1 Teaching the Lesson materials - Everyday Math - Login

236 Unit 4 Division

Teaching the Lesson materials

Key ActivitiesStudents review and practice the use of a friendly number paper-and-pencil division algorithmstrategy. They play Division Dash to practice mental division with 1-digit divisors.

Key Concepts and Skills• Use the partial-quotients algorithm for problems.

[Operations and Computation Goal 3]• Apply friendly numbers to identify partial quotients.

[Operations and Computation Goal 3]• Factor numbers to identify partial quotients.

[Operations and Computation Goal 3]

Key Vocabularydividend • divisor • partial quotient • quotient • remainder

Ongoing Assessment: Recognizing Student Achievement Use journal page 101.[Operations and Computation Goal 3]

Ongoing Assessment: Informing Instruction See page 240.

Ongoing Learning & Practice materials

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

Differentiation Options materials

Students review divisibilityrules for 1-digit divisors.

Students find numbers tomeet divisibility criteria.

Students review vocabularyfor the parts of a divisionproblem.

� Student Reference Book, p. 11� Teaching Master

(Math Masters, p. 105)� Class Data Pad

See Advance Preparation

ELL SUPPORTENRICHMENTREADINESS

3

� Math Journal 1, p. 102� Study Link Master

(Math Masters, p. 104)

2

� Math Journal 1, p. 101� Student Reference Book,

pp. 22, 23, and 303� Study Link 4�1� Teaching Aid Master

(Math Masters, p. 415)� Class Data Pad� slates� Per partnership: 4 each of the

number cards 1–9, (from theEverything Math Deck, if available)


1

Additional InformationAdvance Preparation For Part 1, you will need 2 copies of the computation grid (Math Masters, page 415) for each student.

Objective To review the partial-quotients division algorithm with whole numbers.

Technology Assessment Management SystemJournal page 101See the iTLG.

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� Math Message Follow-UpAsk volunteers to share their solution strategies. Expect that somestudents will suggest breaking 127 into friendly numbers.

Survey the class for clues that the Math Message was a divisionproblem. The problem gave the whole (127 days) and asked howmany groups (weeks); because there are 7 days in a week, theproblem was to figure out how many 7s are in 127. Ask volunteersto write a number model for this problem. 127 / 7; �

127

7�; 127 � 7;

and 7��1�2�7�

� Reviewing the Partial-Quotients Algorithm(Math Masters, p. 415)

Given a dividend and a divisor, the partial-quotients algorithmis one pencil-and-paper strategy for division. Model the followingsteps on the Class Data Pad:

1. Write the problem in traditional form: 7��1�2�7�.

2. Draw a vertical line to the right of the problem to separate thesubtraction part of the algorithm from the partial quotients.

7��1�2�7�

WHOLE-CLASS ACTIVITY

WHOLE-CLASSDISCUSSION

1 Teaching the Lesson

Lesson 4�2 237

Getting Started

Math MessageAmy is 127 days older than Bob. How many weeks is that?

Study Link 4�1 Follow-Up Have partners compare answers. Explain that fact family relationships canbe used to check computations. Write 605 � 67 � 528 on the board or atransparency. An addition problem from this fact family will check the subtraction. Write528 � 67 � 605. Ask: Are there any problems with this approach? Most students willrecognize either the subtraction or the addition error. It is important to calculate the checkproblem, not just rewrite the numbers. 528 � 67 � 595, not 605, so the subtraction was incorrect in the initial number sentence. Change the equal sign to not equal, andthen write 605 � 67 � 538. Encourage students to use number relationships to checktheir calculations.

Mental Math andReflexes Pose multiplication and division problemslike the following.

How many 5s are in 45? 9What number times 9 equals 27? 3What is 3 times 120? 360How many 4s are in 32? 8What number times 8 equals 40? 5Multiply 5 times 80. 400What number times 7 equals 35? 5Multiply 12 by 7. 84Multiply 55 by 3. 165

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238 Unit 4 Division

Links to the Future

NOTE When the result of division isexpressed as a quotient and a nonzeroremainder, Everyday Mathematics uses anarrow rather than an equal sign, as in 246 � 12 → 20 R6. Everyday Mathematicsprefers this notation because 246 � 12 � 20R6 is not a proper number sentence. Thearrow is read as is, yields, or results in.Model this expression for students in yourexamples of the partial-quotients algorithm.Label the arrow on the Class Data Pad fordisplay throughout this unit.

Explain that with this notation, students will list their partialquotients on the right of the vertical line and then subtract therelated multiples on the left of the vertical line, until theremaining dividend is smaller than the divisor.

Students will practice the partial-quotients algorithm in Lesson 4-4, using aneasy-multiples strategy to find partial quotients, and in Lesson 4-5 with decimaldividends.

One strategy for finding partial quotients is to use friendlynumbers. Rename the dividend as an expression that containsmultiples of the divisor. Make a name-collection box for 127, andadd the expression 70 � 57. Use this expression to model the algorithm.

3. Ask: How many 7s are in 70? 10, because 10 � 7 � 70. Write70 under 127 and 10 next to it, to the right of the vertical line.Subtract, saying: 127 minus 70 equals 57. Explain that 10 isthe first partial quotient and 57 is what remains to be divided.

7��1�2�7� 127 � 70 � 57– 70 10

57 57 is left to divide.

4. Ask: How many 7s are in 57? 8, because 8 * 7 � 56. Write 56under 57 and 8 next to it, to the right of the vertical line.Subtract, saying: 57 minus 56 equals 1. Explain that 8 is thesecond partial quotient, and 1 is what remains to be divided.

7��1�2�7�– 70 10

57– 56 8


5. Explain that they can stop this process when the number leftto be divided is smaller than the divisor. This number can be written in the quotient as a whole-number remainder.

6. Combine the partial quotients, saying: 10 � 8 equals 18. Write18 above the dividend. Circle the 1 and write R1 next to 18.There are 18 [7s] in 127, with a remainder of 1. So Amy ishow many weeks older than Bob? About 18 weeks older, or 18weeks and 1 day older

18 R17��1�2�7�

– 70 1057

– 56 810 18

127 � 7 → 18 R1

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Links to the Future

Adjusting the ActivityELL

The arrow as a mathematical symbol is used to represent several different concepts. To support English language learners, remind students of therelationship between multiplication and division. Explain that when a quotient iswritten to show a whole number remainder, the remainder is not part of the multiplication expression for that fact family. So we need a different way, thearrow, to show that the division results in the quotient and the whole-numberremainder.

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

Ask students for other ways to rename 127 using multiples of 7.Add these to the name-collection box. Sample answers: 105 � 22;70 � 49 � 7 � 1; 35 � 35 � 35 � 21 � 1 Have students chooseone of these expressions to use with the partial-quotients algorithm. Remind them to write the problem and draw thevertical line, using the problem on the Class Data Pad as a model.To help students remember place value as they write digits, havethem use a computation grid. Circulate and assist.

� Using the Partial-Quotients Algorithm(Math Journal 1, p. 101; Student Reference Book, pp. 22 and 23)

Remind students that pages 22 and 23 in the Student ReferenceBook and the samples on the Class Data Pad can be used to verifycorrect usage of the steps in this algorithm. Have studentscomplete the page. Circulate and assist.

Problem 5 on journal page 101 will provide some information about students’ability to interpret remainders. Interpreting remainders will be covered in Lesson 4-6.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 101 to assess students’ understanding of the partial-quotientsalgorithm. Students are making adequate progress if they demonstrate accurateuse of the notation for the algorithm.


Journal Page 101 �

INDEPENDENTACTIVITY

Lesson 4�2 239

101

The Partial-Quotients Division AlgorithmLESSON

4 �2

Date Time

Use the partial-quotients algorithm to solve these problems.

1. 6�4�9�5�

2. 832 � 15 ∑

3. 3,518 / 32 ∑

4. �5,

53460� ∑

5. Jerry was sorting 389 marbles into bags. He put a dozen in each bag. How many bags does he need? 33 bags

99 R14109 R3055 R7

82 R3 �

Math Journal 1, p. 101

Student Page

Division Algorithms

Different symbols may be used to indicate division. For example,“94 divided by 6” may be written as 94 � 6, 6�9�4� , 94 / 6, or �

964�.

♦ The number that is being divided is called the dividend.

♦ The number that divides the dividend is called the divisor.

♦ The answer to a division problem is called the quotient.

♦ Some numbers cannot be divided evenly. When this happens,the answer includes a quotient and a remainder.

Partial-Quotients MethodIn the partial-quotients method, it takes several steps to findthe quotient. At each step, you find a partial answer (called apartial quotient). These partial answers are then added tofind the quotient.

Study the example below. To find the number of 6s in 1,010 firstfind partial quotients and then add them. Record the partialquotients in a column to the right of the original problem.

Whole Numbers

Four ways to show“123 divided by 4”

123 � 4 123 / 4

4�1�2�3� �1243

�

123 is the dividend.4 is the divisor.

ExampleExample 1,010 / 6 � ?

6�1�,0�1�0� ↓� 600 100

410� 300 50

110� 60 10

50� 48 8

2 168

Write partial quotients in this column.Think: How many [6s] are in 1,010? At least 100.

The first partial quotient is 100. 100 ∗ 6 � 600

Subtract 600 from 1,010. At least 50 [6s] are left in 410.

The second partial quotient is 50. 50 ∗ 6 � 300

Subtract. At least 10 [6s] are left in 110.

The third partial quotient is 10. 10 ∗ 6 � 60


The fourth partial quotient is 8. 8 ∗ 6 � 48

Subtract. Add the partial quotients.

↑ ↑Remainder Quotient

168 R2The answer is 168 R2. Record the answer as 6�1�,0�1�0�or write 1,010 / 6 → 168 R2.

Student Reference Book, p. 22

Student Page


240 Unit 4 Division

102

Date Time

Math Boxes LESSON

4 �2

6.

Circle the best answer.

A. In 1900, more than half of the communities were rural.

B. In 1900, 6 out of 10 communities inthe United States were rural.

C. In 1900, more than �34� of communities

in the United States were rural.

1. Write � or �.

a. 0.45 �34�

b. 0.89 �180�

c. �45� 0.54

d. �13� 0.35

e. �78� 0.9�

�

��

�

9 8389

2. Sasha earns $4.50 per day on her paperroute. She delivers papers every day. Howmuch does she earn in two weeks?

Open sentence:

Solution:

Answer: $63d � 634.50 � 7 � 2 � d

38–40 243

12 207

34–36 125

3. Write the prime factorization of 80. 4. Without using a protractor, find themeasurement of the missing angle.

90°

50° 40°

5. Solve.

a. 209.0 b. $30.49� 73.5 � $8.51

c. 4.339 d. 25.03� 6.671 � 14.58

39.6111.01

$21.98135.5

2 � 2 � 2 � 2 � 5,11or 24 � 5

40%Urban 60%

RuralThe United States

in 1900


Student Page

Division Dash

Materials � number cards 1–9 (4 of each)� 1 score sheet

Players 1 or 2Skill Division of 2-digit by 1-digit numbers Object of the game To reach 100 in the fewest divisions possible.Directions1. Prepare a score sheet like the one shown at the right.2. Shuffle the cards and place the deck number-side

down on the table. 3. Each player follows the instructions below:

♦ Turn over 3 cards and lay them down in a row, from left to right. Use the 3 cards to generate a divisionproblem. The 2 cards on the left form a 2-digit number.This is the dividend. The number on the card at the right is the divisor.

♦ Divide the 2-digit number by the 1-digit number andrecord the result. This result is your quotient. Remaindersare ignored. Calculate mentally or on paper.

♦ Add your quotient to your previous score and record yournew score. (If this is your first turn, your previous scorewas 0.)

4. Players repeat Step 3 until one player’s score is 100 ormore. The first player to reach at least 100 wins. If there isonly one player, the Object of the game is to reach 100 in asfew turns as possible.

Games

ExampleExample Turn 1: Bob draws 6, 4, and 5. He divides64 by 5. Quotient � 12. Remainder is ignored.The score is 12 � 0 � 12.

Turn 2: Bob then draws 8, 2, and 1. Hedivides 82 by 1. Quotient � 82. The score is 82 � 12 � 94.

Turn 3: Bob then draws 5, 7, and 8. He divides 57 by 8. Quotient � 7. Remainder is ignored. The score is 7 � 94 � 101.

Bob has reached 100 in 3 turns and the game ends.

Player 1

Quotient Score

Player 2

Quotient Score

5

5

6

6

4

4

64 is the dividend. 5 is the divisor.

Quotient Score

12 12

82 94

7 101


Student Page

� Introducing Division Dash(Student Reference Book, p. 303)

Division Dash uses randomly generated numbers to obtain valuesfor 1-digit divisors and 2-digit dividends. Encourage students tocalculate mentally, but do not restrict paper-and-pencil use.

Discuss the example on the Student Reference Book page. Thenhave the class play a round of Division Dash together. The wholeclass mentally calculates the division. Remind students that onlythe whole-number part of the quotient is recorded. If the dividendis less than the divisor, the quotient should be recorded as 0.

After students understand the rules, have partners play the game.Circulate and assist.

Ongoing Assessment: Informing InstructionWatch for students who use paper-and-pencil, rather than mental strategies, tocalculate the division. To help them bridge into mental math, ask them to writethe division expression 4��4�9� but then use multiplication facts and friendly parts to calculate mentally.

� Math Boxes 4�2(Math Journal 1, p. 102)

Mixed Review Math Boxes in this lesson are paired withMath Boxes in Lesson 4-4 and 4-6. The skills in Problems5 and 6 preview Unit 5 content.

Writing/Reasoning Have students write a response to the following: Explain why your answer to Problem 4 iscorrect. Sample answer: The sum of the measures of the

angles equals 180°. My answer is correct because 50 � 90 � 140,and the missing angle measure is 40 because 50 � 90 � 40 � 180.

� Study Link 4�2(Math Masters, p. 104)

Home Connection Students practice the partial-quotientsdivision algorithm.

INDEPENDENTACTIVITY

INDEPENDENTACTIVITY

2 Ongoing Learning & Practice



� Reviewing Divisibility Rules(Student Reference Book, p. 11)

To provide experience with identifying factors, have partners readabout divisibility on page 11 of the Student Reference Book andcomplete the Check Your Understanding problems.

� Exploring Divisibility by the Digits(Math Masters, p. 105)

To apply students’ understanding of factors, have themexplore divisibility from another perspective. Studentsexamine 3-digit numbers that meet certain divisibility

criteria. Then they use the same criteria to identify larger numbers.

� Supporting Math Vocabulary DevelopmentTo provide language support for division, have volunteers write adivision number model on chart paper in several different formats.

127 / 7 → 18 R1�12

77

� → 18 R1

127 � 7 → 18 R1

18 R17��1�2�7�

For each number model, label andunderline the dividend in red (thenumber being divided); label andunderline the divisor in blue (thenumber the dividend is beingdivided by); label and circle thequotient in a third color; label andcircle the remainder in a fourthcolor. Emphasize that both thequotient and the remainder arepart of the answer. Display thischart throughout all the divisionlessons.

15–30 Min

SMALL-GROUP ACTIVITY

ELL SUPPORT

5–15 Min

PARTNER ACTIVITYENRICHMENT

5–15 Min

PARTNER ACTIVITYREADINESS

3 Differentiation Options STUDY LINK

4�2 Division

Name Date Time

Here is the partial-quotients algorithm using a friendly numbers strategy.

7�2�3�7� Rename dividend (use multiples of the divisor):237 � 210 � 21 � 6

How many 7s are in 210? 3030 The first partial quotient. 30 � 7 � 210

Subtract. 27 is left to divide.

How many 7s are in 27? 3The second partial quotient. 3 � 7 � 21Subtract. 6 is left to divide.

Add the partial quotients: 30 � 3 � 33

1. Another way to rename 237 with multiples of 7 is 237 � 70 � 70 � 70 � 21 � 6

If the example had used this name for 237, what would the partial quotients have been?

2. 6�1�6�6� 3. 214 / 5

Answer: Answer:

4. 485 � 15 5. 17�4�0�8�

Answer: Answer: 2432 R5

42 R427 R4

10, 10, 10, and 3

�21027

�21

6

3

33

Practice

6. 3,817 � 168 �

Check: � �

7. 52,517 � 281 �

Check: � � 52,51752,236; or 281281; or 52,23652,236

3,817; or 168168; or 3,8173,9853,985

Remainder Quotient Answer: 33 R6

→ →

22 23

Math Masters, p. 104

Study Link Master

LESSON

4�2

Name Date Time

Divisibility by the Digits

Ms. Winters asked Vito and Jacob to make answer cards for a division puzzle. They had to find numbers that met all of the following characteristics.

1. Jacob knew that with divisibility rules, it should be easy. The boys started with 3-digitnumbers and found 123 and 242. Latoya checked their work. What should she tell them?123 is correct because 1 is divisible by 1; 12is correct because it is an even number; and 123 is correct because 1 � 2 � 3 � 6, which isdivisible by 3.242 is not correct because 2 � 4 � 2 � 8, which is not divisible by 3.

Example:

◆ The first digit is divisible by 1. 1

◆ The first two digits are divisible by 2. 12

◆ The first three digits are divisible by 3. 120

◆ The first four digits are divisible by 4. 1,204

◆ The first five digits are divisible by 5. 12,040

◆ The first six digits are divisible by 6. 120,402

◆ The first seven digits are divisible by 7. 1,204,021

◆ The first eight digits are divisible by 8. 12,049,216

◆ The first nine digits are divisible by 9. 120,402,162

Puzzle Numbers

4-digit 5-digit 6-digit 7-digit 8-digit 9-digit

1,472 14,725 147,252 1,472,527 14,725,272 147,252,7261,624 16,240 162,408 1,624,084 16,240,840 162,408,402

2. Use the characteristics listed above to find as many puzzle numbers as you can. Record them in the boxes below.Sample answers:


Teaching Master

Lesson 4�2 241

59 � 7 � 8 R3

598 R3

7

59 / 7 � 8 R3

A Division Problem

Dividend

Quotient

Remainder

Divisor


248 Unit 4 Division

Teaching the Lesson materials

Key ActivitiesStudents play Divisibility Dash to practice recognizing multiples and using divisibility rules.They practice finding partial quotients by using easy multiples of the divisor.

Key Concepts and Skills• Apply division facts and extended facts to identify partial quotients.

[Operations and Computation Goal 2]• Use divisibility rules to identify multiples.

[Operations and Computation Goal 3]• Use the partial-quotients algorithm for problems.


Key Vocabularymultiple • divisor • partial quotient • dividend

Ongoing Assessment: Informing Instruction See page 251.

Ongoing Assessment: Recognizing Student Achievement Use journal page 107.[Operations and Computation Goal 3]

Ongoing Learning & Practice materialsStudents practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students practice finding friendly numbersusing expanded notation and multiples.

Students use lists of multiples of the divisorto solve division problems.

� Teaching Masters (Math Masters, pp. 111 and 112)

� Per partnership: 4 each of numbercards 1–9 (from the Everything MathDeck, if available)


EXTRA PRACTICEREADINESS

3

� Math Journal 1, p. 108� Study Link Master

(Math Masters, p. 110)

2

� Math Journal 1, pp. 106 and 107� Student Reference Book,

pp. 22, 23, and 302� Study Link 4�3� Teaching Master

(Math Masters, p. 109; optional)� Teaching Aid Master

(Math Masters, p. 415)� Class Data Pad� calculator� Per partnership: 4 each of number

cards 0–9; 2 each of number cards2, 3, 5, 6, 9 and 10 (from theEverything Math Deck, if available)


1

Objective To provide practice with strategies for the partial-quotients algorithm.

Technology Assessment Management SystemJournal page 107See the iTLG.

Additional InformationAdvance Preparation For Part 1, you will need a coin for the calculator practice in the MentalMath and Reflexes and 2 copies of the computation grid (Math Masters, page 415) for eachstudent.

For Part 3, prepare Math Masters, page 111 to provide individualized practice as needed.

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� Math Message Follow-UpSurvey the class for their 3-digit numbers. Write studentresponses on the Class Data Pad. Ask students how they mightcheck these numbers without actually dividing by 6. Moststudents will refer to the divisibility rule for 6: A number is divisible by 6 if it is divisible by 2 and 3. Check the numbers as aclass, and discuss students’ strategies for finding their numbers.

� Introducing Divisibility Dash(Student Reference Book, p. 302)

Playing Divisibility Dash provides students with practice recognizing multiples and using divisibility rules in a contextthat also develops speed. The variation is for 3-digit numbers.Discuss the variation example on Student Reference Book, page302, and demonstrate a turn by playing one hand as a class. Thenallow partners time to play at least 3 rounds of Divisibility Dash.

� Reviewing the Partial-Quotients Algorithm(Math Masters, p. 415)

Explain that another approach to finding partial quotients is touse a series of at least...not more than multiples of the divisor. Agood strategy is to start with easy numbers, such as 100 times thedivisor or 10 times the divisor.



WHOLE-CLASSDISCUSSION

1 Teaching the Lesson

Lesson 4�4 249

Getting Started

Math MessageWrite a 3-digit number thatis divisible by 6.

Mental Math and Reflexes For calculator practice, write each problem on the board or a transparency. Use a coin toss to determine whether students express the answer with a whole-number remainder or a fraction remainder.

Ask volunteers to explain the meaning of the fraction remainder. The divisor representshow many are needed in a group or how many groups. The divisor is the denominator. The remainder is the numerator; how many you have. The fraction represents division—the remainder, �

15�, is one divided by 5.

5��3�6� 7 R1; 7�15�

11 � 4 2 R3; 2�34�

34 / 8 4 R2; 4�28� or 4�

14�

6��7�5� 12 R3; 12�36� or 12�

12�

78 � 8 9 R6; 9�68� or 9�

34�

99 / 8 12 R3; 12�38�

11��1�0�2� 9 R3; 9�39� or 9�

13�

25��2�3�0� 9 R5; 9�255� or 9�

15�

680 / 50 13 R30;

13�3500� or 13�

35�

Study Link 4�3 Follow-Up Allow students five minutes to comparetheir answers and resolve any differences. Circulate and assist.

ExampleExample Andrew’s cards: Divisor card:

Andrew uses his cards to make 2 numbers that are multiples of 3:

He discards these 4 cards and holds the 2 and 8 for the next round of play.

Divisibility Dash

Materials � number cards 0–9 (4 of each)� number cards: 2, 3, 5, 6, 9, and 10 (2 of each)

Players 2 or 3 Skill Recognizing multiples, using divisibility testsObject of the game To discard all cards.

Directions1. Shuffle the divisor cards and place them number-side down

on the table. Shuffle the draw cards and deal 8 to eachplayer. Place the remaining draw cards number-side downnext to the divisor cards.

2. For each round, turn the top divisor card number-side up.Players take turns. When it is your turn: ♦ Use the cards in your hand to make 2-digit numbers that

are multiples of the divisor card. Make as many 2-digitnumbers that are multiples as you can. A card used tomake one 2-digit number may not be used again to makeanother number.

♦ Place all the cards you used to make 2-digit numbers in adiscard pile.

♦ If you cannot make a 2-digit number that is a multiple ofthe divisor card, you must take a card from the draw pile.Your turn is over.

3. If a player disagrees that a 2-digit number is a multiple ofthe divisor card, that player may challenge. Players use thedivisibility test for the divisor card value to check thenumber in question. Any numbers that are not multiples ofthe divisor card must be returned to the player’s hand.

4. If the draw pile or divisor cards have all been used, they canbe reshuffled and put back into play.

5. The first player to discard all of his or her cards is the winner.

Games

1

1

2

2

5

5

5

5

7

7

8

8

1

1

5

5

5

5

7

7

3

3

Note

The number cards 0–9 (4 of each) are the draw cards. This set of draw cards is alsocalled the draw pile.

The number cards 2, 3, 5, 6, 9, and 10 (2 of each) are thedivisor cards.


Student Page


250 Unit 4 Division

LESSON

4�4

Name Date Time

Easy Multiples

1,000 º � 1,000 º �

100 º � 100 º �

50 º � 50 º �

20 º � 20 º �

10 º � 10 º �

5 º � 5 º �

1,000 º � 1,000 º �

100 º � 100 º �

50 º � 50 º �

20 º � 20 º �

10 º � 10 º �

5 º � 5 º �

1,000 º � 1,000 º �

100 º � 100 º �

50 º � 50 º �

20 º � 20 º �

10 º � 10 º �

5 º � 5 º �


Teaching Master

1. Write the problem 6��1�,0�1�0�, drawing a vertical line to the rightof the problem.● Are there at least 100 [6s] in 1,010? Yes, because 100 � 6 �

600, which is less than 1,010. Are there at least 200 [6s] in1,010? No, because 200 � 6 � 1,200, which is more than1,010.

● So there are at least 100 [6s] but not more than 200 [6s].Try 100.

Write 600 under 1,010. Write 100 to the right. 100 is thefirst partial quotient.

6��1�,0�1�0�– 600 100 The first partial quotient,

100 � 6 � 600.

2. Next find out how much is left to be divided. Subtract 600from 1,010.


100 � 6 � 600.410 410 is left to divide.

3. Now find the number of 6s in 410. There are several ways todo this:

� Use a fact family and extended facts. 6 � 6 � 36; 60 � 6 � 360, so there are at least 60 [6s] in 410.


100 � 6 � 600.410 410 is left to divide.

– 360 60 The second partial quotient, 60 � 6 � 360.

50 50 is left to divide.– 48 8 The third partial quotient,

8 � 6 � 48.2 2 is left to divide.

� Or continue to use at least...not more than multiples witheasy numbers. For example, ask: Are there at least 100 [6s]in 410? No, because 100 � 6 � 600. Are there at least 50[6s]? Yes, because 50 � 6 � 300.


100 � 6 � 600.410 410 is left to divide.



Division Algorithms

Different symbols may be used to indicate division. For example,“94 divided by 6” may be written as 94 � 6, 6�9�4� , 94 / 6, or �

964�.

♦ The number that is being divided is called the dividend.

♦ The number that divides the dividend is called the divisor.

♦ The answer to a division problem is called the quotient.

♦ Some numbers cannot be divided evenly. When this happens,the answer includes a quotient and a remainder.

Partial-Quotients MethodIn the partial-quotients method, it takes several steps to findthe quotient. At each step, you find a partial answer (called apartial quotient). These partial answers are then added tofind the quotient.

Study the example below. To find the number of 6s in 1,010 firstfind partial quotients and then add them. Record the partialquotients in a column to the right of the original problem.

Whole Numbers

Four ways to show“123 divided by 4”

123 � 4 123 / 4

4�1�2�3� �1243

�

123 is the dividend.4 is the divisor.

ExampleExample 1,010 / 6 � ?

6�1�,0�1�0� ↓� 600 100

410� 300 50

110� 60 10

50� 48 8

2 168

Write partial quotients in this column.Think: How many [6s] are in 1,010? At least 100.

The first partial quotient is 100. 100 ∗ 6 � 600

Subtract 600 from 1,010. At least 50 [6s] are left in 410.

The second partial quotient is 50. 50 ∗ 6 � 300


The third partial quotient is 10. 10 ∗ 6 � 60


The fourth partial quotient is 8. 8 ∗ 6 � 48

Subtract. Add the partial quotients.

↑ ↑Remainder Quotient

168 R2The answer is 168 R2. Record the answer as 6�1�,0�1�0�or write 1,010 / 6 → 168 R2.


Student Page


Subtract 300 from 410, and continue by asking: Are there 10 [6s] in 110? Yes, because 10 � 6 � 60. Are there 20 [6s] in 110? No, because 20 � 6 � 120.


100 � 6 � 600.410 410 is left to divide.


110 110 is left to divide.– 60 10 The third partial quotient,

10 � 6 � 60.50 50 is left to divide.

– 48 8 The fourth partial quotient, 8 � 6 � 48.


4. When the subtraction leaves a number less than the divisor (2 in this example), students should move to the final step andadd the partial quotients.

168 R26��1�,0�1�0�

– 600 100410

– 360 6050

– 48 82 168

1,010 � 6 → 168 R2

Ongoing Assessment: Informing InstructionWatch for students who use only multiples of 10. Encourage them to look forlarger multiples of the divisor, as appropriate. Suggest they first compile a list ofeasy multiples of the divisor. Example: If the divisor is 6, students might make the following list:

200 � 6 � 1,200100 � 6 � 600

50 � 6 � 30020 � 6 � 12010 � 6 � 605 � 6 � 30

Remind students that listing the easy multiples in advance allows them to focuson solving the division problem, rather than looking for multiples. Math Masters,page 109 provides an optional form for writing multiples.

Use cards (1 through 9, 4 of each) to generate random 3- or 4-digitdividends and 1- or 2-digit divisors for the class. Ask partners touse the partial-quotients algorithm to solve these problems.Circulate and assist.

106

The Partial-Quotients AlgorithmLESSON

4 �4

Date Time

Example: 185 / 8 ∑ ?

One way: Another way: Another way:

8�1�8�5� 8�1�8�5� 8�1�8�5��80 10 � 160 20 Rename 185 using 105 25 multiples of 8:

�80 10 �24 3 160 � 24 � 125 1 23 Think: 160 � 20 [8s]

�24 3 24 � 3 [8s]1 23 20 � 3 � 23 [8s] with

1 left over

The answer, 23 R1, is the same for each way.

Use the partial-quotients algorithm to solve these problems.

1. 64 � 8 � 2. 749 / 7 �

3. 2,628 � 36 � 4. 8,190 / 9 �

5. Raoul has 237 string bean seeds. He plants them in rows with 8 seeds in each row. How many complete rows can he plant?

Estimate:

Solution: rows298 � 30 � 240, or 240 � 8 � 30

910731078


Student Page

Lesson 4�4 251

168 R26��1�,0�1�0�

– 600 100410

– 300 50110– 60 10

50– 48 8

2 168


252 Unit 4 Division

108

Math Boxes LESSON

4�4

Date Time

1. Write � or �.

a. �35� 0.70

b. �14� 0.21

c. 0.38 �130�

d. 0.6 �23�

e. 0.95 �19000��

�

�

��

3. Write the prime factorization of 132.2 � 2 � 3 � 11, or

22 � 3 � 11

4. Without using a protractor, find themeasurement of the missing angle.

102°

120°

79°

59°

5. Solve.

a. 2.03 � 0.76 �

b. � 57.97 � 3.03

c. � 691.23 � 507.26

d. 29.05 � 103.94 � 132.991,198.49

611.27

6. Fill in the circle next to the best answer.

A. More than �12� of the students

chose blue. B. 50% of the students chose

yellow or green.C. More than 25% of the

students chose yellow or red.

blue

yellow greenred

Favorite 5th Grade Colors

2. Jamie bikes 18.5 mi per day. How many miles will she ride in 13 days?

Open sentence:

Solution:

Answer: 240.5 mim � 240.518.5 � 13 � m

9 83 89

12

34–36 125

38–40 243

207


Student Page

After students have worked for a few minutes, look for partnerships with solutions that have different partial-quotientslists, and ask them to share their solutions with the class.Emphasize the following:

� Students should use the multiples that are easy for them. Thismight sometimes require more steps, but it will make the workgo faster.

� Students should not be concerned if they pick a multiple that istoo large. If that happens, they will quickly realize that theyhave a subtraction problem with a larger number being subtracted from a smaller number. Students can use this information to revise the multiple they used.

� Using the Partial-Quotients Algorithm(Math Journal 1, pp. 106 and 107; Student Reference Book, pp. 22 and 23)

Have students solve the problems on the journal pages, showingtheir work on the computation grids. Encourage students to usethe Student Reference Book as needed. Circulate and assist.

NOTE Students will continue to practice the partial-quotients algorithm throughoutthis unit and in Math Boxes and Ongoing Learning & Practice activities throughoutthe year.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 107, Problem 10 to assess students’ understanding of division. Students are making adequate progress if they have written a numberstory that can be solved using division.


� Math Boxes 4�4(Math Journal 1, p. 108)

Mixed Review Math Boxes in this lesson are paired withMath Boxes in Lessons 4-2 and 4-6. The skills inProblems 5 and 6 preview Unit 5 content.

INDEPENDENTACTIVITY

2 Ongoing Learning & Practice

JournalPage 107Problem 10

�

INDEPENDENTACTIVITY

107

The Partial-Quotients Algorithm continuedLESSON

4 �4

Date Time

Divide.

6. 823 / 3 ➝

7. 2,815 � 43 ➝

8. 4,290 / 64 ➝

9. Regina put 1,610 math books into boxes. Each box held 24 books. How many boxes did she use?

Estimate:

Solution: boxes

10. Make up a number story that can be solved with division. Solve it using a division algorithm.Answers vary.

Solution: Answers vary.

1,600 / 25 � 64, or 24 � 70 � 1,680

67 R265 R20

274 R1

68

�


Student Page


� Study Link 4�4(Math Masters, p. 110)

Home Connection Students practice the partial-quotientsdivision algorithm.

� Using Expanded Notation to Find Multiples(Math Masters, p. 112)

To explore using extended facts, have students write numbers inexpanded notation. Students then complete Math Masters, page112 by using the expanded notation to find equivalent names.

� Practicing Division(Math Masters, p. 111)

Use Math Masters, page 111 to create division problemsfor individualized extra practice. Encourage students touse multiplication to check their problems. Alternately,have students create problems for partners to solve.

5–15 Min

INDEPENDENTACTIVITYEXTRA PRACTICE

15–30 Min

PARTNER ACTIVITYREADINESS

3 Differentiation Options

INDEPENDENTACTIVITY

LESSON

4�4

Name Date Time

Using Expanded Notation

◆ Work with a partner. Use a deck with 4 each of cards 1–9.◆ Take turns dealing 4 cards and forming a 4-digit number.◆ Write the number in standard notation and expanded notation.◆ Then write equivalent names for the value of each digit.

Sample answers:1. Write a 4-digit number.

2. Write the number in expanded notation.

� � �

3. Write equivalent names for the value of each digit.

4302001,000

1,234

4. Write a 4-digit number.

5. Write the number in expanded notation.

� � �

6. Write equivalent names for the value of each digit.

1st digit 2nd digit 3rd digit 4th digit 1st digit 2nd digit 3rd digit 4th digit

1st digit 2nd digit 3rd digit 4th digit

2 º 500 2 º 100 3 º 10 2 º 210 º 100 50 º 4 15 º 2 3 �1

600 � 400 8 º 25 6 º 5


Teaching Master

LESSON

4�4

Name Date Time

Division Practice

For each division problem, complete the list of multiples of the divisor.Then divide.

1. �2�3�4�5�6�6� 2. �

Answer: Answer:

200 º � 200 º �

100 º � 100 º �

50 º � 50 º �

20 º � 20 º �

10 º � 10 º �

5 º � 5 º �

3. / 4. �

Answer: Answer:

200 º � 200 º �

100 º � 100 º �

50 º � 50 º �

20 º � 20 º �

10 º � 10 º �

5 º � 5 º �


Teaching Master

Lesson 4�4 253

STUDY LINK

4�4 Division

22 23

Name Date Time

Here is an example of the partial-quotients algorithm using an “at least...not more than” strategy.

8�1�8�5� Begin estimating with multiples of 10.

How many 8s are in 185? At least 10.10 The first partial quotient. 10 º 8 � 80


How many 8s are in 105? At least 10.10 The second partial quotient. 10 º 8 � 80


How many 8s are in 25? At least 3.The third partial quotient. 3 º 8 � 243Subtract. 1 is left to divide.

1 23 Add the partial quotients: 10 � 10 � 3 � 23

Remainder Quotient Answer: 23 R1

Solve.

1. 639 � 9 2. 954 � 18

Answer: Answer:

3. 1,990 / 24 4. 972 / 37

Answer: Answer:

5. Robert is making a photo album. 6 photos fit on a page. How many pages will he need for 497 photos? pages83

26 R1082 R22

5371

Practice

6. 2,746 � 68 �

Check: � �

7. 3,461 � 165 �

Check: � � 3,4613,296; 165165; 3,2963,296

68; 2,7462,746; 682,8142,814

� 80�105

� 80�25

� 24�

→ →


Study Link Master


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