chapter 13 lesson2-1 finding all the factors and odd numbers in this category. are all of the...

Lesson2-12-12-1

Chapter 13 • Lesson 2-1CC 72 Common Core Resource Guide

Chapter 13

Lesson Planner

STUDENT OBJECTIVESn• To find factors of a numbern• To use a diagram to find all the factors of a numbern• To recognize that a whole number is a multiple of any of its factorsn• To identify prime and composite numbers

(CCRG p. CC 73)

Open Ended Problem Solving / Headline Story Skills Practice and Review— Dividing in Half

(CCRG pp. CC 74–CC 76)

Categorizing Numbers (CCRG p. CC 74)

Finding All the Factors (CCRG p. CC 75)

Finding all the Factors in a Number (CCRG p. CC 76)

• CCRG: Activity Master, Number Groups

• LAB Masters, CCRG pp. CC 78–CC 79

(CCRG p. CC 77)

Leveled Problem Solving (CCRG p. CC 77)

Intervention Activity (CCRG p. CC 77)

Extension Activity (CCRG p. CC 77)

Practice Master, CCRG p. CC 80

Extension Master, CCRG p. CC 81

Lesson Notes

About the LessonStudents use their knowledge of multiplication facts to find factors of a number. They find and list pairs of factors of a number in an organized way and identify the number as prime, composite, or neither.

About the MathematicsIf two whole numbers a and b can be multiplied to get another whole number c, then a and b are called factors of c. For example, the factors of 24 are all the whole numbers that can be multiplied (by another whole number) to make 24. They are 1, 2, 3, 4, 6, 8, 12, and 24.

It is often easier to find factors in pairs because knowing one factor in a pair gives you a way to find the missing factor. The factor pairs for 24 are 1 and 24; 2 and 12; 3 and 8; 4 and 6.

Finding All the Factors

Lesson 13.2-1 has been added. Use this lesson after Lesson 13.2.

NCTM Standards 1, 2, 4, 6, 7, 8, 9, 10Common Core State Standards 4.OA 4

Common Core Resource Guide CC 73

Developing Mathematical Language Vocabulary: factor, factor pair, prime, composite, multiple

When two whole numbers are multiplied, each are factors of their product. For example 4 and 5 are factors of 20. A number that has only two factors, 1 and itself, is a prime number. A number that has more than two factors is a composite number. Every whole number is a multiple of each of its factors. For example 10 is a multiple of 1 (ten ones make 10), a multiple of 2 (five twos make 10), and a multiple of 5 (two fives make 10).

Familiarize students with the terms factor, prime, and composite.

Beginning Write factor diagrams on the board for several prime and composite numbers. Have students circle the diagrams that contain only two factors.

Intermediate On the board, list the numbers 2, 25, 24, 9, 7, 13, 50, 100. Ask students to identify which of the numbers are prime.

Advanced Ask students to name the prime numbers between 4 and 10; between 10 and 15; between 15 and 20.

Open-Ended Problem SolvingRead the Headline Story to the students. Encourage them to create problems that can be solved using information from the story.

Your school’s fourth grade

classrooms are sharing 48

new books equally. What

can you say about the

number of fourth grade

classrooms and the number

of books each class got?

Possible responses:

If there are 4 classes, each class could get 12 books. If there are 6 classes, each class could get 8 books. The more classrooms there are, the fewer books each classroom gets.

Skills Practice and ReviewDividing in Half

Ask students to find the number that is exactly half of the given number. Keep the pace quick so that all students have a chance to respond to at least one problem. Record the questions and the answers on the board. Some examples:

Half of 10 is 5

Half of 20 is 10

Half of 4 is 2

Half of 16 is 8

Half of 22 is 11

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CC 74 Common Core Resource Guide Chapter 13 • Lesson 2-1

Name Date Activity Master X

Number Groups

Activity Master

16 3 9 17 32 25

11 21 23 30 37 2

9 1625

30 32

3 1311 17

23 372

Number Group A

Group A Characteristics:

Number Group B

Group B Characteristics:

Decide which number group each of these numbers belongs to. What characteristics do the numbers in each number group has in common?

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CC 82 Common Core Resource Guide

More than 1 factor pair Only 1 factor pair

Characteristics may vary.

Odd and even numbers 2 is the only even number

Activity Master: Number Groups

Concept AlertSome students may have heard of or used the term factor prior to this lesson. If the term comes up while completing Activity Master: Number Groups let all students know that the multiplication facts they are writing are made up of factors of the number. The term will be addressed formally later in the lesson.

15MIN

whole class

Materials• For the teacher:

transparency of AM: Number Groups (optional)

• For each student: AM: Number Groups

Categorizing Numbers

Purpose To categorize numbers into two main categories: numbers that have only two factors, and numbers that have more than two factors

Introduce Distribute Activity Master: Number Groups to students. Read the directions to the class and ask for suggestions for how they could categorize these numbers. Students may suggest that numbers can be split up into even and odd, or numbers with one digit and numbers with two digits.

Encourage students to study each number one at a time and find different multiplication facts that will result in that number. Display each number on the board and write all multiplication facts under the number. For example:

16 1 3 16 2 3 8 4 3 4

In this case, 8 3 2 and 2 3 8 would be considered the same multiplication fact, such as with 3.

3 1 3 3

Task Help students with the first few numbers and then allow pairs or small groups to work on the rest. When students have had time to list multiplication facts for all of the numbers ask them to suggest how to fill in the boxes on Activity Master: Number Groups. One box should be filled in with all of the numbers that have only one multiplication fact (prime) and the other box should be filled in with the numbers that have more than one multiplication fact (composite).

Talk MathThese questions assume that you have placed composite numbers in Group A and prime numbers in Group B.

What do the numbers in Group A have in common? Possible answer: These numbers can be made with more than one multiplication fact.

What do the numbers in Group B have in common? Possible answer: These numbers can be made with only one multiplication fact.

Are all of the numbers in Group A even? Possible answer: No, there are even and odd numbers in this category.

Are all of the numbers in Group B even? Possible answer: No. There are even and odd numbers in this category.

What are some other numbers that could belong to Group A? Possible answer: 6, 10, 40 (any composite number)

What are some other numbers that could belong to Group B? Possible answer: 5, 7, 13 (any prime number)

As students name multiplication facts that result in each number you may wish to ask them for strategies for making sure they have found all possible multiplication facts for that number.

NCTM Standards 1, 2, 4, 6, 7, 8, 9, 10CCSS 4.OA 4


Chapter 13 • Lesson 2-1 Common Core Resource Guide CC 75

25MIN

whole class Finding All the Factors

Purpose To list all the factors of a number

Introduce Using the number 32 from Activity Master: Number Groups, point out that 32 is a multiple of each number used in the multiplication facts you wrote: 32 is a multiple of 1, 32 is a multiple of 2, 32 is a multiple of 4, and so on.

Introduce the term factor to your students. When two whole numbers are multiplied each are factors of their product. An organized way to find all of the factors of a number is to find factor pairs. Show your students this method for finding all possible factors using the number 32 from Activity Master: Number Groups. Write the pair that consists of 1 and the number itself, (in this case, 1 and 32) as shown below. Leave plenty of room if you think the number has lots of other factor pairs.

Ask students to decide if 2 is a factor yes, and if so, ask them to name the other member of the factor pair 16. Write this pair in the diagram and connect the factors as shown. Continue filling in the factor pairs, listing the factors in increasing order from left to right. Students will see that as one factor increases, the other factor in the pair decreases, and therefore they keep moving toward the center of the diagram. So, one of the next two factors will be greater than 2, while the other factor will be less than 32.

Task What are the factor pairs of 23? Ask students to name them and draw the factor diagram for 23 on the board.

Talk MathAsk these questions as the factors are named.

What factors of 23 should we name first? Explain? 1 and 23 Possible explanation: When we name the factor pairs, it is easy to begin with 1 and the number itself because 1 is a factor for all numbers and 1 is the smallest possible factor.

How did you make sure that there were no other factors? Possible answer: I tried other numbers, going up from 1 and none worked. I stopped when I got to 5 because I realized that 5 would have to be multiplied by a smaller number to get a number less than 23.

Practice Repeat this exercise with more numbers, including numbers with only 2 factors and numbers with more than 2 factor pairs. For example, you might have students find the factors of 2 1 and 2, 5 1 and 5, 8 1, 2, 4, and 8 21 1, 3, 7, and 21 and 30 1, 2, 3, 5, 6, 10, 15, and 30.

Share Point out to students that there is a special name for numbers that only have 1 factor pair (1 arc in the factor diagrams). They are called prime numbers. Numbers that have more than 1 factor pair (and therefore more than 1 arc in the factor diagrams) are called composite numbers. Ask students to use the factor diagram patterns to find all of the prime numbers that have been factored so far 23, 2, and 5, and all of the composite numbers that have been factored 32, 8, 21, and 30.

Extend If you have time, have students think of other numbers that are prime or composite, and draw the factor diagrams. Examples of other prime numbers include 3, 7, and 11. Examples of composite numbers include 4, 9, 10, and 14.

Possible Discussion A whole number will always be a multiple of its factors. Have students check this with one or more of the composite numbers used on Activity Master: Number Groups. For example, 21 is a multiple of 1, of 3, and of 7.

Concept Alert Students may ask if 1 is a prime number. A prime number is defined as any whole number greater than 1 that has only two whole number factors: 1 and itself.

321

321 2 164 8



Reflect and Summarize the Lesson

How can you find all the factors of a number?

Possible answer: You could begin by writing 1 on the left and the number itself on the right and connecting them, since 1 and the number itself are always factors. Then, you could try 2, 3, and so on as factors. If some of these numbers are factors, write them and connect them to their factor pair. Continue checking numbers to see if they are factors until they become as big as the numbers on the right of the list.

Chapter 13 • Lesson 2-1CC 76 Common Core Resource Guide

20MIN

individuals

Finding All the Factors in a Number

LAB Masters, CCRG pp. CC 78–CC 79

Purpose To list the factors for a number and to determine whether the number is prime, composite, or neither


Teaching Notes for LAB Master, CCRG page CC 78Students should be able to complete this page independently. In Problems 1 to 4, students are shown the number of factors they need to find for each pair of numbers. This will help them know when they have found all of the factors. The story problem in Problem 5 is related to this lesson’s Headline Story.

Ongoing AssessmentThis LAB page should give you a sense of whether students are finding the factors in an organized way. If one factor pair is given, do students begin by finding the other factor in the pair?

Teaching Notes for LAB Master, CCRG page CC 79Students may complete this page independently. Problems 6 to 11 focus on the use of factoring to decide if a number is prime, composite, or neither. Encourage students to find factors and use arcs to join factor pairs to be sure they have found all factors of a number.

Challenge ProblemThe challenge problem asks students to find two composite numbers that have no common factors other than 1. Such numbers are called relatively prime.

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Name Date

Finding All the FactorsNCTM Standards 1, 6, 7, 8, 9, 10 Common Core State Standards 2.NBT 1, 2, 3, 6, 7, 8, 9

Lesson2-12-1Chapter 13

12

12

1 2

24

31 24

41

Write all the factors of each number in the diagram. Connect the factor pairs.

Mrs.Levinhas30bookstodonatetoclassroomsatschool.Howmanybookswilleachclassroomgetifthereare

2classrooms? 5classrooms? 15classrooms?


Explainapatternyouseeinthenumberofclassroomsandthenumberofbooks.

1

10

102

15

10

6

3

2

5

Possible answer: The number of classrooms and number books form factor pairs.

563 4

84 6 122411

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Challenge List two composite numbers that do not share any factors other than 1. Explain your answer.

List the factors and draw lines to connect factor pairs. Write P for prime, C for composite, or N for neither.

Number Factors P, C, or N

16

17

25

1

19

42

Are all even numbers composite numbers? Explain your answer.

No, 2 is a prime number. The only factors are 1 and 2.

C

P

C

N

P

C

Possible answer: 25 and 24 Possible explanation: The factors of 25 are 1, 5, and 25. The

factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The only factor that they share is 1.

84 161 2

1 17

2551

1

1 19

146 7 212 3 421

Lesson Activity Book Master, CCRG p. CC 78 Lesson Activity Book Master, CCRG p. CC 79


Chapter 13 • Lesson 2-1 Common Core Resource Guide CC 77

Leveled Problem Solving

Javier is playing a number puzzle problem. He gives clues to find his number.

Basic LevelJavier’s number is a prime number less than 20. The sum of its digits is 4. 13

On LevelJavier’s number is a composite number. It is an odd number less than 25 and has 3 factors. 9

Above LevelJavier’s number is a composite number. It is one more than a prime number and one less than a prime number. It has 6 factors. Possible Answers: 12, 18

Intervention Activity Extension Activity

Number Line Jump

Draw a number line from 0 to 20 on the board. Ask these questions.

• What size jumps allow you to land at both 10 and 20? 1, 2, 5, 10

• What size jumps allow you to land at even and odd numbers? 1, 3, 5, 7, 9

• Complete this statement: If you make jumps of 3 spaces and a friend makes jumps of spaces, you will both land on 15 but not 18. 5 or 15

What’s My Number?

Have students create number puzzles for a partner. Puzzles should include at least three clues. For example:

1. This number is prime.

2. This number is less than 20.

3. 21 is a multiple of this number. 7 or 3

Practice Master, CCRG p. CC 80 Extension Master, CCRG p. CC 81ExtensionLesson 2-1

Prime Numbers Less Than 100 �Put�a�box�around�1�because�it�is�neither�prime��

nor�composite.

Circle�2�and�cross�out�the�other�multiples�of�2.


�� Continue�in�the�same�way�with�each�remaining��number�(in�order)�that�has�not�been�crossed�out.

The circled numbers are prime. The crossed-out numbers are composite.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Name� � Date�

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Finding All the FactorsList the factors and draw lines to connect factor pairs. Write P for prime, C for composite, or N for neither.

Name Date PracticeLesson 2-1

Test Prep


50

22

1

5

45

Which group contains all of the factors of 18?

A. 1 18 B. 1 2 6 9 18 C. 1 2 3 6 9 18 D. 1 3 6 9 18

Which pair of numbers has no factors in common except 1.

A. 12, 50 B. 15, 28 C. 25, 10 D. 15, 12

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C

C

N

P

C

255 10 501 2

222 111

1

1 5

455 91


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Name Date

Finding All the FactorsNCTM Standards 1, 6, 7, 8, 9, 10 Common Core State Standards 2.NBT 1, 2, 3, 6, 7, 8, 9

Lesson2-12-1Chapter 13

12

12

1 2

24

31 24

41

Write all the factors of each number in the diagram. Connect the factor pairs.

Mrs.Levinhas30bookstodonatetoclassroomsatschool.Howmanybookswilleachclassroomgetifthereare



Explainapatternyouseeinthenumberofclassroomsandthenumberofbooks.

1

10

102


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Challenge List two composite numbers that do not share any factors other than 1. Explain your answer.

List the factors and draw lines to connect factor pairs. Write P for prime, C for composite, or N for neither.


16

17

25

1

19

42

Are all even numbers composite numbers? Explain your answer.


Finding All the FactorsList the factors and draw lines to connect factor pairs. Write P for prime, C for composite, or N for neither.

Name Date PracticeLesson 2-1

Test Prep


50

22

1

5

45

Which group contains all of the factors of 18?

A. 1 18 B. 1 2 6 9 18 C. 1 2 3 6 9 18 D. 1 3 6 9 18

Which pair of numbers has no factors in common except 1.

A. 12, 50 B. 15, 28 C. 25, 10 D. 15, 12

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ExtensionLesson 2-1

Prime Numbers Less Than 100 �Put�a�box�around�1�because�it�is�neither�prime��

nor�composite.



�� Continue�in�the�same�way�with�each�remaining��number�(in�order)�that�has�not�been�crossed�out.

The circled numbers are prime. The crossed-out numbers are composite.

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

Name� � Date�©

�Sch

ool�S

peci

alty


TMG4CCRG_CH13_L2-1_Extension_p81.indd 81 12/28/10 1:00:25 PM

Name Date Activity Master X

Number Groups

Activity Master

16 3 9 17 32 25

11 21 23 30 37 2

9 1625

30 32

3 1311 17

23 372

Number Group A

Group A Characteristics:

Number Group B

Group B Characteristics:

Decide which number group each of these numbers belongs to. What characteristics do the numbers in each number group has in common?

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TMG4CCRG_CH13_L2-1_TN_p83.indd 83 12/28/10 2:04:46 PM

chapter 13 lesson2-1 finding all the factors and odd numbers in this category. are all of the...

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