p. 418

Teaching Aid Master