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Decimal Long Division EM3TLG1 G5 466Z-NEW.Qx 6/20/08 11:42 AM Page 557 EM3TLG1_G5_466Z-NEW.qx 6/20/08 11:42 AM Page 556 JE PRO CT Objective To extend the long division algorithm to problems in which both the divisor and the dividend are decimals. 1 Doing the Project materials Recommended Use During or after Lesson 4-6 and Project 5. ٗ Math Journal, p. 16 ,Key Activities ٗ Student Reference Book Students explore the meaning of division by a decimal and extend long division to pp. 37, 54G, 54H, and 60 decimal divisors. Key Concepts and Skills • Use long division to solve division problems with decimal divisors. [Operations and Computation Goal 3] • Multiply numbers by powers of 10. [Operations and Computation Goal 3] • Use the Multiplication Rule to find equivalent fractions. [Number and Numeration Goal 5] • Explore the meaning of division by a decimal. [Operations and Computation Goal 7] Key Vocabulary decimal divisors • dividend • divisor 2 Extending the Project materials Students express the remainder in a division problem as a whole number, a fraction, an ٗ Math Journal, p. 17 exact decimal, and a decimal rounded to the nearest hundredth. ٗ Student Reference Book, p. 54I Technology See the iTLG. 466Z Project 14 Decimal Long Division EM3TLG1_G5_466Z-NEW.qx 6/20/08 11:42 AM Page 557 Student Page 1 Doing the Project Date PROJECT 14 Dividing with Decimal Divisors WHOLE-CLASS 1. Draw lines to connect each number model with the number story that fits it best. Number Model Number Story ▼ Exploring Meanings for DISCUSSION What is the area of a rectangle 1.75 m by ?cm 50 0.10 ء Decimal Division 2 Sales tax is 10%. What is the sales tax on a $2 ء (Math Journal, p. 16) 1.75 0.50 purchase? How many dimes are there in $2? 2 / 0.10 Have partners solve Problem 1 on journal page 16. When most A rectangle’s area is 8 cm2 and its width is students have finished, have volunteers explain their solutions. 1.75 / 0.50 0.25 cm. What is its length? Use student responses to emphasize the following ideas: How many 50-cm pieces can be cut from 8 / 0.25 1.75 m of string? ᭟ One way to think about division is as “How many ___ Find equivalent problems with no decimals in the divisors and solve. 2. 38 / 0.5 ϭ 380 / 5 3. 0.84 / 0.07 ϭ 84 / 7 are in ___?” equivalent problem equivalent problem 76 12 ᭟ Think of missing factors. Given the area and one dimension of Solution: Solution: 4. 501 / 0.03 ϭ 50,100 / 3 5. 0.465 / 1.5 ϭ 4.65 / 15 a rectangle, for example, we can use division to find the other equivalent problem equivalent problem 16,700 0.31 dimension. Solution: Solution: 6. Indy cars use about 1.3 gallons of fuel for each 2.5-mile lap. How many miles per gallon is that? ᭟ Numbers in problems can be represented one way in the 1.92 miles/gallon (unit) problem’s number story and another way in the matching number model. For example, 10% in one of the number stories corresponds to 0.10 in the matching number model. Math Journal, p. 16 ᭟ The idea that multiplication makes bigger and division makes smaller, which many students have formed from their work with whole numbers, does not apply to multiplication and division by numbers less than 1. WHOLE-CLASS ▼ Dividing with Decimal Divisors DISCUSSION (Math Journal, p. 16) Remind students of two key facts that may be used to solve division problems with decimal divisors: ● A fraction can be interpreted as a division problem, and vice versa. ● Multiplying the numerator and denominator of a fraction by the same nonzero number results in an equivalent fraction. Project 14 466AA EM3TLG1_G5_466Z-NEW.qx 6/20/08 1:51 PM Page 558 Student Page Write 27 / 0.3 ϭ ? on the board or a transparency. Using division Date methods such as partial quotients and long division is PROJECT 14 Representing Remainders as Decimals cumbersome with decimal divisors. Ask students how they might You might write the answer to a problem such as 17 / 6 in two ways: 17 / 6 → 2 R 5, or use multiplication to rename the problem to an equivalent by rewriting the remainder as a fraction: 17 / 6 = 2ᎏ5ᎏ. 6 With decimal long division, you can show the quotient as a decimal: 17 / 6 = 2.83ෆ. The problem that is easier to solve. repeat bar means that the 3s repeat forever. But, in most situations, having infinitely many 3s is not practical, so answers are often rounded to some reasonable number of decimal 27 places, usually two or three: 17 / 6 = 2.83, or 17 / 6 = 2.833. 27 / 0.3 can be thought of as ᎏᎏ. 0.3 Notice that 17 / 6 = 2.83 is not actually true. You can check this by multiplying 2.83 by 6. You won’t get 17. But, for most practical purposes, 2.83 or 2.833 is close enough to 17 / 6 ᎏ27ᎏ ϭ ᎏ27ᎏ 1ᎏ0ᎏ ϭ ᎏ27ᎏ0 that most people aren’t bothered by a rounded number model. 0.3 0.3 * 10 3 Complete the table. 270 Answer as But ᎏ can be thought of as 270 / 3. Decimal 3 Quotient and Mixed Exact Rounded to Problem Remainder Number Decimal Hundredths So, 27 / 0.3 ϭ 270 / 3. 5 1. 17 / 6 2 R5 2ᎏᎏ 2.83 2.83 6 3 The resulting problem, 270 / 3, is easier to solve than the original 2. 15 / 4 3 R3 3ᎏᎏ 3.75 3.75 4 problem, 27 / 0.3, but has the same answer. 5 3. 5 / 8 0 R5 ᎏᎏ 0.625 0.63 8 2 Work through similar problems until students understand 4. 17 / 3 5 R2 5ᎏᎏ 5.66ෆ 5.67 3 the principle: 2 5. 56 / 9 6 R2 6ᎏᎏ 6.222ෆ 6.22 9 Multiplying the dividend and the divisor in a division problem by the same nonzero number does not change the quotient. Suggestions: Math Journal, p. 17 ᭟ 45 / 0.9 45 / 0.9 ϭ ᎏ45ᎏ 1ᎏ0ᎏ ϭ ᎏ45ᎏ0 ϭ 450 / 9 ϭ 50 0.9 * 10 9 ᭟ 12 / 0.03 12 / 0.03 ϭ ᎏ1ᎏ2 1ᎏ0ᎏ0 ϭ ᎏ12ᎏ00 ϭ 1200 / 3 ϭ 400 0.03 * 100 3 NOTE Student Reference Book pages ᭟ 105 / 0.015 105 / 0.015 ϭ ᎏ10ᎏ5 1ᎏ,0ᎏ00 ϭ 105ᎏ,ᎏ000 ϭ 0.015 * 1,000 15 54G and 54H provide a detailed step-by-step 105,000 / 15 7,000 explanation of how to “clear decimals” in the divisor so that the long division algorithm can Have students complete journal page 16. be applied. Other relevant Student Reference Book pages include page 37 (multiplication by Circulate and assist. powers of 10) and 60 (using multiplication to find equivalent fractions). Student Page 2 Extending the Project Decimals and Percents U.S. Traditional Long Division Method: PARTNER ▼ Using Long Division to Rename Renaming Fractions as Decimals Any whole number can ACTIVITY The U.S. traditional long division method can be used to be written as a decimal by attaching a decimal rename fractions as decimals. point and one or more Fractions as Decimals 0s; the value of the number remains the same: 5 ϭ 5.0. (Student Reference Book, p. 54I; Math Journal, p. 17) With all decimal numbers, attaching one or more zeros to the right of the digit that is furthest to the right will not change the value of the number: Have partners read Student Reference Book, page 54I and then ϭ 8.3 8.3000. complete journal page 17. When students have finished, Use the U.S. traditional long division method to rename ᎏ5ᎏ as a decimal. 8 discuss any difficulties or curiosities they encountered. ᎏ5ᎏ Step 1: Write 8 as a division problem. Write 5 with several 0s after the decimal point: 5.000. (You can always add more 0s if you need them.) 8ͤ5ෆ.0ෆ0ෆ0ෆ Step 2: Solve the division problem. Stop when the remainder is 0, or when you have enough precision for your purposes, or when you notice a repeating pattern. .625 8ͤ5ෆ.0ෆ0ෆ0ෆ Ϫ4 8 20 Ϫ16 40 Ϫ40 0 This division problem divided evenly in three decimal places. ᎏ5ᎏ ϭ 0.625 8 54I Student Resource Book, p. 54I 466BB Project 14 Decimal Long Division.
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