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Dividing Decimals Getting Ready for the Dance ACTIVITY ACTIVITY 2.7 Investigative 2.7 Dividing Decimals SUGGESTED LEARNING STRATEGIES: Close Reading, Use Manipulatives, Look for a Pattern, Create Representations Activity Focus My Notes • Dividing decimals by whole numbers • Dividing decimals by decimals • Repeating decimals The middle school students at Montgomery Middle School are planning a dance. They will decorate the gym, hire a deejay, and provide refreshments. The students have listed their expenses and will share the cost of the dance. Expense Decorations Deejay Refreshments Cost $272.64 $168.64 $113.28 Materials • BLM 2: Hundred grid Chunking the Activity Thirty-two students will attend the dance. What is each student’s share of the total cost? You will have to divide to solve this problem, so first explore some ideas about division. Think about a simpler problem: 0.72 ÷ 9. Remember: 0.72 means 72 hundredths, or 72 out of 100. #1–6 Ex 1 #7–10 #11 #12 100 or 1 whole. Shade ____ 72 of the squares in 1. This grid represents ____ 100 100 the grid. #13–14 Ex 2 #15 Ex 3 #16–17 Ex 4 #18 Ex 5 Paragraph Close Reading © 2010 College Board. All rights reserved. 1 Use Manipulatives, Create Representations This model will give students a concrete look at dividing a decimal by a whole number. Make sure that the students understand that the entire chart represents one unit. 2. To divide by 9, you need to make 9 groups. Mark the shaded sections into 9 groups. How many squares are in each group? 8 squares W We write 0.72 instead o of .72 for readability purposes purposes. Encourage students to use the leading zero before the decimal point when decimals have a value less than one. TEACHER TO TEACHER 3. What is the value of one small square? 0.01 4. What is the value of each group of squares that you made? 0.08 5. Complete this number sentence. 0.72 divided by 9 = 0.08 2 Use Manipulatives Not all students will mark the same 9 groups. Allow students to share their representations. Unit 2 • Operations with Numbers © 2010 College Board. All rights reserved. 103-110_SB_MS1_2-7_SE.indd 103 103 Differentiating Instruction 12/16/09 5:55:33 PM Some students may benefit by using different colored pencils to shade each group. 34 Look for a Pattern Unit 2 • Operations with Numbers 103 ACTIVITY 2.7 continued Dividing Decimals Getting Ready for the Dance 6 Quickwrite Students have studied long division in earlier grades. The important concept for them in this unit is the placement of the decimal point in the quotient. EXAMPLE 1 Close Reading, Activating Prior Knowledge Ask students to read through the example on their own, and then have a classroom discussion to make sure that all students understand the process. You may want to preclude this example with a few review problems on dividing with whole numbers. SUGGESTED LEARNING STRATEGIES: Quickwrite, Close Reading My Notes 6. Look at 0.72 ÷ 9 set up as if dividing whole numbers. 0.08 ____ 9 0.72 MATH TERMS An algorithm is a set of steps or a procedure used to carry out a computation or to solve a problem. a. Place the answer from Question 5 above the dividend. b. How is the decimal point in the quotient placed in relationship to the decimal point in the dividend? Answers may vary. Sample answer: The decimal point in the quotient is placed directly above the decimal point in the dividend. READING MATH quotient ________ divisor dividend A quotient is calculated by dividing a dividend by a divisor. EXAMPLE 1 The decorations for the dance will cost $272.64. What is each student’s share of the cost of the decorations? Step 1: Step 2: TRY THESE A Think/Pair/Share Determine the total number of students. 32 (given on the previous page) Divide the cost of $272.64 by 32. The algorithm is the one used for dividing whole numbers. 8.52 ______ 32 272.64 - 256 166 - 160 64 - 64 0 Solution: Each student will pay $8.52 for dance decorations. Notice that the decimal point is placed in the answer directly above the decimal point in the dividend. P Page 105 may be the first eexposure to estimation com using compatible numbers for some students. Have a class discussion to help students understand the difference between using compatible numbers and rounding. Help students clarify when to use each method. TEACHER TO TEACHER © 2010 College Board. All rights reserved. ACTIVITY 2.7 Continued TRY THESE A Find the quotient. ______ a. 25 168.75 6.75 _____ b. 7 339.5 48.5 104 SpringBoard® Mathematics with Meaning™ Level 1 MINI-LESSON: Compatible Numbers Estimation using compatible numbers is often used with division of decimals. Guide a discussion with students so that it leads to an agreement on a good definition of compatible numbers. Ask students to rewrite the following division problems using compatible numbers and then estimate the quotient. 132 ÷ 54 878 ÷ 32 839.45 ÷ 11.6 104 SpringBoard® Mathematics with Meaning™ Level 1 12/16/09 5:55:45 P 1 © 2010 College Board. All rights reserved. 103-110_SB_MS1_2-7_SE.indd 104 Dividing Decimals ACTIVITY 2.7 Getting Ready for the Dance ACTIVITY 2.7 Continued continued 7 Create Representations SUGGESTED LEARNING STRATEGIES: Guess and Check, Create Representations, Think/Pair/Share, Quickwrite, Group Presentation 8 Think/Pair/Share My Notes It is helpful to estimate a quotient beforehand to see if your solution makes sense. One method of estimating is to use compatible numbers. For example, to divide 272.64 by 32, think of the closest compatible numbers and find the quotient mentally. 9 Create Representations Ask students if their estimates are close to the actual quotient. Have a discussion about why some students make estimates that are closer than others. The exact quotient for question 9 is 3.543125. Students may have questions about what to do with the remainder as they work through the problem. MATH TERMS 272.64 270 32 30 270 ÷ 30 = 90 7. The deejay will cost $168.64. What is each student’s share of the cost? Use compatible numbers to estimate, then solve. Estimates will vary. Sample answer: 160 ÷ 40 = 4; $5.27 Compatible numbers are close in value to the original numbers in an arithmetic problem and are used in place of the original numbers to make it easier to estimate an answer to the problem. 8. Is your answer in Question 7 an exact number of dollars and cents? If not, what should you do with the remainder? It is exact; there is no remainder. 0 Think/Pair/Share Some students may decide to “round down” since the remainder after carrying out the division process 10 through two decimal places is ___ 32 5 ___ or 16 , which is less than one-half. 9. Refreshments for the dance will total $113.38. Estimate first, then find each student’s share of the cost. Estimates may vary. Sample estimate: 120 ÷ 30 = 4. Answers may vary. Two possible answers: $3.54 or $3.55. 10. Is your answer in Question 9 an exact number of dollars and cents? If not, what should you do with the remainder? When a division problem that involves a decimal has a zero remainder, we say that the quotient is a terminating decimal. © 2010 College Board. All rights reserved. Answers may vary. Sample answer: No; round down or round up. a Quickwrite Have a guided discussion about when you might round up, down, or maybe not round at all when dealing with money. Ask whether the group of students will have enough money for refreshments if they round down in Question 9. 11. When there is a remainder in a division problem involving money, how should you round the quotient? Answers may vary. Sample answer: In a case like Question 9, the answer should be rounded up to make sure that the entire cost is covered. ($3.54 × $32 = $113.28 and $3.55 × 32 = $113.60) 12. The students want to get the best buys on the refreshments. They found three brands of pretzels at the grocery store. Brand Bob’s Pretzels Crunchy Pretzels Kettle Pretzels Size 12 oz 14 oz 16 oz Cost $1.68 $1.89 $2.08 Cost of 1 Ounce $0.14 b Group Presentation, Debriefing This question is an introduction to rates, which are taught in unit 4. Debrief the class on their responses to 12b. $0.135 $0.13 a. For each brand, list the cost of an ounce of pretzels. b. Which brand do you think will be the best buy? Explain. Kettle Pretzels; they cost the least per ounce Suggested Assignment Unit 2 • Operations with Numbers © 2010 College Board. All rights reserved. PM 103-110_SB_MS1_2-7_SE.indd 105 105 CHECK YOUR UNDERSTANDING p. 110, #1–3 12/16/09 5:55:48 PM UNIT 2 PRACTICE p. 136, #42–43 Ask students if they have A eever noticed the small signs unde underneath the products at the grocery store that tell how much one unit of the product costs. Ask a grocery store manager for some of those signs to use in your class discussion. TEACHER TO TEACHER Unit 2 • Operations with Numbers 105 ACTIVITY 2.7 continued Dividing Decimals Getting Ready for the Dance c Use Manipulatives Guide students through this exercise. You may want to ask them what they think the solution is before marking the counter. Many students may think that the solution is 3.5. SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Quickwrite, Close Reading My Notes 13. The counter in the concession stand needs to be painted before the dance. The counter is seven feet long and the students want to alternate the colors blue and green in 0.5 ft sections. 0 Use inverse operations by asking students what they would multiply 0.5 by to get 7. 2 3 4 5 6 7 a. How many 0.5 ft. sections will be painted? Mark the counter to show the sections. 14 sections d Quickwrite This question b. Complete this number sentence: 7 ÷ 0.5 = shows the students the value of moving from making a model to the using of an algorithm when dividing by a decimal. EXAMPLE 2 Close Reading Guide students through Example 2. Make sure that they understand that you can always change the divisor to a whole number by multiplying by some power of ten. Ask the students to tell you the smallest power of 10 that they can use to make 1.2 a whole number. 1 Counter 14 14. Suppose you wanted to divide 28.56 by 2.3. Why would it be difficult to use a pictorial model to answer this question? Answers may vary. Sample answer: It would be difficult to draw a model exactly 28.56 units long and divide it into sections 2.3 units long. EXAMPLE 2 WRITING MATH 48 ÷ 1.2 can also be written as ___ 48 ___ or 1.2 48. 1.2 Divide 48 by 1.2. Step 1: Step 2: Step 3: Recall that multiplying a fraction by 1 does not change its value because of the Property of One. If 7 3 __ you multiply the fraction __ 5 by 7 : 7 ___ 3 __ 21 , the resulting __ × = 5 7 35 3 fraction is equivalent to __ 5. Write the division problem as a fraction. 48 ___ 1.2 Rewrite the denominator so that it is a whole number. The smallest number that we can use to do this is 10. 1.2 × 10 = 12 Since you multiplied the denominator by 10, you multiply the numerator by 10 so that you have equivalent fractions. © 2010 College Board. All rights reserved. ACTIVITY 2.7 Continued 48 × ___ 10 = ____ 480 ___ 1.2 Step 4: 10 12 Divide 480 by 12. 40 ____ 12 480 - 48 00 Solution: The quotient is 40. 106 SpringBoard® Mathematics with Meaning™ Level 1 12/16/09 5:55:52 P 1 © 2010 College Board. All rights reserved. 103-110_SB_MS1_2-7_SE.indd 106 106 SpringBoard® Mathematics with Meaning™ Level 1 Dividing Decimals ACTIVITY 2.7 Getting Ready for the Dance ACTIVITY 2.7 Continued continued TRY THESE B Guess and Check (a), Think/Pair/Share (b, c) Students should multiply by the smallest power of ten possible in order to write the denominator as a whole number. SUGGESTED LEARNING STRATEGIES: Guess and Check, Think/Pair/Share, Create Representations, Close Reading My Notes TRY THESE B 5.8364 , find the smallest number by which you a. For the fraction ______ 2.173 can multiply the denominator to make it a whole number. Then write an equivalent fraction with a whole-number denominator. 5836.4 a. 1000; ______ 2173 e Create Representations, Write each division problem as a fraction, rewrite it as an equivalent fraction with a whole-number denominator, and then divide. ______ b. 2.7 13.041 4.83 EXAMPLE 3 Close Reading Students are introduced to adding zeros as placeholders in the dividend. Make sure that students are moving the decimal point in the divisor and dividend correctly. Ask them to estimate their quotient and check to see if their answer is reasonable. ______ c. 0.52 6.5676 12.63 15. The decorating committee spent $22.95 on ribbon, which costs $0.85 per meter. How many meters of ribbon did the committee buy? 27 meters Instead of multiplying the dividend and divisor by the same power of 10 to make them both whole numbers, you can move the decimal points. EXAMPLE 3 ____ Divide: 0.13 72.8 © 2010 College Board. All rights reserved. Step 1: _____ 0.13. 72.8 Step 2: First move the decimal point in this divisor 2 places to the right to make it a whole number. Next move the decimal point in this dividend the same number of places. Notice that a zero must be inserted in the dividend. Step 3: Then divide. 560 _____ 13 7280 65 78 78 00 Solution: • Multiplying by 10 moves the decimal point one place to the right. • Multiplying by 100 moves the decimal point two places to the right. If students are allowed to u use calculators, then you may need to show them how to enter fractions and long division problems into the calculator correctly. The dividend is always entered first, followed by the operation (÷), followed by the divisor. TEACHER TO TEACHER _______ 0.13. 72.80. 560 Unit 2 • Operations with Numbers 107 12/16/09 5:55:56 PM © 2010 College Board. All rights reserved. PM 103-110_SB_MS1_2-7_SE.indd 107 Some students may benefit by making “loops” as they move the decimal point in the divisor and dividend. Recall the rules for multiplying by powers of 10: Unit 2 • Operations with Numbers 107 ACTIVITY 2.7 Continued ACTIVITY 2.7 continued Dividing Decimals Getting Ready for the Dance TRY THESE C Group Presentation SUGGESTED LEARNING STRATEGIES: Group Presentation, Create Representations, Marking the Text My Notes Suggested Assignment TRY THESE C CHECK YOUR UNDERSTANDING p. 110, #4–5 Find each quotient. _____ a. 0.45 103.5 230 ____ b. 0.31 682 2200 UNIT 2 PRACTICE p. 136, #44–46 16. Sharon wants to decorate each table with confetti. A bag of confetti holds 546 grams. She will scatter 45.5 grams on each table. How many tables will she be able to decorate? 12 tables f Create Representations g Create Representations 17. The deejay will play music for 2.5 hours. Each song lasts approximately 3.75 minutes. What is the total number of songs that can be played during the dance? 40 songs Students will have to convert the hours to minutes to work this problem. Sometimes the dividend is not evenly divided by the divisor. EXAMPLE 4 ___ Find the quotient 0.31 42 to the nearest tenth. Step 1: Move the decimal point to create a whole number. Then divide. Notice the remainder of 15. Step 2: Continue dividing by adding two more zeros after the decimal point. Step 3: There is still a remainder, but now there are enough places to round the quotient to the nearest tenth. 0.3 = 0.30 = 0.300 because 30 ___ 10 = ____ 3 · ___ 300 ___ · 10 = _____ 10 10 100 10 1000 ______ 0.31. 42.00. Solution: The rounded quotient is 135.5. 135. _____ 31 4200. - 31 110 - 93 170 - 153 15 © 2010 College Board. All rights reserved. EXAMPLE 4 Marking the Text Students will find quotients when the remainder is not zero by adding zeros to the dividend so that they have enough decimal places (2) to round their answer to the nearest tenth. Discuss how to predict how many zeros to annex to the right of the decimal point when rounding to the nearest tenth, hundredth, and so on. 135.48 _______ 31 4200.00 - 31 110 - 93 170 - 153 150 - 124 260 - 248 12 108 SpringBoard® Mathematics with Meaning™ Level 1 12/16/09 5:55:59 P 1 © 2010 College Board. All rights reserved. 103-110_SB_MS1_2-7_SE.indd 108 108 SpringBoard® Mathematics with Meaning™ Level 1 Dividing Decimals ACTIVITY 2.7 Getting Ready for the Dance ACTIVITY 2.7 Continued continued SUGGESTED LEARNING STRATEGIES: Group Presentation, Create Representations, Quickwrite, Close Reading, Look for a Pattern TRY THESE D Group Presentation Ask students to share their answers on white boards. Make sure that they are rewriting the dividend and rounding correctly. My Notes TRY THESE D Find each quotient to the nearest tenth. ___ a. 0.22 1.3 ___ h Create Representations b. 6.2 27 5.9 4.4 EXAMPLE 5 Close Reading Help students understand that we sometimes round repeating decimals instead of writing them with the bar over the repeating digits. Ask them why they think it might be more beneficial to round the repeating decimal. 18. After spending two hours planning the dance, Mark, LaNita, and Sam decide to go to the cafeteria and get a snack. The cafeteria sells three containers of yogurt for $1.00. a. How much will Mark, LaNita, and Sam each pay for one container of yogurt? Answers may vary. Sample answers: Two will pay $0.33 and one will pay $0.34, or all three may have to pay 34¢ each. b. What do you notice about the quotient? Answers may vary. Sample answer: The 3 repeats in the quotient. MATH TERMS A repeating decimal is a decimal that has one or more digits to the right of the decimal point that repeat forever. SStudents may be familiar 22 as an approximaw with ___ 7 π If not, they will be tion for π. introduced to the concept of π in Unit 5. In either case, you may want to challenge students to 22 as a repeating decimal. express ___ 7 TEACHER TO TEACHER Numbers like these are called repeating decimals. Repeating decimals can either be written with a bar over the digits that repeat or can be rounded. _______ © 2010 College Board. All rights reserved. EXAMPLE 5 [3.142857]. Be sure students Divide 7 by 11. Step 1: 22 is an approxiunderstand that ___ 7 22 mation for π and that, while ___ 7 repeats, π is a nonrepeating, non-terminating decimal. Set up the problem and divide. 0.6363 ______ 11 7.0000 - 66 40 - 33 70 - 66 40 - 33 7 Step 2: Notice the repeating digits in the quotient. Solution: Th _e digits 63 repeat in the quotient, so it can be written with a bar as 0.63 or rounded to 0.64. Unit 2 • Operations with Numbers 12/16/09 5:56:01 PM © 2010 College Board. All rights reserved. 103-110_SB_MS1_2-7_SE.indd 109 PM 109 Unit 2 • Operations with Numbers 109 ACTIVITY 2.7 Continued ACTIVITY 2.7 continued Dividing Decimals Getting Ready for the Dance TRY THESE E Look for a Pattern Suggested Assignment My Notes TRY THESE E CHECK YOUR UNDERSTANDING p. 110, #6–8 __ a. 3 5 − 1.6 or 1.67 __ b. 11 2 — 1.18 or 1.182 UNIT 2 PRACTICE p. 136, #47–50 CHECK YOUR UNDERSTANDING 1. There are 5 groups of two-tenths in 1. Write your answers on notebook paper. Show your work. 5. Sam paid $5.75 for 2.3 pounds of apples. What was the cost for one pound? 1. Use a model to show how many groups of 0.2 are in 1. 6. Sam is saving $5.75 per week to buy a CD player that costs $46. How many weeks will he have to save before he can buy the CD player? 5 and ___ 14 . 7. Compare the quotients of __ 4 11 a. How are they alike? 2. Estimate the quotient: 384.72 ÷ 19.475. Explain your process. 3. Elliott is training for a 17.5 km race. His track coach has separated the course into 2.5 km sections. Into how many sections is the course separated? 8. MATHEMATICAL Why does moving the R E F L E C T I O N decimal point in the divisor and the dividend when dividing by a decimal work? © 2010 College Board. All rights reserved. 4. A jar contains 189 grams of mustard. How many 4.5-gram portions can be made? b. How are they different? 2. Answers may vary. Sample answer: By using compatible numbers, the estimate of the quotient is 380 ÷ 20 = 19 or 380 ÷ 19 = 20. 3. 7 sections 4. 42 portions 5. $2.50 6. 8 weeks a. Answers may vary. Sample answer: Both are decimals. 110 SpringBoard® Mathematics with Meaning™ Level 1 103-110_SB_MS1_2-7_SE.indd 110 b. Answers may vary. Sample 5 answer: __ is a terminating deci4 14 is a repeating decimal. mal; ___ 11 8. Answers may vary. Sample answer: Moving the decimal point the same number of places in the divisor and the dividend is the same as multiplying them both by the same number. This is the Property of One at work in a different way. 110 SpringBoard® Mathematics with Meaning™ Level 1 12/16/09 5:56:04 P © 2010 College Board. All rights reserved. 5 14 = 1.27 7. __ = 1.25; ___ 4 11