Whole Number US Traditional Long Division
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PROJECT
Objective
To review and practice the U.S. traditional long division method for whole numbers and decimals.
1
Doing the Project
Recommended Use: Part A: After Lesson 2-7; Part B: After Lesson 2-8; Part C: After Lesson 8-2
Key Activities Students review the long division algorithm for whole numbers (Part A), decimal dividends (Part B), and decimal divisors (Part C).
materials Math Journal, pp. 12–14 Student Reference Book, pp. 24E–24H and 60E–60I
Key Concepts and Skills • Use long division to divide whole numbers and decimals. [Operations and Computation Goal 2]
• Use long division to rename common fractions 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
Students learn the short division algorithm for single-digit divisors.
Additional Information This project has three parts, each of which is structured as follows: 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.
materials Math Journal, p. 15
Technology See the iTLG.
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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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 to the right in the dividend.
Student Page Date
Time
PROJECT
Whole Number Long Division
13
1. The four sixth-grade classes at Linda Vista Elementary School held a
book sale to raise money for their classroom libraries. The sale raised $464. How much should each class get?
24E–24H
$116
Use long division to solve Problems 2–5. 2. $395 / 5 ?
3. $908 / 22 ?
$79
$41.27
The other reason for using sharing money problems in early work with long division is that money naturally models place value, including decimal place value through hundredths, so using money emphasizes important place-value aspects of the long 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 have little prior experience with long division, be sure to continue to use sharing 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 ?
The directions provide an outline for each part of this project; you will need to adjust your approach depending on your students’ experience with long division.
5. 975 / 75 ?
279
13
1 Doing the Project ▼
Math Journal, p. 12
Part A: Whole Number U.S. Traditional Long Division
PARTNER ACTIVITY
(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
Step 2: Trade 4
$100 s.
$100 s
for 40 $10 s.
That makes 45 $10 s in all.
1 苶5 苶7 苶 5冄9 5 4
Ò Each person gets 1 $100 . Ò 1 $100 each for 5 people Ò 4 $100 s are left.
Step 3: Share the
$10
1 苶5 苶7 苶 5冄9 5 45
Ò 45 $10 s are to be shared.
Step 4: Share the
s.
19 Ò Each person gets 9 $10 s. 苶5 苶7 苶 5冄9 5 45 45 Ò 9 $10 s each for 5 people 0 Ò 0 $10 s are left.
191 苶5 苶7 苶 5冄9 5 45 45 07 5 2
$1
s.
Ò Each person gets 1
Ò7
$1
Ò1
$1
Ò2
$1
$1
.
s are to be shared. each for 5 people s are left.
$957 / 5 ∑ $191 R$2 Each person gets $191; $2 is left over.
Divide. 1. 840 / 7 ?
2. 6冄9 苶8 苶4 苶
3. 4冄5 苶3 苶9 苶
4. 5,280 / 6 ?
Check your answers on page 424.
24E
Student Resource Book, p. 24E
441AA
Project 13 Long Division
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Discuss solutions as a class. Expect that students will use several different methods, including partial-quotients division, informal paper-and-pencil approaches, and the U.S. traditional long division method. Discuss methods other than long division first, and then work through the long division solution step-by-step. Highlight the connections between steps in the U.S. traditional long division method and the process of sharing money. See Student Reference Book, page 24E for an example of the connections. As necessary, use long division to solve one or more similar problems with the whole class, including problems with multidigit divisors. (See Student Reference Book, pages 24G and 24H for a discussion of the U.S. traditional long division method with multidigit divisors.) As students solve the problems related to money, ask them to evaluate the reasonableness of their solutions in the context of the original situation. Suggestions:
• Share $359 among 6 people. • Share $8,295 among 5 people. • Share $2,859 among 25 people.
Student Page Whole Numbers The U.S. traditional long division method is not limited to dividing money.
Note The “leading” 0 in the quotient is shown in the problem to help you understand the long division method. It should not be included in the answer.
3,628 / 5 ? Think about the problem as dividing 3,628 into 5 equal shares. Step 1: Start with the thousands.
Step 2: So trade 3 thousands for 30 hundreds.
Ò There are not enough thousands 0 to share 5 ways. 5冄3 苶6 苶2 苶8 苶
07 苶6 苶2 苶8 苶 5冄3
Ò Each share gets 7 hundreds. Ò 36 hundreds
35
Ò 7 hundreds 5 shares
Share the hundreds.
1
Step 3: Trade 1 hundred for 10 tens.
Ò 1 hundred is left.
Step 4: Trade 2 tens for 20 ones.
Share the tens.
Share the ones.
072 Ò Each share gets 2 tens. 5冄3 苶6 苶2 苶8 苶 35 12 Ò 10 tens 2 tens 10 Ò 2 tens 5 shares 2 Ò 2 tens are left.
0725 Ò Each share gets 5 ones. 苶6 苶2 苶8 苶 5冄3 35 12 10 28 Ò 20 ones 8 ones 25 Ò 5 ones 5 shares 3 Ò 3 ones are left.
3,628 / 5 ∑ 725 R3
2. 6冄8 苶,5 苶8 苶6 苶
1. 5,376 / 6 = ?
3. 4冄6 苶,9 苶2 苶3 苶
4. 8,029 / 3 = ?
Check your answers on page 424.
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 Resource Book, p. 24F
Student Page
Student Page Whole Numbers Whole Numbers 7720 / 25 ?
U.S. Traditional Long Division Method: Multidigit Divisors
Make a table of easy multiples of the divisor. 1 25
You can use the U.S. traditional long division method to divide by larger numbers.
2 25
Share $681 among 21 people.
1 21
21
2 21
42
3 21
63
4 21
84
5 21
105
6 21
126
8 21
168
10 21
210
75 100
5 25
125 150
8 25
200
10 25
250
Double 25. Add 2 25 and 1 25. Double 2 25. Halve 10 25. Double 3 25. Double 4 25. Move the decimal point one place to the right.
Step 1: There are not enough thousands to Double 21. Add 2 21 and 1 21. Double 2 21. Halve 10 21. Double 3 21. Double 4 21. Move the decimal point one place to the right.
Step 1: There are not enough [$100]s to
share 25 ways, so trade the thousands for hundreds. Share the hundreds.
3 苶7 苶2 苶0 苶 25冄7 75 2
Step 2: Trade the 5 [$10]s for 50 [$1]s.
share 21 ways, so trade 6 [$100]s for 60 [$10]s.
Ò Each person gets 3 [$10]s. Ò There are 68 [$10]s to share. Ò 3 [$10]s 21 Ò 5 [$10]s are left.
32 21冄6 苶8 苶1 苶 63 51 42 9
Ò Ò Ò Ò
Each share gets 3 hundreds. 77 hundreds 3 hundreds 25 shares 2 hundreds are left.
Step 2: Trade the hundreds for tens. Share the tens.
30 Ò There are not enough tens to share. 苶7 苶2 苶0 苶 25冄7 75 22 Ò 20 tens 2 tens
Step 3: Trade the tens for ones.
Share the 51 [$1]s.
Share the ones.
Share the 68 [$10]s.
3 21冄6 苶8 苶1 苶 63 5
50
4 25 6 25
Make a table of easy multiples of the divisor. This can help you decide how many to share at each step.
25
3 25
308 Ò Each share gets 8 ones. 苶7 苶2 苶0 苶 25冄7 75 220 Ò 22 tens 0 ones
Ò Each person gets 2 [$1]s.
200 Ò 8 ones 25 shares 20 Ò 20 ones are left.
Ò 50 [$1]s 1 [$1] Ò 2 [$1]s 21 Ò 9 [$1]s are left.
Beginning in the late 1920s and early 1930s, the U.S. Treasury issued a small number of large bills, including $500, $1,000, $5,000, $10,000, and $100,000 bills. By the mid-1940s, the Treasury stopped making these bills, and in 1969 President Nixon removed them from circulation because they were rarely used and were attractive to counterfeiters.
7720 / 25 ∑ 308 R20
$681 / 21 ∑ $32 R$9 Divide 1. 650 / 25 ?
2. 7,720 / 25 ?
3. 13冄5 苶,8 苶1 苶9 苶
4. 48冄5 苶,2 苶8 苶6 苶
Check your answers on page 424.
24H
24G
Student Resource Book, p. 24H Student Resource Book, p. 24G
Project 13
441BB
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11:56 AM
Part B: U.S. Traditional Long Division with Decimal Dividends
Student Page Date
Time
PROJECT
Long Division with Decimal Dividends
13
1. Three friends bought some supplies for a school project for $14.07. 60E, 60F, 60I
How much should each friend pay?
Page 530
▼
EM3TLG1_G6_441Z-DD_NEW.qx
PARTNER ACTIVITY
(Math Journal, p. 13; Student Reference Book, pp. 60E, 60F, and 60I)
$4.69
2. $25.86 / 6 ?
Ask students to solve Problem 1 on journal page 13. 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.
3. 1.071 / 7 ?
$4.31
0.153
Discuss solutions as a class, again starting with methods other than long division and then working through the U.S. traditional long division solution step-by-step. See Student Reference Book, page 60E for a step-by-step solution of a similar problem.
3 5. Rename as a decimal. 8
4. 7.0000 / 11 ?
0.636
0.375
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. As students solve the problems, ask them to evaluate the reasonableness of their solutions in the context of the original situation. Suggestions:
Math Journal, p. 13
• 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?
• Rename 79 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.
Date
▼
Student Page
Part C: U.S. Traditional Long Division with Decimal Divisors
Time
PROJECT
Long Division with Decimal Divisors
13
1. Donuts cost $0.89 each at the Farmer’s Market. How many donuts
can be bought for $11.50?
60G 60H
12
For Problems 2–5, find equivalent problems with no decimals in the divisors. Then solve the equivalent problems. 2. 24 / 0.8
240 / 8 30 equivalent problem
4. 28.8 / 1.8
288 / 18 16 equivalent problem
3. 27.090 / 0.06
2,709 / 6 451.5 equivalent problem
5. 0.0084 / 0.3
0.084 / 3 0.028 equivalent problem
Math Journal, p. 14
441CC
Project 13 Long Division
PARTNER ACTIVITY
(Math Journal, p. 14; Student Reference Book, pp. 60G and 60H)
Ask students to solve Problem 1 on journal page 14. Once students have solved the problem individually, they should check their work with a partner’s work. As students work, circulate to help and note what methods they are using.
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Discuss solutions as a class, starting with methods other than long division and then working through the U.S. traditional long division solution step-by-step. See Student Reference Book, pages 60G and 60H for step-by-step solutions of similar problems.
Student Page Decimals and Percents
U.S. Traditional Long Division Method: Decimal Dividends 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. Suggestions:
Share $5.29 among 3 people. Step 1: Share the dollars. 1 苶5 苶.2 苶9 苶 3冄$ 3
• 732 / 0.6 ? • 45.05 / 0.5 ? • 603 / 0.0009 ?
2
Ò Each person gets 1 dollar. Ò 1 dollar each for 3 people Ò 2 dollars are left.
Step 2: Trade the dollars for dimes. Share the dimes. 1.7 3冄$ 苶5 苶.2 苶9 苶 3 22 2 1 1
When students are ready, ask them to solve Problems 2–5 on journal page 14.
Ò Each person gets 7 dimes. Write a decimal point to show amounts less than a dollar. Ò 20 dimes 2 dimes Ò 7 dimes each for 3 people Ò 1 dime is left.
Step 3: Trade the dime for pennies. Share the pennies. 1.76 3冄$ 苶5 苶.2 苶9 苶 3 22 2 1 19 18 1
Ò Each person gets 6 pennies.
Ò 10 pennies 9 pennies Ò 6 pennies each for 3 people Ò 1 penny is left.
Each person gets $1.76. There is 1¢ left. $5.29 / 3 ∑ $1.76 R1¢
2 Extending the Project
Divide. 2. 7冄$ 苶8 苶.6 苶1 苶
1. $7.26 / 6 = ?
3. 7冄$ 苶5 苶.6 苶2 苶
4. $8.04 / 3 = ?
▼
Check your answers on page 424A.
PARTNER ACTIVITY
Exploring Short Division
60E
Student Resource Book, p. 60E
(Math Journal, p. 15)
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 arithmetic involved is complicated, so short division is normally used only with single-digit divisors. Students should study the examples on journal page 15. Once they understand the method, they use it to solve Problems 1–6. Ask students to discuss their solution strategies.
Student Page Decimals and Percents You can use the U.S. long division method to divide decimals that do not represent money. 3.97 / 5 ? Step 1: Trade the ones for tenths and share the tenths. .7 5冄3 苶.9 苶7 苶 35 4
Student Page Date
.79 5冄3 苶.9 苶7 苶 35 47 45 2
Short Division
13
Each share gets 7 tenths. Write a decimal point in the quotient. 3 ones 9 tenths 39 tenths. 7 tenths 5 35 tenths. 4 tenths are left.
Step 2: Trade the remaining tenths for hundredths. Share the hundredths.
Time
PROJECT
Ò Ò Ò Ò
Ò Each share gets 9 hundredths.
Ò 4 tenths 7 hundredths 47 hundredths. Ò 9 hundredths 5 45 hundredths. 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 multiplying and subtracting is done mentally. Short division works best with single-digit divisors.
At this point, you can either round 0.79 to 0.8 and write 3.97 / 5 5 0.8, 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: Long Division
Short Division
3 5 8
Example 2: Long Division 3 0 7
3 5 8 R2
5 1 7 9 2
5 1 7
2
9 42
3 0 7
3 9 2 1
1 5
.794 5冄3 苶.9 苶7 苶0 苶 35 47 45 20 20 0
Short Division 3 9 2
2
1
9
2 9
0 2
2 5
0
4 2
2 1
4 0
2 1
2
0
1.
5
3
7
8 R2
8
39
42
3.
Ò 2 hundredths 0 thousandths 20 thousandths. Ò 4 thousandths 5 20 thousandths. Ò No thousandths are left.
3.97 / 5 0.794
Divide.
2.
7
Ò Each share gets 4 thousandths. Ò 3.97 3.970
7
1
2
2
5 R4
8
15
17
39
9
4
6 R5
6
28
41
1. 8.28 / 4 ?
2. 4冄9 苶.6 苶4 苶
3. 6冄8 苶.6 苶7 苶
4. 38.65 / 5 = ?
Check your answers on page 424A.
4.
3
2
2
9
6 R2
6
8
29
20
5.
6
5
9
3
60F
Student Resource Book, p. 60F
6.
4
1
4
2
5 R2
5
17
10
22
4
2
2 R3
8
20
21
Math Journal, p. 15
Project 13
441DD
6/20/08
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Page 527
PROJECT
Objective
To review and practice the U.S. traditional long division method for whole numbers and decimals.
1
Doing the Project
Recommended Use: Part A: After Lesson 2-7; Part B: After Lesson 2-8; Part C: After Lesson 8-2
Key Activities Students review the long division algorithm for whole numbers (Part A), decimal dividends (Part B), and decimal divisors (Part C).
materials Math Journal, pp. 12–14 Student Reference Book, pp. 24E–24H and 60E–60I
Key Concepts and Skills • Use long division to divide whole numbers and decimals. [Operations and Computation Goal 2]
• Use long division to rename common fractions 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
Students learn the short division algorithm for single-digit divisors.
Additional Information This project has three parts, each of which is structured as follows: 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.
materials Math Journal, p. 15
Technology See the iTLG.
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
Project 13
441Z
EM3TLG1_G6_441Z-DD_NEW.qx
6/20/08
11:56 AM
Page 528
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 to the right in the dividend.
Student Page Date
Time
PROJECT
Whole Number Long Division
13
1. The four sixth-grade classes at Linda Vista Elementary School held a
book sale to raise money for their classroom libraries. The sale raised $464. How much should each class get?
24E–24H
$116
Use long division to solve Problems 2–5. 2. $395 / 5 ?
3. $908 / 22 ?
$79
$41.27
The other reason for using sharing money problems in early work with long division is that money naturally models place value, including decimal place value through hundredths, so using money emphasizes important place-value aspects of the long 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 have little prior experience with long division, be sure to continue to use sharing 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 ?
The directions provide an outline for each part of this project; you will need to adjust your approach depending on your students’ experience with long division.
5. 975 / 75 ?
279
13
1 Doing the Project ▼
Math Journal, p. 12
Part A: Whole Number U.S. Traditional Long Division
PARTNER ACTIVITY
(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
Step 2: Trade 4
$100 s.
$100 s
for 40 $10 s.
That makes 45 $10 s in all.
1 苶5 苶7 苶 5冄9 5 4
Ò Each person gets 1 $100 . Ò 1 $100 each for 5 people Ò 4 $100 s are left.
Step 3: Share the
$10
1 苶5 苶7 苶 5冄9 5 45
Ò 45 $10 s are to be shared.
Step 4: Share the
s.
19 Ò Each person gets 9 $10 s. 苶5 苶7 苶 5冄9 5 45 45 Ò 9 $10 s each for 5 people 0 Ò 0 $10 s are left.
191 苶5 苶7 苶 5冄9 5 45 45 07 5 2
$1
s.
Ò Each person gets 1
Ò7
$1
Ò1
$1
Ò2
$1
$1
.
s are to be shared. each for 5 people s are left.
$957 / 5 ∑ $191 R$2 Each person gets $191; $2 is left over.
Divide. 1. 840 / 7 ?
2. 6冄9 苶8 苶4 苶
3. 4冄5 苶3 苶9 苶
4. 5,280 / 6 ?
Check your answers on page 424.
24E
Student Resource Book, p. 24E
441AA
Project 13 Long Division
EM3TLG1_G6_441Z-DD_NEW.qx
6/20/08
11:56 AM
Page 529
Discuss solutions as a class. Expect that students will use several different methods, including partial-quotients division, informal paper-and-pencil approaches, and the U.S. traditional long division method. Discuss methods other than long division first, and then work through the long division solution step-by-step. Highlight the connections between steps in the U.S. traditional long division method and the process of sharing money. See Student Reference Book, page 24E for an example of the connections. As necessary, use long division to solve one or more similar problems with the whole class, including problems with multidigit divisors. (See Student Reference Book, pages 24G and 24H for a discussion of the U.S. traditional long division method with multidigit divisors.) As students solve the problems related to money, ask them to evaluate the reasonableness of their solutions in the context of the original situation. Suggestions:
• Share $359 among 6 people. • Share $8,295 among 5 people. • Share $2,859 among 25 people.
Student Page Whole Numbers The U.S. traditional long division method is not limited to dividing money.
Note The “leading” 0 in the quotient is shown in the problem to help you understand the long division method. It should not be included in the answer.
3,628 / 5 ? Think about the problem as dividing 3,628 into 5 equal shares. Step 1: Start with the thousands.
Step 2: So trade 3 thousands for 30 hundreds.
Ò There are not enough thousands 0 to share 5 ways. 5冄3 苶6 苶2 苶8 苶
07 苶6 苶2 苶8 苶 5冄3
Ò Each share gets 7 hundreds. Ò 36 hundreds
35
Ò 7 hundreds 5 shares
Share the hundreds.
1
Step 3: Trade 1 hundred for 10 tens.
Ò 1 hundred is left.
Step 4: Trade 2 tens for 20 ones.
Share the tens.
Share the ones.
072 Ò Each share gets 2 tens. 5冄3 苶6 苶2 苶8 苶 35 12 Ò 10 tens 2 tens 10 Ò 2 tens 5 shares 2 Ò 2 tens are left.
0725 Ò Each share gets 5 ones. 苶6 苶2 苶8 苶 5冄3 35 12 10 28 Ò 20 ones 8 ones 25 Ò 5 ones 5 shares 3 Ò 3 ones are left.
3,628 / 5 ∑ 725 R3
2. 6冄8 苶,5 苶8 苶6 苶
1. 5,376 / 6 = ?
3. 4冄6 苶,9 苶2 苶3 苶
4. 8,029 / 3 = ?
Check your answers on page 424.
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 Resource Book, p. 24F
Student Page
Student Page Whole Numbers Whole Numbers 7720 / 25 ?
U.S. Traditional Long Division Method: Multidigit Divisors
Make a table of easy multiples of the divisor. 1 25
You can use the U.S. traditional long division method to divide by larger numbers.
2 25
Share $681 among 21 people.
1 21
21
2 21
42
3 21
63
4 21
84
5 21
105
6 21
126
8 21
168
10 21
210
75 100
5 25
125 150
8 25
200
10 25
250
Double 25. Add 2 25 and 1 25. Double 2 25. Halve 10 25. Double 3 25. Double 4 25. Move the decimal point one place to the right.
Step 1: There are not enough thousands to Double 21. Add 2 21 and 1 21. Double 2 21. Halve 10 21. Double 3 21. Double 4 21. Move the decimal point one place to the right.
Step 1: There are not enough [$100]s to
share 25 ways, so trade the thousands for hundreds. Share the hundreds.
3 苶7 苶2 苶0 苶 25冄7 75 2
Step 2: Trade the 5 [$10]s for 50 [$1]s.
share 21 ways, so trade 6 [$100]s for 60 [$10]s.
Ò Each person gets 3 [$10]s. Ò There are 68 [$10]s to share. Ò 3 [$10]s 21 Ò 5 [$10]s are left.
32 21冄6 苶8 苶1 苶 63 51 42 9
Ò Ò Ò Ò
Each share gets 3 hundreds. 77 hundreds 3 hundreds 25 shares 2 hundreds are left.
Step 2: Trade the hundreds for tens. Share the tens.
30 Ò There are not enough tens to share. 苶7 苶2 苶0 苶 25冄7 75 22 Ò 20 tens 2 tens
Step 3: Trade the tens for ones.
Share the 51 [$1]s.
Share the ones.
Share the 68 [$10]s.
3 21冄6 苶8 苶1 苶 63 5
50
4 25 6 25
Make a table of easy multiples of the divisor. This can help you decide how many to share at each step.
25
3 25
308 Ò Each share gets 8 ones. 苶7 苶2 苶0 苶 25冄7 75 220 Ò 22 tens 0 ones
Ò Each person gets 2 [$1]s.
200 Ò 8 ones 25 shares 20 Ò 20 ones are left.
Ò 50 [$1]s 1 [$1] Ò 2 [$1]s 21 Ò 9 [$1]s are left.
Beginning in the late 1920s and early 1930s, the U.S. Treasury issued a small number of large bills, including $500, $1,000, $5,000, $10,000, and $100,000 bills. By the mid-1940s, the Treasury stopped making these bills, and in 1969 President Nixon removed them from circulation because they were rarely used and were attractive to counterfeiters.
7720 / 25 ∑ 308 R20
$681 / 21 ∑ $32 R$9 Divide 1. 650 / 25 ?
2. 7,720 / 25 ?
3. 13冄5 苶,8 苶1 苶9 苶
4. 48冄5 苶,2 苶8 苶6 苶
Check your answers on page 424.
24H
24G
Student Resource Book, p. 24H Student Resource Book, p. 24G
Project 13
441BB
6/20/08
11:56 AM
Part B: U.S. Traditional Long Division with Decimal Dividends
Student Page Date
Time
PROJECT
Long Division with Decimal Dividends
13
1. Three friends bought some supplies for a school project for $14.07. 60E, 60F, 60I
How much should each friend pay?
Page 530
▼
EM3TLG1_G6_441Z-DD_NEW.qx
PARTNER ACTIVITY
(Math Journal, p. 13; Student Reference Book, pp. 60E, 60F, and 60I)
$4.69
2. $25.86 / 6 ?
Ask students to solve Problem 1 on journal page 13. 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.
3. 1.071 / 7 ?
$4.31
0.153
Discuss solutions as a class, again starting with methods other than long division and then working through the U.S. traditional long division solution step-by-step. See Student Reference Book, page 60E for a step-by-step solution of a similar problem.
3 5. Rename as a decimal. 8
4. 7.0000 / 11 ?
0.636
0.375
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. As students solve the problems, ask them to evaluate the reasonableness of their solutions in the context of the original situation. Suggestions:
Math Journal, p. 13
• 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?
• Rename 79 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.
Date
▼
Student Page
Part C: U.S. Traditional Long Division with Decimal Divisors
Time
PROJECT
Long Division with Decimal Divisors
13
1. Donuts cost $0.89 each at the Farmer’s Market. How many donuts
can be bought for $11.50?
60G 60H
12
For Problems 2–5, find equivalent problems with no decimals in the divisors. Then solve the equivalent problems. 2. 24 / 0.8
240 / 8 30 equivalent problem
4. 28.8 / 1.8
288 / 18 16 equivalent problem
3. 27.090 / 0.06
2,709 / 6 451.5 equivalent problem
5. 0.0084 / 0.3
0.084 / 3 0.028 equivalent problem
Math Journal, p. 14
441CC
Project 13 Long Division
PARTNER ACTIVITY
(Math Journal, p. 14; Student Reference Book, pp. 60G and 60H)
Ask students to solve Problem 1 on journal page 14. Once students have solved the problem individually, they should check their work with a partner’s work. As students work, circulate to help and note what methods they are using.
EM3TLG1_G6_441Z-DD_NEW.qx
6/20/08
11:56 AM
Page 531
Discuss solutions as a class, starting with methods other than long division and then working through the U.S. traditional long division solution step-by-step. See Student Reference Book, pages 60G and 60H for step-by-step solutions of similar problems.
Student Page Decimals and Percents
U.S. Traditional Long Division Method: Decimal Dividends 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. Suggestions:
Share $5.29 among 3 people. Step 1: Share the dollars. 1 苶5 苶.2 苶9 苶 3冄$ 3
• 732 / 0.6 ? • 45.05 / 0.5 ? • 603 / 0.0009 ?
2
Ò Each person gets 1 dollar. Ò 1 dollar each for 3 people Ò 2 dollars are left.
Step 2: Trade the dollars for dimes. Share the dimes. 1.7 3冄$ 苶5 苶.2 苶9 苶 3 22 2 1 1
When students are ready, ask them to solve Problems 2–5 on journal page 14.
Ò Each person gets 7 dimes. Write a decimal point to show amounts less than a dollar. Ò 20 dimes 2 dimes Ò 7 dimes each for 3 people Ò 1 dime is left.
Step 3: Trade the dime for pennies. Share the pennies. 1.76 3冄$ 苶5 苶.2 苶9 苶 3 22 2 1 19 18 1
Ò Each person gets 6 pennies.
Ò 10 pennies 9 pennies Ò 6 pennies each for 3 people Ò 1 penny is left.
Each person gets $1.76. There is 1¢ left. $5.29 / 3 ∑ $1.76 R1¢
2 Extending the Project
Divide. 2. 7冄$ 苶8 苶.6 苶1 苶
1. $7.26 / 6 = ?
3. 7冄$ 苶5 苶.6 苶2 苶
4. $8.04 / 3 = ?
▼
Check your answers on page 424A.
PARTNER ACTIVITY
Exploring Short Division
60E
Student Resource Book, p. 60E
(Math Journal, p. 15)
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 arithmetic involved is complicated, so short division is normally used only with single-digit divisors. Students should study the examples on journal page 15. Once they understand the method, they use it to solve Problems 1–6. Ask students to discuss their solution strategies.
Student Page Decimals and Percents You can use the U.S. long division method to divide decimals that do not represent money. 3.97 / 5 ? Step 1: Trade the ones for tenths and share the tenths. .7 5冄3 苶.9 苶7 苶 35 4
Student Page Date
.79 5冄3 苶.9 苶7 苶 35 47 45 2
Short Division
13
Each share gets 7 tenths. Write a decimal point in the quotient. 3 ones 9 tenths 39 tenths. 7 tenths 5 35 tenths. 4 tenths are left.
Step 2: Trade the remaining tenths for hundredths. Share the hundredths.
Time
PROJECT
Ò Ò Ò Ò
Ò Each share gets 9 hundredths.
Ò 4 tenths 7 hundredths 47 hundredths. Ò 9 hundredths 5 45 hundredths. 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 multiplying and subtracting is done mentally. Short division works best with single-digit divisors.
At this point, you can either round 0.79 to 0.8 and write 3.97 / 5 5 0.8, 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: Long Division
Short Division
3 5 8
Example 2: Long Division 3 0 7
3 5 8 R2
5 1 7 9 2
5 1 7
2
9 42
3 0 7
3 9 2 1
1 5
.794 5冄3 苶.9 苶7 苶0 苶 35 47 45 20 20 0
Short Division 3 9 2
2
1
9
2 9
0 2
2 5
0
4 2
2 1
4 0
2 1
2
0
1.
5
3
7
8 R2
8
39
42
3.
Ò 2 hundredths 0 thousandths 20 thousandths. Ò 4 thousandths 5 20 thousandths. Ò No thousandths are left.
3.97 / 5 0.794
Divide.
2.
7
Ò Each share gets 4 thousandths. Ò 3.97 3.970
7
1
2
2
5 R4
8
15
17
39
9
4
6 R5
6
28
41
1. 8.28 / 4 ?
2. 4冄9 苶.6 苶4 苶
3. 6冄8 苶.6 苶7 苶
4. 38.65 / 5 = ?
Check your answers on page 424A.
4.
3
2
2
9
6 R2
6
8
29
20
5.
6
5
9
3
60F
Student Resource Book, p. 60F
6.
4
1
4
2
5 R2
5
17
10
22
4
2
2 R3
8
20
21
Math Journal, p. 15
Project 13
441DD
