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Texas Essential Knowledge and Skills

Algebraic Reasoning—5.4.A Identify prime and composite numbers

MATHEMATICAL PROCESSES5.1.C Select tools, technology, and techniques5.1.D Communicate mathematical ideas and reasoning5.1.F Analyze mathematical relationships

Are You Ready?Access Prior KnowledgeUse the Are You Ready? 7.1 in the Assessment Guide to assess students’ understanding of the prerequisite skills for this lesson.

Vocabularycomposite number, divisible

Go to Multimedia eGlossary at thinkcentral.com

7.1 Factors and Divisibility How can you tell whether one number is a factor of another number?

Essential Question?

Lesson OpenerMaking ConnectionsInvite students to tell you what they know about division of whole numbers.

When can you use division to solve a problem? (to find how many equal-numbered groups of objects there are) What do the numbers in a division problem tell us? (the total number of objects, the number of groups, and the number of objects in each group) How are the numbers in a division problem related to multiplication? (Accept all reasonable answers.)

Using the Digital LessonHave students use base-ten blocks to brainstorm how they can group sets of equal numbers. Ask them to compare their groupings to those of their classmates.

Learning TaskWhat is the problem the students are trying to solve? Connect the story to the problem.

• What is the total number of golf balls? (88)

• What does the problem tell you about how to separate the golf balls into the buckets? (There must be the same number in each bucket.)

• What will the number of buckets used tell you? (how many equal groups there are)

Literacy and MathematicsChoose one or more of the following activities.

• Have students work with a partner to ask and answer questions about the information given in the problem.

• Have students work in pairs to create a similar problem in which they need to find how many groups are needed to divide a set of objects into equal numbers.

How can you tell whether one number is a

factor of another number?

Lesson 7.1 291A

Unlock the ProblemUnlock the Problem

Essential Question?

Mathematical ProcessesMath Talk

Factors and Divisibility

Algebraic Reasoning—5.4.A

MATHEMATICAL PROCESSES5.1.C, 5.1.D, 5.1.F7.1

How can you tell whether one number is a factor of another number?

Name

Students in Carlo’s art class painted 32 square tiles

for a mosaic. They will arrange the tiles to make a

rectangle. Can the rectangle have 32 tiles arranged

into 3 equal rows, without gaps or overlaps?

A composite number is a whole number greater than

1 that has more than two factors. You can use models

and division to find factors of composite numbers.

One Way Draw a model.

Think: Try to arrange the tiles into 3 equal rows to make a rectangle.

A rectangle __ have 32 tiles arranged into 3 equal rows.

▲ Mosaics are decorative patterns made with pieces of glass or other materials.

Think: Divide to see whether the unknown factor is a whole number.

Another Way Use division.

If 3 is a factor of 32, then the unknown factor in 3 × ■ = 32 is a whole number.

The unknown factor in 3 × ■ = 32 __ a whole number.

So, a rectangle __ have 32 tiles arranged in 3 rows.

3 ) ‾ 3 2

Explain how the model relates to the quotient and remainder

for 32 ÷ 3.

Math IdeaA factor of a number divides the number evenly. This means the quotient is a whole number and the remainder is 0.

Possible answer: the model shows 3 rows with 10 tiles and 2 rows each with an extra tile. The quotient is 10 with a remainder of 2.

is not

cannot

cannot

– 30 2– 0

2

_

_

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Module 7 291

English Language Learners Language SupportELL

36 is divisible by 2.

36

ELPS 2.C.2, 3.F.1, 3.G.2Leveled Activities ELPS

Beginning: Activity 20 1.A.1, 3.G.2, 4.C.3

Intermediate: Activity 26 3.G.1, 4.D, 4.F.2

Advanced: Activity 27 2.I.3, 3.B.3, 4.D

Advanced High: Activity 43 4.F.8, 4.G.2, 4.G.4

Go to thinkcentral.com for the ELL Activity Guide containing these leveled activities.

Strategy: Model LanguageMaterials: index cards, markers

• Students learn the language of divisibility when it is modeled.• Have students write 2-digit numbers on four cards.• Explain the divisibility rules for 2, 3, 5, 6, and 9. • Shuffle the cards and then draw the top card. Tell

students by what numbers it is divisible and how you know.

• Draw another card.• Is this number divisible by 2? How do you know?• Have partners take turns drawing a new card and asking the

modeled questions to determine divisibility.

Visual / AuditoryPartners

Unlock the Problem Have students read the problem and the caption beneath the picture of the mosaic. Ask students if they have ever made a mosaic at school or at home.

One WayDiscuss how actual tiles could be arranged in a rectangle. Point out that arranging tiles requires materials that may not always be available. Drawing a model is a simpler method to use.

Another WayFor greater numbers, using tiles or drawing a model can be time–consuming. Using division is a more efficient method.

• Is there a remainder when 32 is divided by 3? yes

• Why does the remainder help you answer the question? Possible answer: because there is a remainder in the division, I know that 32 tiles cannot be arranged into 3 rows without gaps or overlaps.

Math Talk Use Math Talk to help students relate the model to the division when determining whether one number is a factor of another number.

Mathematical Processes

291 Module 7

The number is even.

The sum of the digits

is divisible by 3.

The last digit is 0 or 5.

The number is even

and divisible by 3.

The sum of the digits is

divisible by 9.

2

3

5

6

9

Divisibility Rules

Number Divisibility Rule

Mathematical ProcessesMath Talk

Share and ShowShare and Show

1. Is 4 a factor of 28? Draw a model to help.

Think: Can you make a rectangle with 28 squares in 4 equal rows?

4 _ a factor of 28.

Is 5 a factor of the number? Write yes or no.

Is 6 a factor of 72?

Think: If 72 is divisible by 6, then 6 is a factor of 72.

Test for divisibility by 6:

Is 72 even? _

What is the sum of the digits of 72?

_ + _ = _

Is the sum of the digits divisible by 3?

_______

72 is divisible by _ .

So, 6 is a factor of 72.

Divisibility Rules A number is divisible by another

number if the quotient is a counting number and the

remainder is 0.

Some numbers have a divisibility rule. You can use a

divisibility rule to tell whether one number is a factor

of another.

2. 27

_

3. 30

_

4. 36

_

5. 53

_

How are divisibility and factors related? Explain.

Possible explanation: a number is divisible by each of its factors.

6

7 2 9

Yes; 9 ÷ 3 = 3, and the remainder is 0.

yes

no nonoyes

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Go to Go to thinkcentral.com for additional enrichmentactivities in the Enrich Activity Guide.

Enrich

1

2

3

a student misses the checked exercises

Quick Check

IF

THENDifferentiate Instruction withRtI Tier 1 Lesson 45

Divisibility RulesRead and discuss the divisibility rules. Then ask:

• Suppose that you have to divide two numbers. Give an example of how divisibility rules can be used to check your answer. Possible answer: I know that every even number is divisible by 2. So, if I divide an even number by 2 and have a remainder in my answer, I know to recheck my answer.

• Can a number be divisible by more than one number? Give an example to support your answer. Yes; possible example: 10 is divisible by 2, by 5, and by 10.

• Why isn’t there a divisibility rule for 1? All nonzero numbers are divisible by 1.

• Why isn’t there a divisibility rule for 0? You cannot divide by 0.

Share and ShowThe first problem connects to the learning model. Have students use the MathBoard to explain their thinking.

Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard.

• Have students find numbers that are divisible by 8 by looking at multiples of 8. (Students may need to choose greater numbers, like 8 × 56, for the activity.)

• Have students consider the divisibility rules shown in the lesson. Have students make conjectures of what a divisibility rule for 8 would be. Encourage them to use the strategy guess, check, and revise. If you can divide a number by 2 three times, then the number is divisible by 8. For example, 64 ÷ 2 = 32, 32 ÷ 2 = 16, 16 ÷ 2 = 8. So, 64 is divisible by 8.

• Challenge students to explain why the divisibility rule works. The inverse of dividing by 2 three times is multiplying by 2 three times. 2 × 2 × 2 is 8. So, you are basically dividing by 8.

LogicalSmall Group

COMMON ERRORSError Students look only at the last digit of a number to decide if it is divisible by 3 or 9.

Example 19 is divisible by 3 and by 9 because the last digit in the number is divisible by 3 and by 9.

Springboard to Learning To show that 19 is not divisible by 3 or 9, have students complete the division problems shown below.

19 ÷ 3 = ■ 19 ÷ 9 = ■

Point out that since none of the answers are whole numbers, 19 is not divisible by 3 or by 9.

CE

Lesson 7.1 292

Stamps SetsCountry Number of stamps

Germany 90

Sweden 78

Japan 63

Canada 25

Problem SolvingProblem Solving


Name

List all the factor pairs in the table. Use a model or paper and pencil to help.

10. What’s the Error? George said if 2 and 4 are factors of

a number, then 8 is a factor of the number. Is he correct? Explain.

Factors of 24

_ × _ = _ _ , _

_ × _ = _ _ , _

_ × _ = _ _ , _

_ × _ = _ _ , _

Factors of 39

_ × _ = _ _ , _

_ × _ = _ _ , _

6. 7.

8. 56 9. 64

List all the factor pairs for the number. Make a table to help.

Use the table to solve 11–12.

11. Multi-Step Dirk bought a set of stamps. The number of stamps

in the set he bought is divisible by 2, 3, 5, 6, and 9. Which set is it?

12. Multi-Step Geri wants to put 6 stamps

on some pages in her stamp book and 9 stamps on other

pages. Explain how she could do this with the stamp set

for Sweden.

6

8

1312

3924

4

3

32

11

24

24

3924

3924

6

8

1312

3924

4

3

32

11

1 and 64, 2 and 32, 4 and 16,

8 and 8

1 and 56, 2 and 28, 4 and 14,

7 and 8

No; possible explanation: 2 and 4 are factors of 20, but 8 is not a factor.

Possible answer: Geri could break apart 78 into

60 + 18, since 60 is divisible by 6 and 18 is divisible by 9. She could make 10 pages

with 6 stamps each (60 ÷ 6 = 10) and 2 pages with 9 stamps each (18 ÷ 9 = 2).

Germany

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Module 7 • Lesson 1 293

Problem SolvingProblem

For Problem 10, some students may find it helpful if you remind them that when one number is divisible by another number, the division of those numbers does not produce a remainder.

To solve Problem 11, students should infer that because the correct answer is divisible by 5, two numbers (78 and 63) can immediately be eliminated as possible answers.

Go Deeper• Ask students to divide 24 by 2 and then divide

the quotient, 12, by 2 again. Is 24 divisible by 4? Explain. Yes; 24 can be evenly divided by 4.

• Now try it with 46. Divide 46 by 2. Then divide the quotient by 2. What do you notice? 46 ÷ 2 = 23; 23 ÷ 2 = 11 r1; 46 is not divisible by 4 because there is a remainder.

• If a number can be divided by 2 twice without a remainder, then that number is divisible by 4.

Math on the Spot Video Tutor

Through the Math on the Spot Video Tutor, students will be guided through an interactive solving of this type of H.O.T. problem. Use this video to also help students solve the H.O.T. problem in the Interactive Student Edition. With these videos and H.O.T. problems, students will build skills needed in the TEXAS assessment.

MV

Math on the Spot videos are in theInteractive Student Edition and atthinkcentral.com.

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A number is divisible by another number if the quotient is a counting number and the remainder is 0. You can decide if a number is divisible by 2, 3, 5, 6, or 9 by using divisibility rules instead of dividing. Divisibility rules help you decide if one number is a factor of another.

Is 39 divisible by 2, 3, 5, 6, or 9?

Result Conclusion Divisibility Rules

39 ÷ 2 19 r1 39 is not divisible by 2. The last digit, 9, is not even, so 39 is not divisible by 2.

39 ÷ 3 13 r0 39 is divisible by 3. The sum of the digits, 3 + 9 = 12, is divisible by 3, so 39 is divisible by 3.

39 ÷ 5 7 r4 39 is not divisible by 5. The last digit, 9, is not a 0 or 5, so 39 is not divisible by 5.

39 ÷ 6 6 r3 39 is not divisible by 6. 39 is not divisible by both 2 and 3, so it is not divisible by 6.

39 ÷ 9 4 r3 39 is not divisible by 9. The sum of the digits, 3 + 9 = 12, is not divisible by 9, so 39 is not divisible by 9.

39 is divisible by 3. So, 3 is a factor of 39. A number is divisible by each of its factors. A whole number that has more than two factors is a composite number.

Use the chart to tell whether 30 is divisible by each divisor. Explain.

1.

2.

3.

4.

5.

Is 4 a factor of the number? Write yes or no.

6. 81 7. 24 8. 56

5.4.ALESSON

45Factors and DivisibilityOBJECTIVE Determine whether a number is a factor of a given number.

Result Conclusion (yes/no) Explanation

30 ÷ 2

30 ÷ 3

30 ÷ 5

30 ÷ 6

30 ÷ 9

no

15 yes 30 is an even number.10 yes 3 + 0 = 3; 3 is divisible by 3.6 yes The last digit is a 0.5 yes 30 is divisible by both 2 and 3.3 r3 no 3 + 0 = 3; 3 is not divisible by 9.

yes yes

Algebraic Reasoning 89 © Houghton Mifflin Harcourt Publishing Company

Name

E43Enrich

Enrich 43

Invisible Divisible

Use the clues to find all possibilities for the unknown digit in each

number.

1. The number below has 2 as a factor.

What could the unknown digit be?

5,83



3,2 6



1,9 5



6,30



7,71



4, 11

7. The number below has 3 and 5 as

factors. What could the unknown digit

be?

6,1 5

8. The number below has 2 and 9 as

factors. What could the unknown digit

be?

2,3 6

9. Stretch Your Thinking A number is divisible by 2 if the last

digit is divisible by 2. A number is divisible by 4 if the last two

digits form a number divisible by 4. A number is divisible by 8 if the

last three digits form a number divisible by 8. Describe a possible

pattern in the divisibility rules. Then test each of the following

numbers for divisibility by 8.

3,488 5,614 4,320 3,052

0, 2, 4, 6, 8

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

0, 6

0, 3, 6, 9

1, 3, 5, 7, 9

0, 3, 6, 9

Possible answer: 4 5 2 3 2 and 8 5 2 3 2 3 2. So, for

each successive time 2 is a factor, you need to consider

one more digit; 3,488 and 4,320 are divisible by 8.

0, 9

7

1

2

3

RtI Tier 1 Lesson 45 Enrich 43

293 Module 7

Daily Assessment TaskDaily Assessment Task

Mathematical Processes

Fill in the bubble completely to show your answer.

13. There are 54 people attending a party. Carla is arranging chairs and

tables. How should she arrange the chairs so that there will be an equal

number of people seated at each table?

A 12 chairs at each table

B 8 chairs at each table

C 9 chairs at each table

D 10 chairs at each table

14. Analyze Jake organizes 48 marbles into packs. He places the same

number of marbles into each pack. How could he arrange the marbles?

A 8 or 10 in each pack

B 9 or 12 in each pack

C 10 or 12 in each pack

D 8 or 12 in each pack

15. Multi-Step Megan has 34 rocks in her rock collection. She wants to put

5 rocks in some cases and 7 rocks in other cases. How could she arrange

the rocks?

A Have 6 cases of 5 rocks, and 1 case of 7 rocks.

B Have 4 cases of 5 rocks, and 2 cases of 7 rocks.

C Have 2 cases of 5 rocks, and 4 cases of 7 rocks.

D Have 5 cases of 5 rocks, and 2 cases of 7 rocks.

TEXAS Test Prep16. Mrs. Mastrioni bought a set of 80 stamps. She wanted to give all the stamps

to her students as a reward. She could give equal numbers of stamps to

A 2 or 3 students.

B 2 or 6 students.

C 2, 4, 5, or 8 students.

D 2, 4, 8, or 9 students.

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NO

Daily Assessment Task 1

2

3

Games

Differentiated Centers Kit

• Soar to Success MathWarm-Up 31.30

ActivitiesFlowering FactorsStudents complete orange Activity Card 17 by identifying the factors of whole numbers.

GamesFactor FarmStudents practice determining factors of whole numbers.

TEXAS Test Prep CoachIn the Test Prep exercise, if students selected:

A They thought that 80 was divisible by 3.

B They thought that 80 was divisible by 6.

D They thought that 80 was divisible by 9.

Essential Question? WriteMathWriteMath

How can you tell whether one number is a factor of another number? Possible answer: I can use a divisibility rule to check if a number is a factor of another number.

• Enrich 43

• Homework and Practice Lesson 7.1

Can students tell whether one number is a factor of another number?

Lesson 7.1 294

TEXAS Test PrepLesson CheckLesson Check

9. Which number is divisible by 2, 3, and 6?

A 60

B 33

C 46

D 21

10. Jo Beth organizes 63 party favors into gift bags.

She places the same number of favors in each

bag. How could she arrange the favors?

A 3 or 6 in each bag

B 6 or 9 in each bag

C 7 or 8 in each bag

D 7 or 9 in each bag

11. A package of marbles contains 20 blue

marbles, 38 red marbles, 48 green marbles,

and 16 yellow marbles. Which color of marbles

can be divided evenly between 6 people?

A blue

B red

C green

D yellow

12. The fifth-grade students participating in the

math competition will be evenly divided into

teams of 3 students each. How many students

could be participating?

A 25

B 21

C 23

D 26

13. Multi-Step The librarian has 94 books to

organize. He wants to put 5 books on some

shelves and 6 books on other shelves. How

could he arrange the books?

A Have 8 shelves of 5 books, and 9 shelves

of 6 books.

B Have 9 shelves of 5 books, and 8 shelves

of 6 books.

C Have 5 shelves of 6 books, and 6 shelves

of 5 books.

D Have 7 shelves of 5 books, and 6 shelves

of 6 books.

14. Multi-Step A group of 72 students went on a

field trip. The students traveled in vans, with

an equal number of girls and boys in each van.

If 42 boys went on the trip, how many vans

were there?

A 9

B 8

C 7

D 6

Fill in the bubble completely to show your answer.

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Homeworkand Practice

Factors and Divisibility

Algebraic Reasoning—5.4.AMATHEMATICAL PROCESSES 5.1.C, 5.1.D, 5.1.F

7.1List all the factor pairs in the table.

1. 2.

3. 18

5. 36

4. 45

6. 63

7. Planners for the city’s botanical garden are

planting a rose garden. They want to plant

56 rose bushes. Can they arrange the rose bushes

into 4 equal rows? Explain.

8. A marching band with 75 members makes two

rectangular marching formations. One rectangle

has 6 rows and the other rectangle has 9 rows.

Explain how the members can form the two

rectangles.

List all the factor pairs for the number. Make a table to help.

Factors of 30

_ × _ = _ _ , _

_ × _ = _ _ , _

_ × _ = _ _ , _

_ × _ = _ _ , _

Factors of 15

_ × _ = _ _ , _

_ × _ = _ _ , _

6

10

515

1530

5

3

32

11

30

30

1530

1530

6

10

515

1530

5

3

32

11

1 and 18, 2 and 9, 3 and 6 1 and 45, 3 and 15, 5 and 9

1 and 63, 3 and 21, 7 and 91 and 36, 2 and 18, 3 and 12, 4 and 9,

6 and 6

Yes; Possible explanation: 4 × 14 = 56,

so they can plant 4 rows of 14 rose bushes.

Possible answer: Break apart 75 into

30 + 45. They can make 6 rows of 5

members (30 ÷ 6 = 5) and 9 rows of 5

members (45 ÷ 9 = 5).

Module 7 • Lesson 1 295

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Homework and PracticeUse the Homework and Practice pages to provide students with more practice on the concepts and skills of this lesson.

295-296 Module 7

texas essential knowledge and skills - district / homepage · reasoning—5.4.a mathematical...

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