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

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1 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

2 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 What number times 7 equals 3? Multiply 12 by 7. 8 Multiply by 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 on the board or a transparency. An addition problem from this fact family will check the subtraction. Write 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 , not 60, so the subtraction was incorrect in the initial number sentence. Change the equal sign to not equal, and then write 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; ; 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: Draw a vertical line to the right of the problem to separate the subtraction part of the algorithm from the partial quotients Lesson 2 237

3 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 R6. Everyday Mathematics prefers this notation because 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 Use this expression to model the algorithm. 3. Ask: How many 7s are in 70? 10, because 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 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 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 R R1 238 Unit Division

7 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

9 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 The first partial quotient is Subtract 600 from 1,010. At least 0 [6s] are left in 10. The second partial quotient is Subtract. At least 10 [6s] are left in 110. The third partial quotient is Subtract. At least 8 [6s] are left in 0. The fourth partial quotient is Subtract. Add the partial quotients. Remainder Quotient 168 R2 The answer is 168 R2. Record the answer as 61,010 or write 1,010 / 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 / 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 , which is less than 1,010. Are there at least 200 [6s] in 1,010? No, because ,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, The first partial quotient, Next find out how much is left to be divided. Subtract 600 from 1, , The first partial quotient, 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 ; , so there are at least 60 [6s] in , The first partial quotient, is left to divide The second partial quotient, is left to divide. 8 8 The third partial quotient, 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 Are there at least 0 [6s]? Yes, because , The first partial quotient, is left to divide The second partial quotient, is left to divide. 20 º 20 º 10 º 10 º º º Math Masters, p Unit Division

10 Subtract 300 from 10, and continue by asking: Are there 10 [6s] in 110? Yes, because Are there 20 [6s] in 110? No, because , The first partial quotient, is left to divide The second partial quotient, is left to divide The third partial quotient, is left to divide. 8 8 The fourth partial quotient, 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, , R2 168 R2 61, 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: , 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: Rename 18 using 10 2 multiples of 8: Think: [8s] [8s] [8s] with 1 left over The answer, 23 R1, is the same for each way. Use the partial-quotients algorithm to solve these problems / , ,190 / 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: , or 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

11 Date Divide / , ,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 ,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 b c d e Write the prime factorization of , or Jamie bikes 18. mi per day. How many miles will she ride in 13 days? Open sentence: Solution: m mi m. Without using a protractor, find the measurement of the missing angle Ongoing Learning & Practice Math Boxes (Math Journal 1, p. 108). Solve. a b c. 1, d Math Journal 1, p 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 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

12 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 The first partial quotient. 10 º 8 80 Subtract. 10 is left to divide. How many 8s are in 10? At least 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 Add the partial quotients: Remainder Quotient 23 R ,990 / / R22. Robert is making a photo album. 6 photos fit on a page. How many pages will he need for 97 photos? Practice Study Link Master 2, ,76 68 Check: 2,81 2,76; 68 3, ,61 16 Check: 16; 3,296 3,296; 16 3,61 Math Masters, p R ; 2,76 3 pages 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 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, Write equivalent names for the value of each digit. 1st digit 2nd digit 3rd digit th digit 2 º 00 2 º º 10 2 º 2 10 º º 1 º º2 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

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