Math and FUNDRAISING. Ex. 73, p


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1 Standards Preparation Connect 2.7 KEY VOCABULARY leading digit compatible numbers For an interactive example of multiplying decimals go to classzone.com. Multiplying and Dividing Decimals Gr. 5 NS 2.1 Add, subtract, multiply, and divide with decimals; add with negative integers; subtract positive integers from negative integers, and verify the reasonableness of the results. Before you multipled and divided fractions. Now you will multiply and divide decimals to prepare for Grade 6 Standard NS 1.4. You can represent 1 whole with a grid. The width of each row or column is 0.1. Each small square represents The area selected at the right represents seven tenths of 1.3, or There are 91 squares selected, so KEY CONCEPT Multiplying Decimals Words Multiplying Decimals 2 decimal places 12 decimal places 4 decimal places Multiply decimals as you would whole numbers. Then place the decimal point in the product. The number of decimal places in the product is equal to the sum of the number of decimal places in the factors. Numbers E XAMPLE 1 CHECK REASONABLENESS To check that the product in Example 1 is reasonable, round each factor to the place value of the leading digit, and then multiply. The leading digit of a number is the first nonzero digit Round to the nearest whole number Round to the nearest tenth. Because , the product in Example 1 is reasonable. Students should be familiar with the rules for multiplying and dividing whole numbers in order to multiply and divide decimals. Multiply decimals as you would whole numbers. Count the total number of decimal points in the factors. Place this number of decimal places in the product. When dividing decimals, the divisor must be a whole number. If necessary, multiply the divisor by a power of ten to make it a whole number. Multiply the dividend by the same power of ten. Line up the decimal point in the quotient with the one in the dividend and divide as you would with whole numbers. Math and FUNDRAISING Ex. 73, p. 111 For Your Notebook 2.7 Multiplying and Dividing Decimals Transparency Available Find the product or quotient ,500, , How much does Sam earn for mowing 5 lawns if he earns $18 for each lawn? $90 Transparency Available Promotes interactive learning and notetaking skills, pp Suggested Number of Days Basic Average Advanced 2 Days 2 Days 2 Days Block: 1 Block See Teaching Guide/Lesson Plan in Chapter 2 Resource Book, pp Big Idea 3, p. 66 How can you tell how many decimal places are in the product ? Tell students they will learn how to answer this question by finding the sum of the number of decimal places in the factors. Use decimal squares to model the product in the lesson opener. The digits and the number of decimal places in a decimal product are determined by the products of the numerators and denominators of corresponding fractions. For example, } 3 3 } 21 5 } }
2 E XAMPLE 2 Multiplying Decimals Find the product. Check that your answer is reasonable In checking the answer, why should you not round both factors to the nearest whole number? 0.41 would round to 0, giving an estimate of 0. An Animated Math activity in which students choose the appropriate operator tile to form an expression equal to a decimal is available online for Example 1. This activity is also available on the Power Presentations CDROM. Find the product. Check that your answer is reasonable. a. classzone.com Math b Why can you drop the ending zero in Part (a)? The cost of 18 tickets to attend a concert is $ What is the cost of each ticket? $30.25 Why is a zero placed in the dividend? You must divide to the thousandths place to round to the nearest cent. Multiple Representations In Example 3, students should realize that and are different representations of the same number. ANOTHER WAY You can also multiply decimals by rewriting them as fractions: } } } 100, About the Standards Note that Grade 6 Standard MR 2.6 expects you to round as specified. With money, round to the nearest hundredth to find the nearest cent. a decimal places 1 0 decimal places 2 decimal places After you place the decimal point, you can drop any zeros at the end of an answer. b decimal places 1 2 decimal places 5 decimal places c Answer c Answer GUIDED PRACTICE for Examples 1 and 2 Write a zero before the 7 as a placeholder so that the number has five decimal places. Find the product. Then check that your answer is reasonable DIVIDING DECIMALS To divide decimals by whole numbers, you can use long division. Line up the decimal points in the quotient and the dividend. E XAMPLE 3 Dividing a Decimal by a Whole Number TEAM PHOTOS Your baseball team is ordering photo trading cards of each player on your team. A set of 25 cards costs $ Find the price of each card. Round to the nearest cent. SOLUTION To find the cost per card, you need to divide. STEP 1 Divide $15.95 by 25. STEP Line up decimal point in 25q }}} quotient with decimal point in dividend Divide as you would with whole numbers. Round to the nearest cent. $.638 $ Chapter 2 Fraction and Decimal Operations Example 2 In this example, students apply Standard to multiply with decimals. They can also verify the reasonableness of the results. Use estimation to check reasonableness. In part(a), estimate using In part(b), estimate using c Answer The price of each card is about $.64. Catcher Catcher Kevin Baker Challenge advanced students to use decimal squares to create a model depicting the product
3 About the Standards Note that Grade 6 Standard MR 2.1 asks you to verify the reasonableness of results. AVOID ERRORS Don t forget to write zeros as placeholders and line up the decimal point in the quotient with the decimal point in the dividend. To check that the quotient in Example 3 is reasonable, you can use leading digits and compatible numbers. Compatible numbers are numbers that make a calculation easier. 25q } q } Because , the quotient in Example 3 is reasonable. DIVIDING BY A DECIMAL Notice the pattern in the equations below The quotient remains the same when the divisor and the dividend are multiplied by the same power of 10. This is because the quotients can be written as equivalent fractions multiplied by } 10, or 1. This fact can 10 help you to divide decimals. KEY CONCEPT Dividing by a Decimal Words E XAMPLE 4 a. 3.17q } b. 1.9q } Dividing Decimals q } q } GUIDED PRACTICE for Examples 3 and 4 For Your Notebook When you divide by a decimal, multiply both the divisor and the dividend by a power of ten that will make the divisor a whole number. 0.7 Numbers 12.5q } q } 87.5 Round divisor to place value of leading digit. Round dividend to a compatible number. Multiply the divisor and dividend by 100 by moving both decimal points two places to the right. Line up decimal points in dividend and quotient. Multiply the divisor and dividend by 10 by moving both decimal points one place to the right. Find the quotient. Round to the nearest hundredth if necessary Multiplying and Dividing Decimals 109 Find the quotient. a b If a decimal divisor has n decimal places, what power of ten must you multiply the divisor by to make it a whole number? n Encourage students to insert a zero as the leading digit in products that are less than 1. When doing long division, suggest that students place the decimal point in the correct location immediately after moving the decimal points in the divisor and dividend. This will help them avoid placing the decimal incorrectly in the quotient. As students include examples in their notebooks, encourage them to include a note for each example that they moved decimal points by multiplying the divisor and dividend by a power of ten. Students have learned to multiply and divide decimals. To bring closure, have students answer these questions: 1. Essential Question: How can you tell how many decimal places are in the product ? Find the sum of the number of decimal places in the factors. There are 3 decimal places in the product. 2. By what power of 10 do you multiply the divisor and dividend to divide 1.68 by 0.7? Find the quotient of ; 2.4 Encourage students to draw arrows as shown in Example 4 when they are dividing decimals.
4 Answer Transparencies available for all exercises Basic: Day 1: Exs. 1 14, 23, 40 Day 2: EP p. 693 Exs Exs , 72 80, Average: Day 1: MCP p. 65 Exs Exs. 1 6, 14 18, 23, 43 46, Day 2: MCP p. 65 Exs Exs , 41, 42, 51 54, 59, 64 70, 76, Advanced: Day 1: MRSPS p. 26 Exs. 4, 5 Exs. 1 5, 19 23, 47 50, Day 2: MRSPS p. 26 Exs. 6 8 Exs , 42, 55 59, 65 71, 76, 78, 80 88* Block: MCP p. 65 Exs Exs. 1 6, 14 18, 23, 32 35, 41 46, 51 54, 59 70, 76, For a quick check of student understanding of key concepts, go over the following exercises: Basic: 8, 12, 26, 28, 76 Average: 16, 18, 34, 76, 82 Advanced: 20, 22, 38, 82, 84 Student Edition, p. 693 Chapter 2 Resource Book: Practice Levels A, B, C, pp An easily readable reduced practice page (with answers) for this lesson can be found on pp. 66E 66F. 2.7 EXERCISES A SEE EXAMPLES 1 AND 2 on pp for Exs SEE EXAMPLES 3 AND 4 on pp for Exs The decimal point must move four places; The decimal point must move the same number of places in the divisor and the dividend; HOMEWORK KEY SKILLS PROBLEM SOLVING REASONING VOCABULARY Round to the place value of the leading digit. 5 MULTIPLE CHOICE PRACTICE Exs. 23, 42, 76, 78, HINTS AND HOMEWORK HELP for Exs. 7, 25, 79 at classzone.com VOCABULARY Using compatible numbers, what dividend and divisor would you use to estimate the quotient ? 56 and 7 6. WRITING Explain how you can use fractions to multiply decimals. Give an example. Write the decimals as fractions, and multiply. Sample answer: } 1 3 } 7 5 } 11 3 } 7 5 } } FINDING PRODUCTS Find the product. Check that it is reasonable MULTIPLE CHOICE What is the value of the expression ? B A B C D FINDING QUOTIENTS Find the quotient. Round your answer to the nearest hundredth if necessary ERROR ANALYSIS Describe and correct the error made in finding the product or quotient MULTIPLE CHOICE Which quotient is not equal to the others? q } q } A B C D ESTIMATING PRODUCTS Round to the place value of the leading digit to estimate the product Chapter 2 Fraction and Decimal Operations For Exercises 7 22, some students may need help when counting decimal places. Make sure that all students understand that they should count from the right when placing the decimal point in a product. Because students read numbers from the left, they may find this confusing. Remind them that the rightmost digit determines the place value of the decimal. A
5 B 69. Multiply the previous number by 5; 75, 375, Divide the previous number by 2; , , Multiply the previous number by 1.5; , , SEE EXAMPLE 3 on p. 108 for Exs pages; 36 min; pages in 1 hour pages left, more minutes. ESTIMATING QUOTIENTS Use compatible numbers to estimate the quotient MODELING Copy the number line below. Use your copy to show how to represent Use the fact that division is repeated subtraction. See margin xy ALGEBRA Evaluate the expression when p and q p p p q p p 5.08 p p 63. q p 1.12 p p q p p q q OPENENDED Describe the pattern Position n n used to relate the value of a term to its position n in the list. Create Value of term ? and describe your own pattern. Multiply n by 1.5; check students work. DESCRIBING PATTERNS Describe the pattern. Write the next three numbers , 0.6, 3, 15, , 0.5, 0.25, 0.125, , 3, 4.5, 6.75,... CONNECT SKILLS TO PROBLEM SOLVING Exercises will help you prepare for problem solving. Write the expression you would use to solve the problem. 72. Grapes are on sale for $1.25 per pound. You buy 1.8 pounds. How much do you pay for the grapes? Your class holds a car wash to raise money for a field trip. You charge $4.75 per car and earn $779. How many cars does your class wash? You have a wooden board that is 30 inches long. You cut it into pieces that are 3.75 inches long. How many pieces of wood do you have? Wanda buys 3 CDs. With tax, each CD costs $ How much does she spend? MULTIPLE CHOICE You knit a scarf using 4 balls of yarn. The yarn costs a total of $ How much does each ball of yarn cost? B A $.16 B $6.22 C $20.88 D $99.52 California for problem solving help at classzone.com 77. READING RATE Tyler can read about 1.25 pages per minute. Can he finish reading a 65page book in one hour? About how long would it take him to finish? How many pages would he have left after reading a 120page book for one hour? How much more time would he need? Explain. California for problem solving help at classzone.com Exercises 7 22 Students should realize that when using estimation to check for reasonableness, if they round one factor to 0, the estimate of the product will be 0. Thus, it is advisable to avoid rounding a factor to 0. Exercises Some students may multiply the divisor by a power of 10, but not the dividend. Remind students that they must multiply the divisor and the dividend by the same power of 10. Exercises Use compatible numbers to estimate the quotient means students should use numbers that will make it easy to estimate the quotient. You may wish to ask students to explain their answers to see if they used different compatible numbers and why Multiplying and Dividing Decimals 111 Exercise 68 In this exercise, students apply Standard to develop a generalization of the results to fi nd the value of the nth term. In positions 1, 2, 3, and 4, the value of each term is one and a half times the position number. Generalize to the value of the nth position. The value of the nth term is one and a half times n, or 1.5n.
6 Exercise 78 Students should reason that the number of minutes it will take to complete the marathon is greater than The division in choices A and B will give numbers that are less than The subtraction in choice C would not make sense because you would be subtracting minutes from miles. Exercise 82 To complete this exercise, students must understand the meaning of rates. To find the cost per ounce of cereal means to find the cost of 1 ounce of cereal. 85. Sample answer : No, when dividing, the quotient times the divisor must equal the dividend, so if the quotient is a whole number and the divisor b is a whole number, whole number times whole number is always equal to a whole number, yes, ; yes, ; yes, Start at 1.5 and then move back 0.2 spaces until you reach 0. The quotient is the number of times you have to take away 0.2 from 1.5 to reach 0; Trade one dollar for 10 dimes. Separate the dimes into groups of 2. The quotient is the number of groups. 83. Yes; no; $27.75; $3.50 per gallon, $3.70 per gallon, 3.19 p $ a mm; mm; mm 78. MULTIPLE CHOICE A wheelchair racer takes about 5.95 minutes to travel 1 mile. Which expression shows how long it would take the racer to finish a 26.2mile marathon? D A B C D COMETS Encke s comet takes about 3.3 years to orbit the sun. Halley s comet takes about times as long. About how long does Halley s comet take to orbit the Sun? Round to the nearest tenth yr 8 0. SHORT RESPONSE Copy the division problem below. Explain how to use estimation to place the decimal point in the quotient q } WRITING Explain how to use a number line model to find Then explain how to use dollars and dimes to model $ MULTISTEP PROBLEM It costs $4.75 for a 25 ounce box of cereal and $3.45 for a 15 ounce box. a. What is the cost per ounce of each box of cereal? b. How many times the cost per ounce of the smaller box is the cost per ounce of the larger box? Explain. about 1.21 times; $.23 4 $.19 ø GASOLINE At one time, the price of gasoline ranged from $3.19 to $3.59 per gallon. A car needs 8.7 more gallons of gasoline to have a full tank. Is it possible that the cost to fill the tank is $30.45? What is the least possible cost to fill the tank? Explain how you solved the problem. 84. MEASUREMENT The objects below have been magnified. Their actual width w is much smaller. Measure each object in millimeters. Pollen w width shown Sample answer: 10 p 3 is 30, so the decimal should be placed between the 3 and the 2. Algae w width shown Athletes use racing chairs. $.19, $.23 Blood cell w width shown C a. Use your measurements to find the actual width w of each object. b. List the actual objects in order from least to greatest width. blood cell, pollen, algae 85. xy REASONING The quotient of a and b is a whole number. Can a be a decimal number if b is a whole number? Can b be a decimal number if a is a whole number? Can both a and b be decimal numbers? Can the quotient be greater than a? Justify your answers and give examples. See margin. 112 Chapter 2 Fraction 5 MULTIPLE and Decimal CHOICE Operations PRACTICE 5 HINTS AND HOMEWORK HELP at classzone.com
7 86. decimals less than 5.29; decimals greater than 5.29; 5.29; 5.29 divided by 2.3 is 2.3, so decimals smaller than 5.29 will be less than 2.3 when divided, and decimals greater than 5.29 will be greater than 2.3 when divided. NS 2.1 Gr. 5 NS 1.2 Gr. 4 NS REASONING Which decimal number(s) can you divide by 2.3 to make the quotient less than 2.3? greater than 2.3? equal to 2.3? Explain your reasoning. 87. CHALLENGE Runner A runs at a pace of 6.54 miles per hour for 1.3 hours. Runner B runs at a pace of 7.08 miles per hour for 1.1 hours. Who runs farther? How much longer must the other runner continue to run to match the distance of the first runner? Round to the nearest hundredth of an hour. Explain how you solved the problem. Runner A; 0.10 h; and , , then, CHALLENGE A store sells crafts. The first week it prices 5 baskets at $20 each. After each week, the price of each unsold basket drops to 0.9 times the previous week s price. All 5 baskets sell in the same week for a total of at least $53. In what week(s) could they have been sold? week 7 or before CALIFORNIA STANDARDS SPIRAL REVIEW 89. What is the value of } 5 1 } 7? (p. 75) C 6 10 A 3 } 4 B 6 } 5 C 23 } 15 D 49 } Which number is 6 } 1 wr itten as a decimal? (p. 40) A 8 A B 6.18 C 6.81 D During 4 days of basketball practice, you shoot a total of 140 free throws. You shoot the same number of free throws each day. How many free throws do you shoot per day? (p. 671) C A 25 B 30 C 35 D 40 QUIZ for Lessons Find the product or quotient. (p. 88, 94) 7 1. } } } 2. 2 } } 5 4 } } 15 7 } } } 2 5 } } } } 6. 5 } 5 4 } 1 81 } } } } 3 5 } 4 6 } Find the sum or difference. Then check your answer. (p. 102) Find the product or quotient. Round quotients to the nearest hundredth if necessary. Then check your answer. (p. 107) BOOKS Ryan had 180 books. He gave away } 1 of his books and sold } 3 of 3 4 the rest. How many books did he sell? Explain your reasoning. (p. 88) 90 books; } 1 ( 18 0 ) 5 120, } Find the product or quotient Estimate the product of by rounding each factor to the place value of the leading digit Jose bought 2.3 pounds of fish for $ Use compatible numbers to estimate the cost per pound. $9.50 Online Quiz classzone.com An alternate quiz for Lessons is available online in multiple choice format. Practice A, B, C in Chapter 2 Resource Book, pp Study Guide in Chapter 2 Resource Book, pp Practice Workbook, pp Additional challenge is available in the Chapter 2 Resource Book, p. 77. An easily readable reduced copy of the quiz on Lessons (with answers) from the Assessment Book can be found on pp. 66G 66H. EXTRA PRACTICE for Lesson 2.7, p Multiplying ONLINE and QUIZ Dividing at classzone.com Decimals 113 } 1 } D } } } } } D A
8 MIXED REVIEW of Skills and Problem Solving California classzone.com Students who need more review and practice should see the following lessons in the Exercise 1: Lesson 2.5 Exercise 2: Lesson 2.7 Exercise 3: Lesson 2.7 Exercise 4: Lesson 2.7 Exercise 5: Lesson 2.6 Exercise 6: Lesson 2.7 Exercise 7: Lesson 2.4 Exercise 8: Lesson 2.5 Multiple Choice Practice for Lessons You have a piece of wood that is 51 inches long. For a craft project, you need pieces that are each 4 1 } 2 inches long. 51 in. How many pieces that are 4 1 } 2 inches long can you cut from the wood? NS 2.1 A A 11 B 12 C 229 D What is the value of the expression x p when x ? Gr. 5 NS 2.1 B A B C D What is rounded to the nearest hundredth? Gr. 5 NS 2.1 D A B 0.03 C D The table shows the amount of money raised by 3 homerooms at a school fair. How much more was the total raised by homerooms 6A and 6C than by homeroom 6B? Gr. 5 NS 2.1 B Homeroom 6A 6B 6C Money raised $ $ $ A $14.71 B $ C $ D $ What is the value of ? Gr. 5 NS 2.1 B A B C D The recipe shown makes enough garlic mashed potatoes for 5 servings. You want to make 150 servings for a garlic festival. How many cups of water and cups of milk do you need? NS 2.1, MR 3.1 B Garlic Mashed Potatoes 4. In the table below, which description relates a position n to its value? NS 2.2, MR 1.2 C Position n Value 1 2} Garlic Mashed Potatoes 1 box garlic mashed potato mix 1 tbsp butter 2 1/3 c water 1/3 c milk 3 7} n? A Multiplyn by 1 } 1. 2 B Dividen by 1 } 1. 2 A 67} 2 cups water and 9 } 2 cups milk 3 3 B 70 cups water and 10 cups milk C 116} 2 cups water and 16 } 2 cups milk 3 3 D 350 cups water and 50 cups milk C Multiplyn by 2 1 } 2. D Dividen by 2 1 } What is the value of 2 3 } } 12? NS 2.2 A 192 } 875 B 20 } 21 C C 1 1 } 20 D 52 1 } Chapter 2 Fraction and Decimal Operations Exercise 4 In this exercise, students formulate and justify mathematical conjectures based on a general description, as called for in Standard. Formulate a conjecture. Justify the conjecture using reasoning. The values are greater than the position numbers, so n is multiplied to obtain the value of position n. Each value is more than double its position number. The only reasonable choice is C. The value of each number is 2 1 } 2 times its position. Exercise 7 In this exercise, students evaluate the reasonableness of the solution in the context of the original situation, as called for in Standard. You want to make , or 30 times the given recipe amounts. Calculate } 3 c u p s w a t e r a n d } 3 c u p milk. Check the reasonableness of the answer using estimation: cups water and } cups milk is needed.
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