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Objective To review and practice the U.S. traditional long method for whole numbers and decimals.

1 Doing the Project materials

Recommended Use: Part A: After Lesson 2-7; Part B: After Lesson 2-8; Part C: Math Journal, After Lesson 8-2 pp. 12–14 Key Activities Student Reference Book, pp. 24E–24H and 60E–60I Students review the algorithm for whole numbers (Part A), decimal dividends (Part B), and decimal divisors (Part C). Key Concepts and Skills • Use long division to divide whole numbers and decimals. [Operations and Computation Goal 2] • Use long division to rename common as decimals. [Number and Numeration Goal 5] • Multiply numbers by powers of 10. [Operations and Computation Goal 2] • Use the Multiplication Rule to find equivalent fractions. [Number and Numeration Goal 5] Key Vocabulary U.S. traditional long division method • divisor • dividend • short division

2 Extending the Project materials

Students learn the short for single-digit divisors. Math Journal, p. 15

Additional Information Technology This project has three parts, each of which is structured as follows: See the iTLG. 1. Students work individually to solve a problem using whatever methods they choose. 2. Solutions to the problem are examined in whole-class discussion, including solutions using long division. 3. As necessary, the class works together to use long division to solve one or more similar problems. 4. Students work in partnerships to solve problems with long division. The U.S. traditional long division method for whole numbers and decimals is introduced and practiced in a series of projects in Fourth and Fifth Grade Everyday Mathematics. If students completed those projects, then the work of this project will be review (except for the extension on short division) and students may be able to work with minimal direction from you. If your students did not complete the long division projects in fourth and fifth grades, then you should expect that they will need more support and instruction as they work on this project. In Everyday Mathematics, the U.S. traditional long division method is introduced in situations that involve sharing money equally. There are two reasons for this. One is that the U.S. traditional long division method fits most naturally with what Everyday Mathematics calls equal-sharing situations—situations in which a given amount is shared equally in a known number of shares. In applying long division to such problems, one can think about sharing the larger

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Student Page amounts—those in the left-most places in the dividend—first, and then sharing progressively smaller and smaller amounts as the algorithm moves to places further Date Time to the right in the dividend.

PROJECT 13 Whole Number Long Division The other reason for using sharing money problems in early work with long division

1. The four sixth-grade classes at Linda Vista Elementary School held a is that money naturally models place value, including decimal place value through book sale to raise money for their classroom libraries. The sale raised 24E–24H $464. How much should each class get? hundredths, so using money emphasizes important place-value aspects of the long $116 division algorithm. Of course, long division is not limited to problems involving the equal sharing of money, so after initial work with such situations, students use the algorithm to solve all sorts of division problems. Still, you will notice that the opening problems in Parts A and B of this project involve sharing money. If your students Use long division to solve Problems 2–5. 2. $395 / 5 ? 3. $908 / 22 ? have little prior experience with long division, be sure to continue to use sharing $79 $41.27 money as a primary context until they understand how and why the algorithm works. Student Reference Book pages 24E–24H and 60E–60I are important resources for this project. If your students have significant prior experience with long division, they may be able to understand these pages well enough to do several parts of this project on their own. If your students have less experience with long division, you may want to refer to these pages as background.

4. 837 / 3 ? 5. 975 / 75 ? The directions provide an outline for each part of this project; you will need to adjust 279 13 your approach depending on your students’ experience with long division.

1 Doing the Project Math Journal, p. 12

PARTNER

▼ Part A: Whole Number U.S. ACTIVITY Traditional Long Division (Math Journal, p. 12; Student Reference Book, pp. 24E–24H)

Ask students to solve Problem 1 on journal page 12. Once students have solved the problem individually, they should check their work with a partner’s work and check the reasonableness of the calculated results. As students work, circulate to help and note what methods they are using. Student Page

Whole Numbers

U.S. Traditional Long Division Method: Single-Digit Divisors

U.S. traditional long division is another method you can use to divide.

Share $957 among 5 people.

Step 1: Share the $100 s. Step 2: Trade 4 $100 s for 40 $10 s. That makes 45 $10 s in all.

1 Ò Each person gets 1 $100 . 1 5ͤ9ෆ5ෆ7ෆ 5ͤ9ෆ5ෆ7ෆ 5 Ò 1 $100 each for 5 people 5 4 Ò 4 $100 s are left. 45 Ò 45 $10 s are to be shared.

Step 3: Share the $10 s. Step 4: Share the $1 s.

19 Ò Each person gets 9$10 s. 191 Ò Each person gets 1 $1 . 5ͤ9ෆ5ෆ7ෆ 5ͤ9ෆ5ෆ7ෆ 5 5 45 45 45 Ò 9 $10 s each for 5 people 45 0 Ò 0 $10 s are left. 07 Ò 7$1 s are to be shared. 5 Ò 1 $1 each for 5 people 2 Ò 2 $1 s are left.

$957 / 5 ∑ $191 R$2 Each person gets $191; $2 is left over.

Divide. 1. 840 / 7 ? 2. 6ͤ9ෆ8ෆ4ෆ 3. 4ͤෆ53ෆ9ෆ 4. 5,280 / 6 ?

Check your answers on page 424.

24E

Student Resource Book, p. 24E

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Discuss solutions as a class. Expect that students will use several Student Page different methods, including partial- division, informal Whole Numbers paper-and-pencil approaches, and the U.S. traditional long The U.S. traditional long division method is not limited Note to dividing money. division method. Discuss methods other than long division The “leading” 0 in the is shown in the problem to help you understand the long division first, and then work through the long division solution method. It should not be included in the answer. step-by-step. Highlight the connections between steps in the 3,628 / 5 ? U.S. traditional long division method and the process of sharing Think about the problem as dividing 3,628 into 5 equal shares.

money. See Student Reference Book, page 24E for an example of Step 1: Start with the thousands. Step 2: So trade 3 thousands for 30 hundreds. Share the hundreds.

the connections. 0 Ò There are not enough thousands 07 Ò Each share gets 7 hundreds. 5ͤ3ෆ6ෆ2ෆ8ෆ to share 5 ways. 5ͤ3ෆ6ෆ2ෆ8ෆ Ò 36 hundreds 35 Ò 7 hundreds 5 shares As necessary, use long division to solve one or more similar 1 Ò 1 hundred is left. problems with the whole class, including problems with multidigit Step 3: Trade 1 hundred for 10 tens. Step 4: Trade 2 tens for 20 ones. divisors. (See Student Reference Book, pages 24G and 24H for a Share the tens. Share the ones. 072 Ò Each share gets 2 tens. 0725 Ò Each share gets 5 ones. 5ͤ3ෆ6ෆ2ෆ8ෆ 5ͤ3ෆ6ෆ2ෆ8ෆ discussion of the U.S. traditional long division method with 35 35 12 Ò 10 tens 2 tens 12 multidigit divisors.) As students solve the problems related to 10 Ò 2 tens 5 shares 10 28 Ò 20 ones 8 ones 2 Ò 2 tens are left. 25 Ò 5 ones 5 shares money, ask them to evaluate the reasonableness of their solutions 3 Ò 3 ones are left.

in the context of the original situation. Suggestions: 3,628 / 5 ∑ 725 R3 Share $359 among 6 people. • 1. 5,376 / 6 = ? 2. 6ͤෆ8,5ෆ8ෆ6ෆ 3. 4ͤ6ෆ,9ෆ2ෆ3ෆ 4. 8,029 / 3 = ?

Check your answers on page 424. • Share $8,295 among 5 people. 24F • Share $2,859 among 25 people. Student Resource Book, p. 24F When students are ready, ask them to solve Problems 2–5 on journal page 12. Encourage students to share their solutions and the strategies they utilized.

Student Page Student Page Whole Numbers

Whole Numbers 7720 / 25 ? U.S. Traditional Long Division Method: Make a table of easy multiples of the divisor. Multidigit Divisors 1 25 25 You can use the U.S. traditional long division method to divide 2 25 50 Double 25. by larger numbers. 3 25 75 Add 2 25 and 1 25. 4 25 100 Double 2 25. 5 25 125 Halve 10 25. Share $681 among 21 people. 6 25 150 Double 3 25. Make a table of easy multiples of the divisor. This can help you decide how many to share at 8 25 200 Double 4 25. each step. 10 25 250 Move the decimal point one place to the right.

1 21 21 Step 1: There are not enough thousands to Step 2: Trade the hundreds for tens. 2 21 42 Double 21. share 25 ways, so trade the thousands Share the tens. 3 21 63 Add 2 21 and 1 21. for hundreds. Share the hundreds. 4 21 84 Double 2 21. 5 21 105 Halve 10 21. 3 Ò Each share gets 3 hundreds. 30 Ò There are not enough tens to share. 25ͤ7ෆ7ෆ2ෆ0ෆ Ò 77 hundreds 25ͤ7ෆ7ෆ2ෆ0ෆ 6 21 126 Double 3 21. 75 Ò 3 hundreds 25 shares 75 8 21 168 Double 4 21. 2 Ò 2 hundreds are left. 22 Ò 20 tens 2 tens 10 21 210 Move the decimal point one place to the right.

Step 1: There are not enough [$100]s to Step 2: Trade the 5 [$10]s for 50 [$1]s. Step 3: Trade the tens for ones. share 21 ways, so trade 6 [$100]s Share the 51 [$1]s. Share the ones. for 60 [$10]s. Beginning in the late 1920s and early 1930s, the Share the 68 [$10]s. 308 Ò Each share gets 8 ones. U.S. Treasury issued a small number of large ͤෆෆෆෆ 25 7720 bills, including $500, $1,000, $5,000, $10,000, 75 3 Ò Each person gets 3 [$10]s. 32 Ò Each person gets 2 [$1]s. and $100,000 bills. By the mid-1940s, the 21ͤ6ෆ8ෆ1ෆ Ò There are 68 [$10]s to share. 21ͤ6ෆ8ෆ1ෆ 220 Ò 22 tens 0 ones Treasury stopped making these bills, and in 1969 President Nixon removed them from circulation 63 Ò 3 [$10]s 21 63 200 Ò 8 ones 25 shares because they were rarely used and were 51 Ò 50 [$1]s 1 [$1] 20 Ò 20 ones are left. 5 Ò 5 [$10]s are left. attractive to counterfeiters. 42 Ò 2 [$1]s 21 9 Ò 9 [$1]s are left. 7720 / 25 ∑ 308 R20

$681 / 21 ∑ $32 R$9

Divide 1. 650 / 25 ? 2. 7,720 / 25 ? 3. 13ͤෆ5,8ෆ1ෆ9ෆ 4. 48ͤ5ෆ,2ෆ8ෆ6ෆ

Check your answers on page 424.

24H

24G Student Resource Book, p. 24H Student Resource Book, p. 24G

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PARTNER

Student Page ▼ Part B: U.S. Traditional Long ACTIVITY Date Time Division with Decimal Dividends PROJECT 13 Long Division with Decimal Dividends (Math Journal, p. 13; Student Reference Book, 1. Three friends bought some supplies for a school project for $14.07. 60E, 60F, How much should each friend pay? 60I pp. 60E, 60F, and 60I) $4.69 Ask students to solve Problem 1 on journal page 13. Once students have solved the problem individually, they should check 2. $25.86 / 6 ? 3. 1.071 / 7 ? $4.31 0.153 their work with a partner’s work and check the reasonableness of the calculated results. As students work, circulate to help and note what methods they are using. Discuss solutions as a class, again starting with methods other than long division and then working through the U.S. traditional

3 4. 7.0000 / 11 ? 5. Rename as a decimal. 8 long division solution step-by-step. See Student Reference Book, 0.636 0.375 page 60E for a step-by-step solution of a similar problem. As necessary, use long division to solve similar problems with the whole class, including problems in non-money contexts, problems that involve extending the division into decimal places not present in the original dividend, and renaming-fractions-as-decimals problems. See Student Reference Book, pages 60E, 60F, and 60I. Math Journal, p. 13 As students solve the problems, ask them to evaluate the reasonableness of their solutions in the context of the original situation. Suggestions: • Share $5.79 among 3 people. • A ribbon that is 7.5 m long is to be cut into 6 pieces. How long should each piece be? 7 • Rename 9 as a decimal. When students are ready, ask them to solve Problems 2–5 on journal page 13. Encourage them to share their solutions and the strategies they utilized. Student Page

Date Time PARTNER

▼ Part C: U.S. Traditional Long PROJECT ACTIVITY 13 Long Division with Decimal Divisors Division with Decimal Divisors 1. Donuts cost $0.89 each at the Farmer’s Market. How many donuts

can be bought for $11.50? 60G 60H 12 (Math Journal, p. 14; Student Reference Book, pp. 60G and 60H)

Ask students to solve Problem 1 on journal page 14. Once For Problems 2–5, find equivalent problems with no decimals in the divisors. Then solve the equivalent problems. students have solved the problem individually, they should check 2. 24 / 0.8 3. 27.090 / 0.06 240 / 8 30 2,709 / 6 451.5 their work with a partner’s work. As students work, circulate to equivalent problem equivalent problem help and note what methods they are using.

4. 28.8 / 1.8 5. 0.0084 / 0.3 288/18 16 0.084 / 3 0.028 equivalent problem equivalent problem

Math Journal, p. 14

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Discuss solutions as a class, starting with methods other Student Page than long division and then working through the U.S. traditional long division solution step-by-step. See Student Reference Book, Decimals and Percents U.S. Traditional Long Division Method: Decimal Dividends

pages 60G and 60H for step-by-step solutions of similar problems. You can use the U.S. traditional long division method to divide money in dollars and cents notation.

Solve one or more similar problems with the whole class. Share $5.29 among 3 people.

Step 1: Share the dollars. Suggestions: 1 Ò Each person gets 1 dollar. 3ͤ$ෆ5ෆ.2ෆ9ෆ 3 Ò 1 dollar each for 3 people • 732 / 0.6 ? 2 Ò 2 dollars are left. Step 2: Trade the dollars for dimes. Share the dimes. 1.7 Ò Each person gets 7 dimes. Write a decimal point 45.05 / 0.5 ? 3ͤ$ෆ5ෆ.ෆ29ෆ to show amounts less than a dollar. • 3 2 2 Ò 20 dimes 2 dimes 603 / 0.0009 ? 2 1 Ò 7 dimes each for 3 people • 1 Ò 1 dime is left.

Step 3: Trade the dime for pennies. Share the pennies. When students are ready, ask them to solve Problems 2–5 on 1.76 Ò Each person gets 6 pennies. 3ͤ$ෆ5ෆ.2ෆ9ෆ journal page 14. 3 2 2 2 1 19 Ò 10 pennies 9 pennies 18 Ò 6 pennies each for 3 people 1 Ò 1 penny is left.

Each person gets $1.76. There is 1¢ left. $5.29 / 3 ∑ $1.76 R1¢

2 Extending the Project Divide. 1. $7.26 / 6 = ? 2. 7ͤ$ෆ8ෆ.6ෆ1ෆ 3. 7ͤ$ෆ5ෆ.6ෆ2ෆ 4. $8.04 / 3 = ? Check your answers on page 424A.

PARTNER 60E

▼ Exploring Short Division ACTIVITY (Math Journal, p. 15) Student Resource Book, p. 60E

Students may enjoy learning short division, which is an efficient paper-and-pencil method for solving problems with single-digit divisors. The method can be used with multidigit divisors, but the mental involved is complicated, so short division is normally used only with single-digit divisors. Student Page

Students should study the examples on journal page 15. Once they Decimals and Percents

understand the method, they use it to solve Problems 1–6. You can use the U.S. long division method to divide decimals that do not represent money.

Ask students to discuss their solution strategies. 3.97 / 5 ? Step 1: Trade the ones for tenths and share the tenths. .7 Ò Each share gets 7 tenths. Write a decimal point in the quotient. 5ͤ3ෆ.ෆ97ෆ Ò 3 ones 9 tenths 39 tenths. 3 5 Ò 7 tenths 5 35 tenths. 4 Ò 4 tenths are left.

Student Page Step 2: Trade the remaining tenths for hundredths. Share the hundredths. .79 Ò Each share gets 9 hundredths. 5ͤ3ෆ.9ෆ7ෆ Date Time 3 5 47 Ò 4 tenths 7 hundredths 47 hundredths. PROJECT Short Division 45 Ò 9 hundredths 5 45 hundredths. 13 2 Ò 2 hundredths are left.

Short division is a fast way to divide with paper and pencil. It’s like long division, but all the At this point, you can either round 0.79 to 0.8 and write 3.97 / 5 5 0.8, multiplying and subtracting is done mentally. Short division works best with single-digit divisors. or you can continue dividing into the thousandths. Study the examples below. Then use short division to solve Problems 1–6. Step 3: Continue dividing into the thousandths. Add a 0 to the end of 3.97. (Adding 0s or “padding” a decimal with 0s doesn’t change its value.) Example 1: Example 2: .794 Ò Each share gets 4 thousandths. ͤෆෆෆෆ Long Division Short Division Long Division Short Division 5 3.970 Ò 3.97 3.970 3 5 358 358R2 3 07 3 07 47 5 1 7 92 5 1 7 29 42 329 1 329 21 45 1 5 9 20 Ò 2 hundredths 0 thousandths 20 thousandths. 20 Ò 4 thousandths 5 20 thousandths. 2 9 0 2 0 Ò No thousandths are left. 2 5 0 42 2 1 3.97 / 5 0.794 40 2 1 2 0

Divide. 1. 2. 7 7 8R2 1 2 2 5R4 1. 8.28 / 4 ? 2. 4ͤ9ෆ.6ෆ4ෆ 3. 6ͤෆ8.6ෆ7ෆ 4. 38.65 / 5 = ? Check your answers on page 424A. 5 3 8 39 42 7 8 15 17 39

3. 4. 2 2 9 6R2 9 4 6R5 60F 2 2 2 4 3 6 8 9 0 6 5 6 8 1 Student Resource Book, p. 60F

5. 6. 1 4 2 5R2 4 2 2R3

4 5 17 10 22 9 3 8 20 21

Math Journal, p. 15

Project 13 441DD