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

Objective To review the partial-quotients division algorithm with whole numbers. 1 Teaching the Lesson materials Key Activities Students review and practice the use of a friendly number paper-and-pencil division algorithm strategy. They play Division Dash to practice mental division with 1-digit divisors. Key Concepts and Skills Use the partial-quotients algorithm for problems. Apply friendly numbers to identify partial quotients. Factor numbers to identify partial quotients. Key Vocabulary dividend divisor partial quotient quotient remainder Math Journal 1, p. 101 Student Reference Book, pp. 22, 23, and 303 Study Link 1 Teaching Aid Master (Math Masters, p. 1) Class Data Pad slates Per partnership: each of the number cards 1 9, (from the Everything Math Deck, if available) See Advance Preparation 2 Ongoing Assessment: Recognizing Student Achievement Use journal page 101. Ongoing Assessment: Informing Instruction See page 20. Ongoing Learning & Practice materials Students practice and maintain skills through Math Boxes and Study Link activities. 3 Differentiation Options Math Journal 1, p. 102 Study Link Master (Math Masters, p. 10) materials READINESS Students review divisibility rules for 1-digit divisors. ENRICHMENT Students find numbers to meet divisibility criteria. ELL SUPPORT Students review vocabulary for the parts of a division problem. Student Reference Book, p. 11 Teaching Master (Math Masters, p. 10) Class Data Pad See Advance Preparation Additional Information Advance Preparation For Part 1, you will need 2 copies of the computation grid (Math Masters, page 1) for each student. Technology Assessment Management System Journal page 101 See the itlg. 236 Unit Division

Getting Started Mental Math and Reflexes Pose multiplication and division problems like the following. How many s are in? 9 What number times 9 equals 27? 3 What is 3 times 120? 360 How many s are in 32? 8 What number times 8 equals 0? Multiply times 80. 00 What number times 7 equals 3? Multiply 12 by 7. 8 Multiply by 3. 16 Math Message Amy is 127 days older than Bob. How many weeks is that? Study Link 1 Follow-Up Have partners compare answers. Explain that fact family relationships can be used to check computations. Write 60 67 28 on the board or a transparency. An addition problem from this fact family will check the subtraction. Write 28 67 60. Ask: Are there any problems with this approach? Most students will recognize either the subtraction or the addition error. It is important to calculate the check problem, not just rewrite the numbers. 28 67 9, not 60, so the subtraction was incorrect in the initial number sentence. Change the equal sign to not equal, and then write 60 67 38. Encourage students to use number relationships to check their calculations. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Ask volunteers to share their solution strategies. Expect that some students will suggest breaking 127 into friendly numbers. Survey the class for clues that the Math Message was a division problem. The problem gave the whole (127 days) and asked how many groups (weeks); because there are 7 days in a week, the problem was to figure out how many 7s are in 127. Ask volunteers to write a number model for this problem. 127 / 7; 12 7 7 ; 127 7; and 7127 Reviewing the Partial- Quotients Algorithm (Math Masters, p. 1) WHOLE-CLASS Given a dividend and a divisor, the partial-quotients algorithm is one pencil-and-paper strategy for division. Model the following steps on the Class Data Pad: 1. Write the problem in traditional form: 7127. 2. Draw a vertical line to the right of the problem to separate the subtraction part of the algorithm from the partial quotients. 7127 Lesson 2 237

Explain that with this notation, students will list their partial quotients on the right of the vertical line and then subtract the related multiples on the left of the vertical line, until the remaining dividend is smaller than the divisor. Links to the Future Students will practice the partial-quotients algorithm in Lesson -, using an easy-multiples strategy to find partial quotients, and in Lesson - with decimal dividends. NOTE When the result of division is expressed as a quotient and a nonzero remainder, Everyday Mathematics uses an arrow rather than an equal sign, as in 26 12 20 R6. Everyday Mathematics prefers this notation because 26 12 20 R6 is not a proper number sentence. The arrow is read as is, yields, or results in. Model this expression for students in your examples of the partial-quotients algorithm. Label the arrow on the Class Data Pad for display throughout this unit. One strategy for finding partial quotients is to use friendly numbers. Rename the dividend as an expression that contains multiples of the divisor. Make a name-collection box for 127, and add the expression 70 7. Use this expression to model the algorithm. 3. Ask: How many 7s are in 70? 10, because 10 7 70. Write 70 under 127 and 10 next to it, to the right of the vertical line. Subtract, saying: 127 minus 70 equals 7. Explain that 10 is the first partial quotient and 7 is what remains to be divided. 7127 127 70 7 70 10 7 7 is left to divide.. Ask: How many 7s are in 7? 8, because 8 * 7 6. Write 6 under 7 and 8 next to it, to the right of the vertical line. Subtract, saying: 7 minus 6 equals 1. Explain that 8 is the second partial quotient, and 1 is what remains to be divided. 7127 70 10 7 6 8 1 1 is left to divide.. Explain that they can stop this process when the number left to 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. Write 18 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 is how many weeks older than Bob? About 18 weeks older, or 18 weeks and 1 day older 18 R1 7127 70 10 7 6 8 10 18 127 7 18 R1 238 Unit Division

Adjusting the Activity The arrow as a mathematical symbol is used to represent several different concepts. To support English language learners, remind students of the relationship between multiplication and division. Explain that when a quotient is written 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, the arrow, to show that the division results in the quotient and the whole-number remainder. ELL Date 2 Time The Partial-Quotients Division Algorithm 82 R3 R7 109 R30 99 R1 Use the partial-quotients algorithm to solve these problems. 1. 69 2. 832 1 3. 3,18 / 32., 360. Jerry was sorting 389 marbles into bags. He put a dozen in each bag. How many bags does he need? 33 bags 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: 10 22; 70 9 7 1; 3 3 3 21 1 Have students choose one of these expressions to use with the partial-quotients algorithm. Remind them to write the problem and draw the vertical line, using the problem on the Class Data Pad as a model. To help students remember place value as they write digits, have them use a computation grid. Circulate and assist. Math Journal 1, p. 101 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 Reference Book and the samples on the Class Data Pad can be used to verify correct usage of the steps in this algorithm. Have students complete the page. Circulate and assist. Links to the Future Problem on journal page 101 will provide some information about students ability to interpret remainders. Interpreting remainders will be covered in Lesson -6. Ongoing Assessment: Recognizing Student Achievement Journal Page 101 Use journal page 101 to assess students understanding of the partial-quotients algorithm. Students are making adequate progress if they demonstrate accurate use of the notation for the algorithm. Whole Numbers Division Algorithms Different symbols may be used to indicate division. For example, 9 divided by 6 may be written as 9 6, 69, 9 / 6, or 9 6. 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 Method In the partial-quotients method, it takes several steps to find the quotient. At each step, you find a partial answer (called a partial quotient). These partial answers are then added to find the quotient. Study the example below. To find the number of 6s in 1,010 first find partial quotients and then add them. Record the partial quotients in a column to the right of the original problem. Example 1,010 / 6? Write partial quotients in this column. 61,010 Think: How many [6s] are in 1,010? At least 100. 600 100 The first partial quotient is 100. 100 6 600 10 300 0 Subtract 600 from 1,010. At least 0 [6s] are left in 10. The second partial quotient is 0. 0 6 300 110 60 10 Subtract. At least 10 [6s] are left in 110. The third partial quotient is 10. 10 6 60 0 8 8 Subtract. At least 8 [6s] are left in 0. The fourth partial quotient is 8. 8 6 8 2 168 Subtract. Add the partial quotients. Remainder Quotient 168 R2 The answer is 168 R2. Record the answer as 61,010 or write 1,010 / 6 168 R2. Four ways to show 123 divided by 123 123 / 123 12 3 123 is the dividend. is the divisor. Student Reference Book, p. 22 Lesson 2 239

Division Dash Materials number cards 1 9 ( of each) Player 1 1 score sheet Quotient Score Players 1 or 2 Skill Division of 2-digit by 1-digit numbers Object of the game To reach 100 in the fewest divisions possible. Directions 1. 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 division problem. 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 and record the result. This result is your quotient. Remainders are ignored. Calculate mentally or on paper. Add your quotient to your previous score and record your new score. (If this is your first turn, your previous score was 0.). Players repeat Step 3 until one player s score is 100 or more. The first player to reach at least 100 wins. If there is only one player, the Object of the game is to reach 100 in as few turns as possible. Example Turn 1: Bob draws 6,, and. He divides 6 by. Quotient 12. Remainder is ignored. The score is 12 0 12. Turn 2: Bob then draws 8, 2, and 1. He divides 82 by 1. Quotient 82. The score is 82 12 9. Turn 3: Bob then draws, 7, and 8. He divides 7 by 8. Quotient 7. Remainder is ignored. The score is 7 9 101. Bob has reached 100 in 3 turns and the game ends. Student Reference Book, p. 303 6 6 Games Player 2 Quotient Score 6 is the dividend. is the divisor. Quotient Score 12 12 82 9 7 101 Introducing Division Dash (Student Reference Book, p. 303) Division Dash uses randomly generated numbers to obtain values for 1-digit divisors and 2-digit dividends. Encourage students to calculate mentally, but do not restrict paper-and-pencil use. Discuss the example on the Student Reference Book page. Then have the class play a round of Division Dash together. The whole class mentally calculates the division. Remind students that only the whole-number part of the quotient is recorded. If the dividend is 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 Instruction WHOLE-CLASS Watch for students who use paper-and-pencil, rather than mental strategies, to calculate the division. To help them bridge into mental math, ask them to write the division expression 9 but then use multiplication facts and friendly parts to calculate mentally. 2 Ongoing Learning & Practice Math Boxes 2 (Math Journal 1, p. 102) Date 2 1. Write or. Math Boxes a. 0. 3 8 b. 0.89 1 0 c. 0. d. 1 0.3 3 e. 7 0.9 8 9 83 89 Time 2. Sasha earns $.0 per day on her paper route. She delivers papers every day. How much does she earn in two weeks? Open sentence:.0 7 2 d Solution: d 63 $63 38 0 23 Mixed Review Math Boxes in this lesson are paired with Math Boxes in Lesson - and -6. The skills in Problems and 6 preview Unit content. Writing/Reasoning Have students write a response to the following: Explain why your answer to Problem is correct. Sample answer: The sum of the measures of the angles equals 180. My answer is correct because 0 90 10, and the missing angle measure is 0 because 0 90 0 180. 3. Write the prime factorization of 80.. Without using a protractor, find the measurement of the missing angle. 2 2 2 2,11 or 2 90 0 0 Study Link 2 (Math Masters, p. 10). Solve. a. 209.0 b. $30.9 73. $8.1 13. $21.98 c..339 d. 2.03 6.671 1.8 11.01 39.61 12 207 6. The United States in 1900 Circle the best answer. 0% Urban 60% Rural A. In 1900, more than half of the communities were rural. B. In 1900, 6 out of 10 communities in the United States were rural. C. In 1900, more than 3 of communities in the United States were rural. 3 36 12 Home Connection Students practice the partial-quotients division algorithm. Math Journal 1, p. 102 20 Unit Division

Study Link Master 3 Differentiation Options Name Date Time STUDY LINK 2 Division Here is the partial-quotients algorithm using a friendly numbers strategy. READINESS Reviewing Divisibility Rules (Student Reference Book, p. 11) PARTNER 1 Min 7237 210 27 21 6 Rename dividend (use multiples of the divisor): 237 210 21 6 How many 7s are in 210? 30 30 The first partial quotient. 30 7 210 Subtract. 27 is left to divide. How many 7s are in 27? 3 3 The second partial quotient. 3 7 21 Subtract. 6 is left to divide. 33 Add the partial quotients: 30 3 33 Remainder Quotient 33 R6 22 23 To provide experience with identifying factors, have partners read about divisibility on page 11 of the Student Reference Book and complete the Check Your Understanding problems. ENRICHMENT Exploring Divisibility by the Digits (Math Masters, p. 10) PARTNER 1 Min To apply students understanding of factors, have them explore divisibility from another perspective. Students examine 3-digit numbers that meet certain divisibility criteria. Then they use the same criteria to identify larger numbers. 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? 10, 10, 10, and 3 2. 6166 3. 21 / 27 R 2 R. 8 1. 1708 32 R 2 Practice 3,98 6. 3,817 168 Check: 3,98 168; or 3,817 2,236 3,817; or 168 7. 2,17 281 Check: 281; or 2,236 2,236; or 281 2,17 Math Masters, p. 10 ELL SUPPORT Supporting Math Vocabulary Development SMALL-GROUP 1 30 Min To provide language support for division, have volunteers write a division number model on chart paper in several different formats. 127 / 7 18 R1 12 7 7 18 R1 127 7 18 R1 18 R1 7127 For each number model, label and underline the dividend in red (the number being divided); label and underline the divisor in blue (the number the dividend is being divided by); label and circle the quotient in a third color; label and circle the remainder in a fourth color. Emphasize that both the quotient and the remainder are part of the answer. Display this chart throughout all the division lessons. 9 7 8 R3 7 A Division Problem Dividend 8 R3 9 Divisor Quotient 9 / 7 8 R3 Remainder Name Date Time 2 Teaching Master 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. 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. 1,20 The first five digits are divisible by. 12,00 The first six digits are divisible by 6. 120,02 The first seven digits are divisible by 7. 1,20,021 The first eight digits are divisible by 8. 12,09,216 The first nine digits are divisible by 9. 120,02,162 1. Jacob knew that with divisibility rules, it should be easy. The boys started with 3-digit numbers and found 123 and 22. Latoya checked their work. What should she tell them? 123 is correct because 1 is divisible by 1; 12 is correct because it is an even number; and 123 is correct because 1 2 3 6, which is divisible by 3. 22 is not correct because 2 2 8, which is not divisible by 3. 2. Use the characteristics listed above to find as many puzzle numbers as you can. Record them in the boxes below.sample answers: Puzzle Numbers -digit -digit 6-digit 7-digit 8-digit 9-digit 1,72 1,72 17,22 1,72,27 1,72,272 17,22,726 1,62 16,20 162,08 1,62,08 16,20,80 162,08,02 Math Masters, p. 10 Lesson 2 21

Objective To provide practice with strategies for the partial-quotients algorithm. 1 Teaching the Lesson materials Key Activities Students 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. Use the partial-quotients algorithm for problems. Key Vocabulary multiple divisor partial quotient dividend Ongoing Assessment: Informing Instruction See page 21. Ongoing Assessment: Recognizing Student Achievement Use journal page 107. Math Journal 1, pp. 106 and 107 Student Reference Book, pp. 22, 23, and 302 Study Link 3 Teaching Master (Math Masters, p. 109; optional) Teaching Aid Master (Math Masters, p. 1) Class Data Pad calculator Per partnership: each of number cards 0 9; 2 each of number cards 2, 3,, 6, 9 and 10 (from the Everything Math Deck, if available) See Advance Preparation 2 Ongoing Learning & Practice materials Students practice and maintain skills through Math Boxes and Study Link activities. 3 Differentiation Options Math Journal 1, p. 108 Study Link Master (Math Masters, p. 110) materials READINESS Students practice finding friendly numbers using expanded notation and multiples. EXTRA PRACTICE Students use lists of multiples of the divisor to solve division problems. Teaching Masters (Math Masters, pp. 111 and 112) Per partnership: each of number cards 1 9 (from the Everything Math Deck, if available) See Advance Preparation Additional Information Advance Preparation For Part 1, you will need a coin for the calculator practice in the Mental Math and Reflexes and 2 copies of the computation grid (Math Masters, page 1) for each student. For Part 3, prepare Math Masters, page 111 to provide individualized practice as needed. Technology Assessment Management System Journal page 107 See the itlg. 28 Unit Division

Getting Started 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. 36 7 R1; 7 1 11 2 R3; 2 3 3 / 8 R2; 2 8 or 1 67 12 R3; 12 3 6 or 12 1 2 78 8 9 R6; 9 6 8 or 9 3 99 / 8 12 R3; 12 3 8 11102 9 R3; 9 3 9 or 9 1 3 2230 9 R; 9 2 or 9 1 680 / 0 13 R30; 13 3 0 0 or 13 3 Ask volunteers to explain the meaning of the fraction remainder. The divisor represents how 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, 1, is one divided by. Math Message Write a 3-digit number that is divisible by 6. Study Link 3 Follow-Up Allow students five minutes to compare their answers and resolve any differences. Circulate and assist. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION Survey the class for their 3-digit numbers. Write student responses on the Class Data Pad. Ask students how they might check these numbers without actually dividing by 6. Most students 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 a class, 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 context that also develops speed. The variation is for 3-digit numbers. Discuss the variation example on Student Reference Book, page 302, and demonstrate a turn by playing one hand as a class. Then allow partners time to play at least 3 rounds of Divisibility Dash. Reviewing the Partial- Quotients Algorithm (Math Masters, p. 1) WHOLE-CLASS WHOLE-CLASS Explain that another approach to finding partial quotients is to use a series of at least...not more than multiples of the divisor. A good strategy is to start with easy numbers, such as 100 times the divisor or 10 times the divisor. Games Divisibility Dash Materials number cards 0 9 ( of each) number cards: 2, 3,, 6, 9, and 10 (2 of each) Players 2 or 3 Skill Recognizing multiples, using divisibility tests Object of the game To discard all cards. Directions 1. Shuffle the divisor cards and place them number-side down on the table. Shuffle the draw cards and deal 8 to each player. Place the remaining draw cards number-side down next 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-digit numbers that are multiples as you can. A card used to make one 2-digit number may not be used again to make another number. Place all the cards you used to make 2-digit numbers in a discard pile. If you cannot make a 2-digit number that is a multiple of the 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 of the divisor card, that player may challenge. Players use the divisibility test for the divisor card value to check the number in question. Any numbers that are not multiples of the divisor card must be returned to the player s hand.. If the draw pile or divisor cards have all been used, they can be reshuffled and put back into play.. The first player to discard all of his or her cards is the winner. 1 2 7 8 3 Example Andrew s cards: Divisor card: Andrew uses his cards to make 1 7 2 numbers that are multiples of 3: He discards these cards and holds the 2 and 8 for the next round of play. 1 Student Reference Book, p. 302 2 1 7 8 7 Note The number cards 0 9 ( of each) are the draw cards. This set of draw cards is also called the draw pile. The number cards 2, 3,, 6, 9, and 10 (2 of each) are the divisor cards. 3 Lesson 29

Whole Numbers Division Algorithms Different symbols may be used to indicate division. For example, 9 divided by 6 may be written as 9 6, 69, 9 / 6, or 9 6. 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 Method In the partial-quotients method, it takes several steps to find the quotient. At each step, you find a partial answer (called a partial quotient). These partial answers are then added to find the quotient. Study the example below. To find the number of 6s in 1,010 first find partial quotients and then add them. Record the partial quotients in a column to the right of the original problem. Example 1,010 / 6? 61,010 Think: How many [6s] are in 1,010? At least 100. 600 100 The first partial quotient is 100. 100 6 600 10 300 0 Subtract 600 from 1,010. At least 0 [6s] are left in 10. The second partial quotient is 0. 0 6 300 110 60 10 Subtract. At least 10 [6s] are left in 110. The third partial quotient is 10. 10 6 60 0 8 8 Subtract. At least 8 [6s] are left in 0. The fourth partial quotient is 8. 8 6 8 2 168 Subtract. Add the partial quotients. Remainder Quotient 168 R2 The answer is 168 R2. Record the answer as 61,010 or write 1,010 / 6 168 R2. Name Date Time Teaching Master Easy Multiples Write partial quotients in this column. Student Reference Book, p. 22 1,000 º 1,000 º 100 º 100 º 0 º 0 º 20 º 20 º 10 º 10 º º º 1,000 º 1,000 º 100 º 100 º 0 º 0 º 20 º 20 º 10 º 10 º º º 1,000 º 1,000 º 100 º 100 º 0 º 0 º Four ways to show 123 divided by 123 123 / 123 12 3 123 is the dividend. is the divisor. 1. Write the problem 61,010, drawing a vertical line to the right of 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] in 1,010? No, because 200 6 1,200, which is more than 1,010. 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 the first partial quotient. 61,010 600 100 The first partial quotient, 100 6 600. 2. Next find out how much is left to be divided. Subtract 600 from 1,010. 61,010 600 100 The first partial quotient, 100 6 600. 10 10 is left to divide. 3. Now find the number of 6s in 10. There are several ways to do this: Use a fact family and extended facts. 6 6 36; 60 6 360, so there are at least 60 [6s] in 10. 61,010 600 100 The first partial quotient, 100 6 600. 10 10 is left to divide. 360 60 The second partial quotient, 60 6 360. 0 0 is left to divide. 8 8 The third partial quotient, 8 6 8. 2 2 is left to divide. Or continue to use at least...not more than multiples with easy numbers. For example, ask: Are there at least 100 [6s] in 10? No, because 100 6 600. Are there at least 0 [6s]? Yes, because 0 6 300. 61,010 600 100 The first partial quotient, 100 6 600. 10 10 is left to divide. 300 0 The second partial quotient, 0 6 300. 110 110 is left to divide. 20 º 20 º 10 º 10 º º º Math Masters, p. 109 20 Unit Division

Subtract 300 from 10, 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. 61,010 600 100 The first partial quotient, 100 6 600. 10 10 is left to divide. 300 0 The second partial quotient, 0 6 300. 110 110 is left to divide. 60 10 The third partial quotient, 10 6 60. 0 0 is left to divide. 8 8 The fourth partial quotient, 8 6 8. 2 2 is left to divide.. When the subtraction leaves a number less than the divisor (2 in this example), students should move to the final step and add the partial quotients. 168 R2 61,010 600 100 10 360 60 0 8 8 2 168 1,010 6 168 R2 168 R2 61,010 600 100 10 300 0 110 60 10 0 8 8 2 168 Ongoing Assessment: Informing Instruction Date Time Watch for students who use only multiples of 10. Encourage them to look for larger multiples of the divisor, as appropriate. Suggest they first compile a list of easy multiples of the divisor. Example: If the divisor is 6, students might make the following list: 200 6 1,200 100 6 600 0 6 300 20 6 120 10 6 60 6 30 Remind students that listing the easy multiples in advance allows them to focus on solving the division problem, rather than looking for multiples. Math Masters, page 109 provides an optional form for writing multiples. The Partial-Quotients Algorithm Example: 18 / 8? One way: Another way: Another way: 818 818 818 80 10 160 20 Rename 18 using 10 2 multiples of 8: 80 10 2 3 160 2 1 2 1 23 Think: 160 20 [8s] 2 3 2 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. 8 107 1. 6 8 2. 79 / 7 73 3. 2,628 36. 8,190 / 9 910. 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: 8 30 20, or 20 8 30 Solution: 29 rows Use cards (1 through 9, of each) to generate random 3- or -digit dividends and 1- or 2-digit divisors for the class. Ask partners to use the partial-quotients algorithm to solve these problems. Circulate and assist. Math Journal 1, p. 106 Lesson 21

Date Divide. 6. 823 / 3 7. 2,81 3 8.,290 / 6 The Partial-Quotients Algorithm continued 27 R1 6 R20 67 R2 Time 9. Regina put 1,610 math books into boxes. Each box held 2 books. How many boxes did she use? Estimate: Solution: 68 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 / 2 6, or 2 70 1,680 After students have worked for a few minutes, look for partnerships with solutions that have different partial-quotients lists, and ask them to share their solutions with the class. Emphasize the following: Students should use the multiples that are easy for them. This might sometimes require more steps, but it will make the work go faster. Students should not be concerned if they pick a multiple that is too large. If that happens, they will quickly realize that they have 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) Math Journal 1, p. 107 Have students solve the problems on the journal pages, showing their work on the computation grids. Encourage students to use the Student Reference Book as needed. Circulate and assist. NOTE Students will continue to practice the partial-quotients algorithm throughout this unit and in Math Boxes and Ongoing Learning & Practice activities throughout the year. Ongoing Assessment: Recognizing Student Achievement Journal Page 107 Problem 10 Date Math Boxes Time Use journal page 107, Problem 10 to assess students understanding of division. Students are making adequate progress if they have written a number story that can be solved using division. 1. Write or. a. 3 0.70 b. 1 0.21 3 c. 0.38 1 0 d. 0.6 2 3 90 e. 0.9 1 00 3. Write the prime factorization of 132. 2 2 3 11, or 2 2 3 11 9 83 89 2. Jamie bikes 18. mi per day. How many miles will she ride in 13 days? Open sentence: Solution: m 20. 20. mi 18. 13 m. Without using a protractor, find the measurement of the missing angle. 79 38 0 23 2 Ongoing Learning & Practice Math Boxes (Math Journal 1, p. 108). Solve. a. 2.03 0.76 1.27 b. 61 7.97 3.03 c. 1,198.9 691.23 07.26 d. 29.0 103.9 132.99 Math Journal 1, p. 108 12 120 102 9 6. Fill in the circle next to the best answer. Favorite th Grade Colors blue red yellow green A. More than 1 of the students 2 chose blue. B. 0% of the students chose yellow or green. C. More than 2% of the students chose yellow or red. 3 36 12 207 Mixed Review Math Boxes in this lesson are paired with Math Boxes in Lessons -2 and -6. The skills in Problems and 6 preview Unit content. 22 Unit Division

Study Link (Math Masters, p. 110) READINESS Home Connection Students practice the partial-quotients division algorithm. 3 Differentiation Options Using Expanded Notation to Find Multiples (Math Masters, p. 112) PARTNER 1 30 Min To explore using extended facts, have students write numbers in expanded notation. Students then complete Math Masters, page 112 by using the expanded notation to find equivalent names. Name Date Time STUDY LINK Division Here is an example of the partial-quotients algorithm using an at least...not more than strategy. Solve. 818 Begin estimating with multiples of 10. How many 8s are in 18? At least 10. 10 The first partial quotient. 10 º 8 80 Subtract. 10 is left to divide. How many 8s are in 10? At least 10. 10 The second partial quotient. 10 º 8 80 Subtract. 2 is left to divide. How many 8s are in 2? At least 3. 3 The third partial quotient. 3 º 8 2 Subtract. 1 is left to divide. 1 23 Add the partial quotients: 10 10 3 23 Remainder Quotient 23 R1 1. 639 9 2. 9 18 71 3. 1,990 / 2. 972 / 37 82 R22. Robert is making a photo album. 6 photos fit on a page. How many pages will he need for 97 photos? Practice 80 10 80 2 2 Study Link Master 2,81 6. 2,76 68 Check: 2,81 2,76; 68 3,296 7. 3,61 16 Check: 16; 3,296 3,296; 16 3,61 Math Masters, p. 110 26 R10 83 68; 2,76 3 pages 22 23 EXTRA PRACTICE Practicing Division (Math Masters, p. 111) 1 Min Use Math Masters, page 111 to create division problems for individualized extra practice. Encourage students to use multiplication to check their problems. Alternately, have students create problems for partners to solve. Teaching Master Name Date Time Teaching Master Name Date Time Division Practice Using Expanded Notation For each division problem, complete the list of multiples of the divisor. Then divide. 1. 2366 2. 200 º 200 º 100 º 100 º 0 º 0 º 20 º 20 º 10 º 10 º º º Work with a partner. Use a deck with each of cards 1 9. Take turns dealing cards and forming a -digit number. Write the number in standard notation and expanded notation. Then write equivalent names for the value of each digit. Sample answers: 1,23 1. Write a -digit number. 2. Write the number in expanded notation. 1,000 200 30 3. Write equivalent names for the value of each digit. 1st digit 2nd digit 3rd digit th digit 2 º 00 2 º 100 3 º 10 2 º 2 10 º 100 0 º 1 º 2 3 1 600 00 8º2 6º 3. /. 200 º 200 º 100 º 100 º 0 º 0 º 20 º 20 º. Write a -digit number.. Write the number in expanded notation. 6. Write equivalent names for the value of each digit. 1st digit 2nd digit 3rd digit th digit 10 º 10 º º º Math Masters, p. 111 Math Masters, p. 112 Lesson 23