Divisibility - Everyday Math - Login Lesson 1 5 37 Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 79–83, 267–269 Key Concepts and Skills • Use divisibility

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Lesson 1� 5 37

Advance Preparation

Teacher’s Reference Manual, Grades 4–6 pp. 79–83, 267–269

Key Concepts and Skills• Use divisibility rules to solve problems.

[Number and Numeration Goal 3]

• Explore the relationship between the

operations of multiplication and division.

[Operations and Computation Goal 2]

Key ActivitiesStudents use a calculator to test for

divisibility by a whole number. They learn

and practice divisibility rules.

Key Vocabularyfactor rainbow � divisible by � quotient �

divisibility rule

MaterialsMath Journal 1, pp. 13 and 14

Study Link 1� 4

calculator � overhead calculator (optional)

Playing Factor CaptorStudent Reference Book, p. 306

Math Masters, pp. 453 and 454

counters or centimeter cubes �

calculator

Students practice finding factors of

a number.

Math Boxes 1� 5Math Journal 1, p. 15

Students practice and maintain skills

through Math Box problems.

Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 4. [Number and Numeration Goal 1]

Study Link 1� 5Math Masters, p. 15

Students practice and maintain skills

through Study Link activities.

READINESS

Practicing Divisibility with Countersper partnership: 66 counters, 3 dice

Students use dice and counters to predict

divisibility relationships between 2 numbers.

EXTRA PRACTICE

Practicing Multiplication FactsMath Journal 1, p. 9

Math Masters, p. 11

Students use a multiplication facts routine.

ENRICHMENTExploring a Test for Divisibility by 4Math Masters, p. 16

Students use place-value concepts to

investigate a test for divisibility by 4.

ELL SUPPORT

Building a Math Word BankDifferentiation Handbook, p. 142

Students add the terms divisor, dividend,

quotient, and remainder to their Math

Word Banks.

Teaching the Lesson Ongoing Learning & Practice

132

4

Differentiation Options

DivisibilityObjectives To introduce divisibility rules for division by 2, 3,

5, 6, 9, and 10; and how to use a calculator to test for divisibility

by a whole number.

5

��

eToolkitePresentations Interactive Teacher’s

Lesson Guide

Algorithms Practice

EM FactsWorkshop Game™

AssessmentManagement

Family Letters

CurriculumFocal Points

Common Core State Standards

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38 Unit 1 Number Theory

Getting Started

Mental Math and Reflexes Pose basic and extended multiplication/division facts. Have students write the answers for each set of problems. At the end of each set, ask students to describe the patterns. Suggestions:

Math Message Solve Problems 1 and 2 at the top of journal page 13.

Study Link 1�4 Follow-Up Have partners compare answers. Ask the class how they know that all possible factors have been listed. Have volunteers model using a factor rainbow to pair factors for 25, 28, 42, and 100. If there is an odd number of factors, the middle factor is paired with itself. Explain that this only happens with square numbers.

8 12642 31 482416

1 Teaching the Lesson

▶ Math Message Follow-Up WHOLE-CLASSDISCUSSION

(Math Journal 1, p. 13)

Students share solution strategies. Use students’ responses to emphasize to the class that even numbers are numbers that are divisible by 2.

▶ Using a Calculator to Test

INDEPENDENT ACTIVITY

for Divisibility by a Whole Number(Math Journal 1, p. 13)

Recall for students the class discussion on the review of divisibility in Lesson 1-4. Remind students that a whole number (the dividend) is divisible by a whole number (the divisor) if the remainder in the division is zero. The result or quotient, must be a whole number. If the remainder is not zero, then the number being divided is not divisible by the second number.

If your students use calculators that display answers to division problems as a quotient and a whole number remainder, you might want to demonstrate the procedure. With the TI-15 calculator, this is done by pressing the Int÷ key instead of the ÷ key. With the Casio fx-55, use the key. For example, If you press 27 Int÷ 5

, or 27 5 , the display will show a quotient of 5 with a remainder of 2.

NOTE Some students may benefit from

doing the Readiness activity before you begin

Part 1 of each lesson. See the Readiness

activity in Part 3 for details.

NOTE Factor rainbows are introduced in

the Study Link Follow-Up. This tool helps

students identify all of the factors for a

given number. The rainbow is a visual

representation of the factor pairs and

provides a way to check if the factor list is

complete. Share the factor rainbow in the

Study Link Follow-Up with the class. Factor

rainbows will be used again in Lesson 1-6.

NOTE If possible, use an overhead calculator

to model the keystrokes and calculator

displays for lesson examples.

Interactive whiteboard-ready

ePresentations are available at

www.everydaymathonline.com to

help you teach the lesson.

5 ∗ 5 25

5 ∗ 50 250

5 ∗ 500 2,500

5 ∗ 5,000 25,000

6 ∗ 3 18

60 ∗ 3 180

600 ∗ 3 1,800

6,000 ∗ 3 18,000

8 ∗ 4 32

80 ∗ 40 3,200

800 ∗ 400 320,000

8,000 ∗ 4,000 32,000,000

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DivisibilityLESSON

1�5

Date Time

No

Yes

No

No

Yes

No

Yes

Yes

Example 2: Is 122 divisible by 5?To find out, divide 122 by 5.

122 / 5 � 24.4

The answer, 24.4, has a decimal

part. So 122 is not divisible by 5.

Example 1: Is 135 divisible by 5?To find out, divide 135 by 5.

135 / 5 � 27

The answer, 27, is a whole

number. So 135 is divisible by 5.

Math Message

1. Circle the numbers that are divisible by 2.

28 57 33 112 123,456 211 5,374 900 399 705

2. What do the numbers that you circled have in common?

Suppose you divide a whole number by a second whole number. The answer may be a

whole number, or it may be a number that has a decimal part. If the answer is a whole

number, we say that the first number is divisible by the second number. If the answer

has a decimal part, the first number is not divisible by the second number.

Use your calculator to help you answer these questions.

3. Is 267 divisible by 9? 4. Is 552 divisible by 6?

5. Is 809 divisible by 7? 6. Is 7,002 divisible by 3?

7. Is 4,735 divisible by 5? 8. Is 21,733 divisible by 4?

9. Is 5,268 divisible by 22? 10. Is 2,072 divisible by 37?

They are all even numbers.

Math Journal 1, p. 13

Student Page

Divisibility RulesLESSON

1�5

Date Time

For many numbers, even large ones, it is possible to test for divisibility without

actually dividing.

Here are the most useful divisibility rules:

� All numbers are divisible by 1.

� All even numbers (ending in 0, 2, 4, 6, or 8) are divisible by 2.

� A number is divisible by 3 if the sum of its digits is divisible by 3.

Example: 246 is divisible by 3 because 2 + 4 + 6 = 12, and 12 is divisible by 3.

� A number is divisible by 6 if it is divisible by both 2 and 3.

Example: 246 is divisible by 6 because it is divisible by 2 and by 3.


Example: 51,372 is divisible by 9 because 5 + 1 + 3 + 7 + 2 = 18, and

18 is divisible by 9.

� A number is divisible by 5 if it ends in 0 or 5.

� A number is divisible by 10 if it ends in 0.

1. Test each number below for divisibility. Then check on your calculator.

2. Find a 3-digit number that is divisible by both 3 and 5.

3. Find a 4-digit number that is divisible by both 6 and 9.

Sample answers: 1,800; 5,454

Sample answers: 735; 540

Divisible. . .

✓ ✓ ✓

✓ ✓ ✓

✓

✓ ✓ ✓ ✓ ✓ ✓

✓ ✓

Number by 2? by 3? by 6? by 9? by 5? by 10?

75 ✓ ✓

7,960

384

3,725

90

36,297


Student Page

Adjusting the ActivityWrite the number model from the first example on journal page 13 on

the board with each number appropriately labeled, including a remainder of zero.

dividend divisor quotient remainder

135 ÷ 5 = 27 R0

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

When testing for divisibility with a calculator that does not display remainders, the first number is not divisible by the second number if the quotient has a decimal part. Ask students to use their calculators to test whether 27 is divisible by 9. 27 is divisible by 9 because the result is 3—a whole number. Test whether 27 is divisible by 5. 27 is not divisible by 5 because the result is 5.4—not a whole number.

Allow 5 to 10 minutes for students to complete Problems 3–10 on the journal page 13.

▶ Introducing Divisibility Rules

WHOLE-CLASS ACTIVITY


Ask: How can you know that a number is divisible by 2 without actually doing the division? Numbers that end in 0, 2, 4, 6, or 8 are divisible by 2. Can you tell whether a number is divisible by 10 without dividing? Yes; numbers that end in 0 are divisible by 10. Can you tell whether a number is divisible by 3 without dividing? Allow students to explore this question before continuing. There are rules that let us test for divisibility without dividing or using a calculator.

1. Go over the divisibility-by-3 rule on journal page 14: A number is divisible by 3 if the sum of its digits is divisible by 3.

2. Illustrate by using the rule to test several examples.

● Is 237 divisible by 3? Yes. 2 + 3 + 7 = 12, and 12 is divisible by 3.

● Is 415 divisible by 3? No. 4 + 1 + 5 = 10, and 10 is not divisible by 3.

3. Ask students to provide examples of a number that is divisible by 3 and a number that is not. Encourage them to apply the divisibility-by-3 test first. Then have them check that it works by carrying out the division on their calculators.

Assign small groups to present examples for the remaining divisibility rules (5, 6, or 9).

Students complete Problems 1–3 independently. Have them check each other’s work.

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Lesson 1�5 39

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Math Boxes LESSON

1�5

Date Time

5. Complete.

a. 70 � 800 �

b. 400 � 5,000 �

c. 6,300 � � 90

d. 21,000 � 70 �

e. 720,000 � 800 � 900

300

70

2,000,000

56,0006. Pencils are packed 18 to a box. How many

pencils are in 9 boxes?

(unit)

162 pencils

4 30 31

19 2018

1. Circle the numbers that are divisible by 3.

221 381 474 922 726

11 4 249

2. Round 3,045,832 to the nearest…

a. million.

b. thousand.

c. ten-thousand. 3,050,000

3,046,000

3,000,000

4. Write an 8-digit numeral with

5 in the hundredths place,

8 in the tens place,

3 in the ones place,

8 in the thousands place,

4 in the hundreds place,

and 6 in all other places.

3. Complete the table.

6 6 8 4 8 3 6 5

Fraction Decimal Percent

�35

� 0.60 60%�14

� 0.25 25%

�12

� 0.50 50%�1

7

0� 0.70 70%

�1

8

0

5

0� 0.85 85%

80 90

.,

�


Student Page

STUDY LINK

1�5 Divisibility Rules

11

Name Date Time

� All even numbers are divisible by 2.


� A number is divisible by 6 if it is divisible by both 2 and 3.


� A number is divisible by 5 if it ends in 0 or 5.

� A number is divisible by 10 if it ends in 0.

1. Use divisibility rules to test whether each number is divisible by 2, 3, 5, 6, 9, or 10.

A number is divisible by 4 if the tens and ones digits form a number that is divisible by 4.

Example: 47,836 is divisible by 4 because 36 is divisible by 4.

It isn’t always easy to tell whether the last two digits form a number that is divisible by 4. Aquick way to check is to divide the number by 2 and then divide the result by 2. It’s the sameas dividing by 4, but is easier to do mentally.

Example: 5,384 is divisible by 4 because 84 / 2 � 42 and 42 / 2 � 21.

2. Place a star next to any number in the table that is divisible by 4.

3. 250 º 7 � 4. 1,931 � 4,763 � 2,059 �

5. (20 � 30) º 5 � 6. 78 � 6 � 132508,7531,750

NumberDivisible…

by 2? by 3? by 6? by 9? by 5? by 10?

998,876

5,890

36,540

33,015

1,098 ✓✓✓✓✓✓

✓✓✓✓✓✓✓✓✓

✓

Practice

�

�

Math Masters, p. 15

Study Link Master

40 Unit 1 Number Theory

2 Ongoing Learning & Practice

▶ Playing Factor Captor PARTNER ACTIVITY

(Student Reference Book, p. 306;

Math Masters, pp. 453–454)

Students practice finding factors of a number by playing Factor Captor. Students have the option of playing any of the two Factor Captor grids. If students are using Grid 2 for the first time, suggest that they omit the last two rows of the gameboard.

▶ Math Boxes 1�5

INDEPENDENT ACTIVITY


Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lessons 1-7 and 1-9. The skills in Problems 5 and 6 preview Unit 2 content.

Writing/Reasoning Have students write a response to the following: Explain how you solved Problem 6. Sample answer: Because there are 18 pencils per box and 9 boxes

total, I multiplied 18 ∗ 9: 10 ∗ 9 is 90 and 8 ∗ 9 is 72; 90 + 72 = 162. There are 162 pencils in all.

Ongoing Assessment: Math Boxes

Problem 4 �Recognizing Student Achievement

Use Math Boxes, Problem 4 to assess students’ understanding of place

value. Students are making adequate progress if they are able to correctly

position and identify digits and their values in whole numbers through the

hundred-thousands and decimals through the hundredths.

[Number and Numeration Goal 1]

▶ Study Link 1�5 INDEPENDENT

ACTIVITY

(Math Masters, p. 15)

Home Connection Students use divisibility rules to test whether numbers are divisible by 2, 3, 5, 6, 9, or 10. They learn the divisibility rule for 4 and recheck the numbers for those that are also divisible by 4.

NOTE As students continue to develop their

strategies for Factor Captor, they will find that

as more numbers are used, the scoring rules

increasingly reward a player for planning

ahead and anticipating an opponent’s moves.


LESSON

1�5

Name Date Time

Divisibility by 4

1. What number is shown by the base-10 blocks?

2. Which of the base-10 blocks could be divided evenly into 4 groups of cubes?

3. Is the number shown by the base-10 blocks divisible by 4?

4. Circle the numbers that you think are divisible by 4.

324 5,821 7,430 35,782,916

Use a calculator to check your answers.

5. Use what you know about base-10 blocks to explain why you only need to

look at the last two digits of a number to decide whether it is divisible by 4.

Sample answer: Because 1,000 and 100are divisible by 4, the numbers that the thousands place and the hundreds place represent are always divisible by 4. So you have to look at only the number formed by the tens and ones digits.

No

1,111

1,000 cubes 100 cubes 10 cubes 1 cube

The groups of 1,000 cubes and100 cubes

Math Masters, p. 16

Teaching Master

3 Differentiation Options

READINESS PARTNER ACTIVITY

▶ Practicing Divisibility 15–30 Min

with CountersTo explore the concept of divisibility using a concrete model, have students use counters to determine whether a number is divisible by the numbers 1–6.

Partners take turns rolling three dice. Make a two-digit number with two of the dice, and count out that number of counters. They predict whether the number of counters is divisible by the number on the third die. Then partners check the prediction by dividing the counters into the number of groups indicated on the third die.

EXTRA PRACTICE SMALL-GROUP

ACTIVITY

▶ Practicing Multiplication Facts 5–15 Min

(Math Journal 1, p. 9; Math Masters, p. 11)

To provide additional practice with basic multiplication facts, have students use the facts routine introduced in Lesson 1-3. See Teacher’s Lesson Guide, pages 28 and 29 to review the procedure.

ENRICHMENT PARTNER ACTIVITY

▶ Exploring a Test 5–15 Min

for Divisibility by 4(Math Masters, p. 16)

To further explore divisibility, have students use place-value concepts to investigate why only the last 2 digits in a number determine whether the number is divisible by 4.

ELL SUPPORT

SMALL-GROUP ACTIVITY

▶ Building a Math Word Bank 5–15 Min

(Differentiation Handbook, p. 142)

To provide language support for division, have students use the Word Bank Template found on Differentiation Handbook, page 142. Ask students to write the terms divisor, dividend, quotient, and remainder; draw a picture representing each term; and write other related words. See the Differentiation Handbook for more information.

Lesson 1�5 41


Copyright

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LESSON

1�3

Name Date Time

Multiplication Facts

11

B List 3 ∗ 3 = 9

3 ∗ 4 = 12

4 ∗ 3 = 12

3 ∗ 5 = 15

5 ∗ 3 = 15

4 ∗ 4 = 16

4 ∗ 5 = 20

5 ∗ 4 = 20

5 ∗ 5 = 25

5 ∗ 6 = 30

6 ∗ 5 = 30

5 ∗ 8 = 40

8 ∗ 5 = 40

6 ∗ 10 = 60

10 ∗ 6 = 60

7 ∗ 10 = 70

10 ∗ 7 = 70

8 ∗ 10 = 80

10 ∗ 8 = 80

9 ∗ 10 = 90

10 ∗ 9 = 90

10 ∗ 10 = 100

A List 3 ∗ 6 = 18

6 ∗ 3 = 18

3 ∗ 7 = 21

7 ∗ 3 = 21

3 ∗ 8 = 24

8 ∗ 3 = 24

3 ∗ 9 = 27

9 ∗ 3 = 27

4 ∗ 6 = 24

6 ∗ 4 = 24

4 ∗ 7 = 28

7 ∗ 4 = 28

4 ∗ 8 = 32

8 ∗ 4 = 32

4 ∗ 9 = 36

9 ∗ 4 = 36

5 ∗ 7 = 35

7 ∗ 5 = 35

5 ∗ 9 = 45

9 ∗ 5 = 45

6 ∗ 6 = 36

6 ∗ 7 = 42

7 ∗ 6 = 42

6 ∗ 8 = 48

8 ∗ 6 = 48

6 ∗ 9 = 54

9 ∗ 6 = 54

7 ∗ 7 = 49

7 ∗ 8 = 56

8 ∗ 7 = 56

7 ∗ 9 = 63

9 ∗ 7 = 63

8 ∗ 8 = 64

8 ∗ 9 = 72

9 ∗ 8 = 72

9 ∗ 9 = 81

Bonus Problems 11 ∗ 11 = 121

11 ∗ 12 = 132

5 ∗ 12 = 60

12 ∗ 6 = 72

7 ∗ 12 = 84

12 ∗ 8 = 80

9 ∗ 12 = 108

10 ∗ 12 = 120

5 ∗ 13 = 65

15 ∗ 7 = 105

12 ∗ 12 = 144

6 ∗ 14 = 84

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Documents

Divisibility - Everyday Math - Login Lesson 1 5 37 Advance Preparation Teacher’s Reference Manual, Grades 4–6 pp. 79–83, 267–269 Key Concepts and Skills • Use divisibility