To estimate products of decimals, round each number. Then multiply.

Transcription

1 Multi-Part Lesson A B C D E Main Idea Estimate the product of decimals and judge the reasonableness of the results. MA.6.A.5.3 Estimate the results of computations with fractions, decimals, and percents and judge the reasonableness of the results. Estimate Products SKATEBOARDING The record for the greatest distance traveled on skateboard in 24 hours was set by James Peters in He traveled about 7.6 miles per hour. 1. Round 7.6 to the nearest whole number Estimate how many miles James Peters traveled in 24 hours. about 24 8 or 192 miles 3. Is your estimate higher or lower than the actual distance he traveled? Explain. The estimate is higher than the actual distance he traveled because 7.6 was rounded up. To estimate products of decimals, round each number. Then multiply. Multi-Part Lesson A Objective A B C D E Estimate Products Students will estimate the product of decimals and judge the reasonableness of the results. MA.6.A.5.3 Estimate the results of computations with fractions, decimals, and percents and judge the reasonableness of the results. Key Concept boxes highlight definitions, formulas, and other important ideas. Multiple representations words, symbols, examples, and models help students understand the concepts. Round Decimals To round a decimal, first underline the digit to be rounded. Then look at the digit to the right of the place being rounded. If the digit is 4 or less, the underlined digit remains the same. If the digit is 5 or greater, add 1 to the underlined digit. After rounding, change all digits after the underlined digit to zeros. Estimate Products Using Rounding Estimate Round to the nearest whole number to make it easier to compute mentally Round 8.7 to 9. Round 2.8 to 3. The product is about FOCUS Activity Review how to round decimals to the nearest whole number. Have students draw a number line from 1 to 10. Then ask them to locate and label 1.5, 2.5, 3.5, and so on through 9.5. Ask: Is 1.2 closer to 1 or to 2? How can you tell? 1; The distance between 1.2 and 1 is less than the distance between 1.2 and 2. Is 4.75 closer to 4 or to 5? How can you tell? 5; The distance between 4.75 and 5 is less than the distance between 4.75 and 5. Use the number line to round the following numbers to the nearest whole number: 5.67; 3.41; ; 3; 10 Estimate each product. a b c ,400 Lesson 27 Tips for New Teachers 7_033_C1L1_ indd 27 1/9/09 12:11:47 PM Rounding If students have trouble remembering the rules for rounding, teach them this rhyme: 5 and above, give it a shove. 4 and below, let it go. 2 TEACH Scaffolding Questions Ask: Suppose James travels at an average of 5.4 miles per hour for 8.25 hours. How would you round the numbers to estimate how far he traveled? Round 5.4 to 5 and 8.25 to 8. About how far would James travel? 40 miles Would your estimate be higher or lower than the actual distance? Explain. Sample answer: Lower; both numbers are rounded down, so the estimated distance is less than the actual distance. Lesson 27

2 Formative Assessment Use the Check Your Progress exercises after each Example to determine students understanding of concepts. Estimate =15 HELICOPTERS One of the first helicopters flew at a speed of 44.7 kilometers per hour. At this speed, about how far could the helicopter travel in 2.5 hours? 135 km RECIPES Anna has $20 to buy 8 pounds of tomatillos for a party. The tomatillos cost $2.89 per pound. Does she have enough money to buy them? Explain your reasoning. No; Sample answer: $2.89 is close to $3. If you round $2.89 to $3, the tomatillos will cost about $24. IWB INTERACTIVE WHITEBOARD READY Focus on Mathematical Content Rounding decimals to the nearest whole number before multiplying is one way to estimate a product. Refer to the digit in the tenths place. If the digit in the tenths place is 5 or higher, the number is rounded to the next whole number. If the digit is less than 5, then the whole number remains the same. Study Tips offer students helpful information about the topics they are studying. Rounding Decimals When rounding decimals, such as to the tenths, the 9 must round up. So, rounded to the nearest tenth is f. Yes; $1 30 = $30 and $2 30 = $60. So, they will earn between $30 and $60. DOGS A greyhound can travel 39.3 miles per hour. At this speed, about how far could a greyhound travel in 6.5 hours? Round 39.3 to 40. Round 6.5 to 7. The greyhound could travel about 280 miles in 6.5 hours. d. MONEY Suppose one U.S. dollar is equal to 5.3 Egyptian pounds. About how many Egyptian pounds would you receive for $48.50? about 250 Egyptian pounds e. ASTRONOMY The Earth is rotating around the Sun about 18.6 miles per second. About how many miles does it travel in 4.8 seconds? about 100 mi SCHOOL SUPPLIES Patrice has $20 to buy 5 binders for her classes. She found binders that cost $4.29 each. Does she have enough money to buy these binders? Explain your reasoning. Estimate. 5 $4 = $20 Estimate 4.29 as 4. 5 $5 = $25 Estimate 4.29 as 5. The actual cost of the binders is between $20 and $25. So, Patrice does not have enough money to buy the binders. f. PROFIT The art club makes a profit of $1.75 on every batch of cookies they sell at a bake sale. The goal is to earn at least $50 selling batches of cookies. They estimate they will need to sell at least 30 batches. Is this estimate reasonable? Explain your reasoning. 28 Chapter 1 Multiply and Divide Decimals 027_033_C1L1_ indd 28 1/21/09 8:59:27 28 Chapter 1 Multiply and Divide Decimals

3 indicates multi-step problem Example 1 (p. 27) Example 2 (p. 28) Example 3 (p. 28) Example 1 (p. 27) = = 2, = 1,100 Example 2 (p. 28) Example 3 (p. 28) 20. Her estimate is not reasonable. Since 12 rounds down to 10 and 15.5 rounds to 16 by a lesser amount, her estimate of = 160 will be too low. Differentiated Homework Options Level Assignment Two-Day Option AL Approaching Level 9 21, 36 38, odd, even, 36, 38, 39, 43, 44 OL On Level 9 25 odd, 26 29, 31, 35, 36, BL Beyond Level Estimate each product Sample answers given = 1, = = = = = MONEY A grocery store sells American cheese for $3.89 per pound. About how much would 1.89 pounds of the cheese cost? about $8 8. MEMORY Greg has 52 megabytes of free space left on his MP3 player. He wants to download 7 songs that each use 7.9 megabytes of memory. He estimates that he will need 56 megabytes of memory. Explain why his estimate is reasonable. If 7.9 is rounded to 8, then the 7 songs will require about 56 megabytes of memory. Since 7.9 was rounded up, the actual amount of memory he will use will be less than 56 megabytes. Media Downloading... = Step-by-Step Solutions begin on page R1. Extra Practice is on page EP2. Estimate each product Sample answers given = = = = = = FRUIT On average, the U.S. produces 36.5 million tons of fruit each year. About how much fruit does it produce in 2.25 years? about 80 million tons 19. SCIENCE A single year on Saturn is equal to 29.4 years on Earth. About how many Earth-years are equal to 3.2 years on Saturn? about CRAFTS Lisha is making headbands using ribbon. She would like to make 12 headbands. Each one requires 15.5 inches of ribbon. She estimates that she will need to buy 160 inches of ribbon. Is her estimate reasonable? Explain your reasoning. 21. GIFT CARDS Miguel received a $50 gift card to a bookstore. He would like to buy 3 books that cost $15.75 each including tax. He estimates that he cannot buy all three books because each book costs about $20, and all three books would cost $60. Is his estimate reasonable? Explain your reasoning. His estimate is not reasonable. He rounded $15.75 to $20. If he rounded $15.75 to $16, the estimate would be $48 for the three books. B Use estimation to determine whether each answer is reasonable. If the answer is reasonable, write yes. If not, write no and provide a reasonable estimate = yes = no; = 50 no; = yes Lesson 29 _033_C1L1_ indd 29 10/1/10 3:19:41 PM 9 21, , 38, 43, 44 3 Check Your Understanding exercises are intended to be completed in class. Example references show students where to look back for review. In homework exercises in Practice and Problem Solving, the Example references function in the same way. AL Alternate Teaching Strategy If students have trouble estimating the product of decimals, Then use one of these reteach options: 1 Quick Review Math Handbook, p Have them use a number line to practice rounding decimals to the nearest whole number. PRACTICE The Alternate Teaching Strategy provides suggestions for remediation for students who did not grasp the concept. Formative Assessment Use Exercises 1 8 to check for understanding. Then use the table below to customize assignments for your students. Odd/Even Assignments Exercises 9 21 are structured so that students practice the same concepts whether they are assigned odd or even problems. Common Error In Exercises 1 6, 9 11, and 15 17, students may have difficulty multiplying in horizontal form. Remind students that they may rewrite the problems in vertical form. Lesson 29

4 AL OL BL Interpersonal Present these questions to a small group. Why is being able to estimate answers in mathematics important? Sample answers: Sometimes you do not need to find an exact answer; estimates can help you check an answer. Describe a situation in which you estimated an answer. How and why was estimation helpful? Sample answer: Before I solved a problem, I estimated the answer. It helped me decide whether my actual solution was reasonable. When in real life is estimating more helpful than finding the exact answer? Sample answer: to determine how much I have spent when I am shopping Name some real-world situations where estimating an answer is not helpful. Sample answer: In sports, such as a race, an exact time is needed to determine the winner. Visit to rewatch the animated version of Money Challenge. 27. More; her wage was rounded up, so the actual total is less than the estimate = 280; 40 8 = 320; Rounding to the ones gives an estimate closer to the actual product = 376; 90 4 = 360; Rounding to the ones gives an estimate closer to the actual product = 354; 60 6 = 360; Rounding to the ones gives an estimate closer to the actual product. 32. Calories: 400; carbohydrates: 100 g; Vitamin C: 400 mg; calcium: 80 mg 30 Chapter 1 Multiply and Divide Decimals 26. GRAPHIC NOVEL Refer to the graphic novel frame below for Exercises a b. I ve already saved $68 for the $200 video game system. a. How much more does Raj need until he has enough to buy the video game system? Raj needs to save $132 more. b. If he works for 25 hours, will he have enough to buy the video game system? Yes; if Raj works 25 hours, he will make $ MONEY Hannah s hourly wage at the ice cream shop is $5.85. The table shows the number of hours she worked. She estimates her earnings to be $120. Without calculating her actual earnings, determine if her estimate is more or less than her actual earnings. Explain your reasoning. 28. FUEL A car releases 19.6 pounds of carbon dioxide for every 1 gallon of gasoline burned. Estimate the number of pounds of carbon dioxide released if 14.5 gallons is burned. about 300 pounds Day Hours Worked Monday 3.5 Tuesday 4.25 Wednesday 3.75 Thursday 2.5 Friday 4.75 REASONING The accuracy of an estimate depends on the place value the numbers are rounded to. For each exercise, round the first factor to the tens and then the ones before estimating. Which way gives an estimate that is closer to the actual product? NUTRITION The table shows some nutritional Orange Juice (1 cup) facts about orange juice. Estimate each value Calories 112 for 1 quart of orange juice. (Hint: 4 cups is Vitamin C 96.9 mg equal to 1 quart.) Carbohydrates 26.8 g 33. TREES A King Palm can grow about 2.1 feet a year. Estimate the height of the King Palm, in yards, after 15 years. about 10 yards Calcium 22.4 mg REASONING Estimate each product to determine if each calculator answer is correct. Explain your reasoning ; ; yes; no; _033_C1L1_ indd 30 1/21/09 8:59:34 30 Chapter 1 Multiply and Divide Decimals

5 36. Sample answer: Practice C 36. OPEN ENDED Name three decimals with a product that is about CHALLENGE A scooter can travel between 22 and 28 miles on each gallon of gasoline. If one gallon of gasoline costs between $3.75 and $3.95 per gallon, about how much will it cost to travel 75 miles? about $ Suppose your friend multiplied 1.2 and 2.6 and got 31.2 as the product. Is your friend s answer reasonable? Justify your response. No; using rounding, the answer should be close to 1 3 or 3. MA.6.A Green peppers are on sale for $2.89 per pound. Mrs. Moseley bought 1.75 pounds of peppers. About how much did she pay for the peppers? B A. less than $4 B. between $5 and $6 C. between $6 and $7 D. more than $7 40. Medina s school lunch menu is shown. Friday Pizza $1.75 Fruit Punch $0.75 Fish and Fries $2.25 Milk $0.80 Salad $1.15 Pudding $0.85 Which of the following is a reasonable estimate for the cost of two slices of pizza, a salad, and fruit punch? G F. $4 H. $8 G. $6 I. $ Mario and Andrew s hourly charge for mowing lawns is shown. Mario $8.25/hr Andrew $5.85/hr Suppose Mario and Andrew each worked 20 hours. About how much more money did Mario earn? B A. $30 B. $40 C. $60 D. $ SHORT RESPONSE Javier bought 4 pencil toppers at the school store for $3.69 each. He estimated how much he needs to pay and gave the cashier $16. Is Javier s estimation reasonable? Explain your reasoning. yes; 4 $ $4 = $16 H.O.T. Problems require students to use Higher Order Thinking skills to solve problems. 4 ASSESS Have students write how to estimate Are students continuing to struggle with estimating the product of decimals and judging the reasonableness of the results? If Yes If No AL Reteach Worksheet CRM (p. 12) OL Skills Practice Worksheet CRM (p. 13) BL Enrich Worksheet CRM (p. 16) Practice DIVING The table gives the scores for the Men s Synchronized Country Points 10-Meter Diving at the 2008 Olympics. (Lesson 0-2) China How many more points did China earn than Germany? Germany Russian Federation What is the difference in the points China and the United Australia States earned? United States Lesson 31 Follow-Up 7_033_C1L1_ indd 31 1/9/09 12:12:07 PM Remind students to keep any notes regarding estimating products under the appropriate tab in their Foldables. Diagnose Student Errors Survey student responses for each item. Class trends may indicate common errors and misconceptions. 39. A. rounded both numbers down instead of rounding up B. correct C. estimated beyond upper limit of range D. guess 40. F. estimated pizza only G. correct H. estimated cost of all items I. estimated cost of all items plus an extra slice of pizza 41. A. subtracted rounded numbers incorrectly B. correct C. rounded $5.85 to $5 instead of $6 D. subtracted rounded numbers incorrectly Lesson 31

6 Multi-Part Lesson B by Whole Numbers Objective Students will use models to multiply a decimal by a whole number. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Resources Manipulatives base-ten blocks Easy-to-Use Manipulatives Hands-On Activity Tools and Resources Base-Ten Models, p. 1 1 FOCUS A B C D E Write these numbers on the board: Ask: Which number has a 4 in the ones place? 4.12 Which number has no tenths? 1.04 Which number has a 1 in the hundredths place? 3.41 Multi-Part Lesson Percents A B C D E Main Idea Use models to multiply a decimal by a whole number. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. by Whole Numbers The table shows the relationship between decimals and base-ten blocks. You can use base-ten blocks to multiply decimals. Ones (1.0) Tenths (0.1) Hundredths (0.01) One 10-by-10 flat represents 1 or 1.0. One 1-by-10 rod represents 0.1. One 1-by-1 cube represents WALKING Louisa walked her dog 0.2 mile on Monday, Wednesday, and Friday. Find the total distance Louisa walked all week. What do you need to find? the total distance Louisa walked Just as 3 2 means 3 groups of 2, means 3 groups of 2 tenths. Model three groups of two tenths using base-ten blocks. Combine the tenths. There are six tenths. So, = 0.6. Louisa walked a total of 0.6 mile. 3 groups and Apply 1 8. Check students work for models. Use base-ten blocks to find each product TEACH 32 Chapter 1 Multiply and Divide Decimals Activity 1 Have students work in pairs to model and check each step. 027_033_C1L1_ indd 32 1/9/09 12:12:13 Interactive WhiteBoard Demonstrate modeling decimals on an Interactive WhiteBoard. Have a student volunteer come to the IWB and use the highlighter tool to identify ones, tenths, and hundredths. 32 Chapter 1 Multiply and Divide Decimals

7 Model to find the product. So, = 3.2. Model two groups of one and six tenths. 2 groups Combine the ones. Then combine the tenths. Combine the 10 tenths into 1 whole. one whole and Apply Check students work for models. Use base-ten blocks to find each product the Results See margin. 17. Compare and contrast 2 16 and How are the products the same? different? 18. MAKE A CONJECTURE Explain how to find without using base-ten blocks. Justify your procedure. Then check your answer using base-ten blocks. 19. MAKE A CONJECTURE Suppose you multiply a whole number by a decimal greater than one. Is the product less than, greater than, or equal to the whole number? Explain your answer. 20. Write a rule that you can use to multiply a whole number by a decimal without using base-ten blocks. Write Math features help students learn to use the language of mathematics. Activity 2 Point out that 10 tenths can be combined to make one whole. 3 ASSESS Use Exercises 9 16 to assess whether students know how to use models to multiply a decimal by a whole number. From Concrete to Abstract Use Exercise 18 to bridge the gap between using models and using an algorithm to multiply a whole number and a decimal. Extending the Concept Have students use base-ten blocks to compare the product of 3 12 with the product of = 36; The product of has the same digits as 3 12, but with one decimal place = 3.6 Additional Answers 17. The factors have the same digits, but one factor in is a decimal. The products, 32 and 3.2, have the same digits, but one is a decimal. 18. Sample answer: Multiply 5 by 0.4 and add to the product of 5 1. So, is 7.0 or 7. Check students work for models. 19. Sample answer: The product is greater than the whole number. For example, = 9 and = Sample answer: Multiply the whole number by the decimal as with whole numbers and place the decimal point the same places to the left as the decimal number. Lesson 33 7_033_C1L1_ indd 33 1/9/09 12:12:24 PM Lesson 33

8 Multi-Part Lesson C Objective Students will estimate and find the product of decimals and whole numbers. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Also addresses MA.6.A.1.3, MA.6.A FOCUS Activity Provide each pair of students with a small object, such as a marker. Have them measure the length of their object to the nearest tenth using a metric ruler. Have students estimate the length of two of their object. Have students find the length of 3, 5, and 10 of their object. 2 Multiply Decimals by Whole Numbers TEACH A B C D E Scaffolding Questions Write the following on the board: = = 3, = 665 How can you use rounding to tell whether the first answer is reasonable? Sample answer: Round 429 to 400 and 11 to 10. Then multiply = 4,000. The answer is not reasonable. Is the second answer reasonable? How do you know? Sample answer: Yes; round 569 to 600 and multiply: = 3,600. How would you round to determine if the third answer is reasonable? Sample answer: Round 13.3 to 13 and then multiply by 5: 13 5 = 65. The answer is not reasonable. What is the error in the third answer? The decimal place was not shown. Multi-Part Lesson Additional Answer A B C D E Main Idea Estimate and find the product of decimals and whole numbers. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Also addresses MA.6.A.1.3, MA.6.A Multiply 6 by 492. Place the decimal point two places from the right. Some parts of lessons have been animated. Use these as a part of the lesson or as a review for your students at a later time. They are available online. 34 Chapter 1 Multiply and Divide Decimals = 14.76; 3 5 = 15; = 14.76; = 19.68; 4 5 = 20; = 19.68; = 24.6; 5 5 = 25; = 24.6 by Whole Numbers PLANTS Bamboo can grow about 4.92 feet in height per day. The table shows different ways to find the total height a bamboo plant can grow in two days. 2. See margin. 1. Use the addition problem and the estimate to find Write an addition problem, an estimate, and a multiplication problem to find the total growth over 3 days, 4 days, and 5 days. 3. MAKE A CONJECTURE Explain how to find Using repeated addition can help you place the decimal point in the product of a whole number and a decimal. The whole number represents the number of times the decimal is used as an addend. So, place the decimal point in the product the same number of places from the right as the decimal factor. Find Estimate 4 1 = two decimal places Place the decimal point two places from the right. Check for Reasonableness Find Estimate 3 14 = one decimal place Place the decimal point one place from the right. Check for Reasonableness Growth of Bamboo Over Two Days Add ft ft = 9.84 ft Estimate. Multiply is about = ft = groups of groups of 14.2 a b c _043_C1L1_ indd 34 1/9/09 12:12:49 34 Chapter 1 Multiply and Divide Decimals

9 Everyday Use Annex to add something Math Use Annex to annex a zero means to place a zero at the beginning or end of a decimal If there are not enough decimal places in the product, you need to annex zeros to the left. Find Annex Zeros in the Product three decimal places Annex a zero on the left of 36 to make three decimal places. Check by Adding = Focus on Mathematical Content Multiplying a decimal by a whole number is similar to multiplying two whole numbers. The only difference is determining where to place the decimal point in the product. Formative Assessment Use the Check Your Progress exercises after each Example to determine students understanding of concepts. d e f Find Find Find Subtracting Decimals When subtracting decimals, remember to line up the decimal points. Examples illustrate all of the concepts taught in the lesson and mirror the exercises in the exercise sets. Check Your Progress exercises give students an opportunity to try a similar problem on their own. TRAIL MIX A batch of trail mix calls for 1.2 pounds of dry cereal. Nigela is making 5 batches of trail mix. She already has 2.2 pounds of cereal. How many more pounds of dry cereal does she need? Step 1 Multiply. Step 2 Subtract. 1.2 one decimal place one decimal place 3.8 So, Nigela will need 3.8 more pounds of dry cereal. g. BIRDS A bee hummingbird has a mass of 1.8 grams. How many grams are 6 hummingbirds and a 4-gram nest? 14.8 g h. MONEY Rafael has saved $45 for a new computer screen. He is saving $8.75 a week for 5 weeks for the remaining amount. What is the total cost of the computer screen? $88.75 Lesson 35 4_043_C1L1_ indd 35 1/9/09 12:12:58 PM MUFFINS A berry muffin recipe calls for 0.75 cups of blueberries. Paul is making 5 batches. He already has 2.5 cups of blueberries. How many more cups does he need? 1.25 cups IWB INTERACTIVE WHITEBOARD READY Subtracting Decimals Remind students to annex zeros if needed to line up decimal places. AL Alternate Teaching Strategy If students have trouble finding the product of decimals and whole numbers, Then use the following reteach option: Remind them to estimate the product first. Have them write their estimates in the margin of their papers before calculating the actual products. Then they can compare their solutions with their estimates. Lesson 35

10 3 PRACTICE Formative Assessment Use Exercises 1 9 to check for understanding. Then use the table below to customize assignments for your students. Odd/Even Assignments Exercises are structured so that students practice the same concepts whether they are assigned odd or even problems. indicates multi-step problem Examples 1 3 (pp ) Example 4 (p. 35) Multiply. 4. $ $ VACATIONS The table shows the number of gallons of gasoline the Beckleys purchased on their road trip. What was Fuel the total cost for gas for the trip? Justify Number of your procedure. $215.27; (12 $4.89) + Gallons (17 $4.72) + (15 $5.09) Cost per Gallon ($) Common Error Students may have trouble placing the decimal point when there are leading or trailing zeros. In Exercise 21, remind students that the trailing zero in the product may be deleted. Finding Area In Exercise 22, remind students to find the area of the poster by multiplying its base by its height. Examples 1 3 (pp ) Multiply. = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP MEASUREMENT Asher recently bought the poster shown at the right. What is its area? (Hint: Use A = bh.) 12.8 ft MEASUREMENT The height of Mount Everest, in meters, can be found by multiplying 8.85 by 1,000. Find the height of Mount Everest. Explain your answer. 8,850 m; 8.85 rounds to 9 and 9 1,000 = 9, ft 4 ft AL OL ELL Visual/Spatial Have students use grid paper to align their columns and keep track of the decimal point. Their work might resemble the following X Example 4 (p. 35) 36 Chapter 1 Multiply and Divide Decimals 24. SCHOOL SUPPLIES Sharon buys 14 folders for $0.75 each. How much change will she receive if she pays with $15? $ TEMPERATURE The hottest temperature recorded in the world, in degrees Fahrenheit, can be found by multiplying by 10. Find this temperature. Justify your procedure F; 1, = 13,460. Since has 2 decimal places, = B 26. MEASUREMENT The thickness of each type of coin is shown in the table. How much thicker is a stack of a dollar s worth of nickels than a dollar s worth of quarters? Explain your answer. 32 mm; = 39; = 7; 39-7 = 32 Coin Thickness (mm) penny 1.55 nickel 1.95 dime 1.35 quarter 1.75 Differentiated Homework Options 034_043_C1L1_ indd 36 1/28/09 3:54:14 The Differentiated Homework Options provide suggestions for the exercises that are appropriate for basic, core, or advanced students. Many of the homework exercises are paired so that students could do the odds one day and the evens the next day. Level Assignment Two-Day Option AL Approaching Level 10 24, 27, odd, 31, even, 27, 29, 30, OL On Level odd, 25 27, , 31, , 29, 30, BL Beyond Level Chapter 1 Multiply and Divide Decimals

11 See Ch. 1 Answer Appendix. C 27. OPEN ENDED Write a real-world problem involving multiplication by a decimal factor. Then solve the problem. 4 ASSESS 28. CHALLENGE Discuss two different ways to find the value of the expression that do not require you to first multiply Ask students to explain the method they used to solve Exercise 15. Practice 29. REASONING Use the product of to find the product of Explain the difference in the two products. 30. Your friend thinks that = 1.20 because you do not count the zero when placing the decimal point. Is your friend correct? Justify your reasoning. No; zero represents the number of hundredths and should be counted. MA.6.A.1.3 Follow-Up Remind students to summarize what they learned in this lesson under the Multiply Decimals by Whole Numbers tab of their Foldables. Encourage them to include several original examples. 31. GRIDDED RESPONSE The school store is selling the following items. Item Price Pennant $2.49 Bumper Sticker $1.79 Magnet $0.89 If Miguel buys two pennants, two bumper stickers, and four magnets, how much in dollars will he spend for all the items? The table shows the admission prices to an amusement park. Admission Prices One-Day Pass Two-Day Pass Adult $39.59 $43.99 Child (ages 3 9) $30.59 $33.99 What is the total price of one-day passes for two adults and three children? B A. $ C. $ B. $ D. $ Are students continuing to struggle with multiplying decimals by whole numbers? If Yes If No AL Reteach Worksheet CRM (p. 18) AL Differentiated Instruction Option 1 (p. 27c) OL Skills Practice Worksheet CRM (p. 19) BL Enrich Worksheet CRM (p. 22) Estimate. (Lesson A) Sample answers given , SPORTS Kaitlyn recorded the number of hours that she practiced soccer drills. What was the total time she practiced? (Lesson 0-5) 5 5_ 12 h 37. GEOMETRY Find the perimeter of a triangle with side lengths 2.12 centimeters, 2.03 centimeters, and 1.98 centimeters. (Lesson 0-2) 6.13 cm Day Hours Practiced Monday 2 1_ 3 Wednesday 1 1_ 3 Friday 1 3_ 4 Lesson 37 Tips for New Teachers 4_043_C1L1_ indd 37 1/21/09 9:00:47 AM New teachers, or teachers new to teaching mathematics, may especially appreciate the Tips for New Teachers. Multiplying Decimals When students multiply by decimals such as , they may write the product as Explain that only one zero is needed to the left of the decimal point, and that they should omit all zeros after the last nonzero digit to the right of the decimal point. Quick Check provides reteaching suggestions for students who continue to struggle. Practice Diagnose Student Errors Survey student responses for each item. Class trends may indicate common errors and misconceptions. 32. A. found the cost of one-day passes for 2 adults and 2 children B. correct C. found the cost of 3 adult and 2 children s one-day passes D. found for two-day passes instead of one-day passes Lesson 37

12 Multi-Part Lesson D by Decimals Objective Students will use decimal models to multiply decimals. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Resources Materials 10-by-10 grids Easy-to-Use Manipulatives Hands-On Activity Tools and Resources Decimal Models, p. 2 1 FOCUS Have each student explain how to use models to multiply a decimal by a whole number. Then review by demonstrating how to use base-ten blocks to model A B C D E TEACH Activity 1 Have students read the multiplication in words: Seven tenths times six tenths. Be sure students understand that each small square represents 0.01, which is one hundredth of the 10-by-10 decimal model. Virtual Manipulatives Use a 10-by-10 grid on the Interactive WhiteBoard and the Virtual Base-Ten Blocks to model the steps students should follow to multiply decimals in the lab. Be sure they understand that each small square in the grid is a hundredth, and each row or column is a tenth. Additional Answers Multi-Part Lesson 0.2 Multipy Decimals A B C D E Main Idea Use decimal models to multiply decimals. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently in each factor and 2 in the product; 1 in each factor and 2 in the product; 1 in each factor and 2 in the product 38 Chapter 1 Multiply and Divide Decimals 6. Sample answer: There is 1 decimal place in each factor, so the product will have 2 decimal places, and you can multiply normally, but add the decimals from right to left; 0.12 by Decimals You can also model decimals on a 10-by-10 grid. Find Use decimal models. Draw a 10-by-10 decimal model. Recall that each small square represents Shade a rectangle that is 7 tenths units wide and 6 tenths units long. Count the number of shaded squares. There are forty-two hundredths in the shaded region. So, = and Apply 1 3. See Ch. 1 Answer Appendix for models. Use decimal models to show each product the Results 4. Tell how many decimal places are in each factor and in each product of Exercises 1 3 above. 5. Is equal to 0.6 or 0.06? Justify your reasoning with a model. 0.06; See margin for model. 6. MAKE A CONJECTURE Use the pattern you discovered in Exercise 4 to find Check your conjecture with a model or a calculator. See margin. 7. Find two decimals with a product of Sample answer: _043_C1L1_ indd 38 1/9/09 12:13: Every effort is made to show answers on the reduced Student Edition page or in the margin of the Teacher Edition. However, answers that do not fit in either of these places can be found in Answer Appendix pages at the end of each chapter. 38 Chapter 1 Multiply and Divide Decimals

13 Model Use an area model. Since you are multiplying tenths by hundredths, think about a 10-by-100 grid. There are 1,000 squares in all, but you will not draw them. 100 units Draw and shade 8 tenths of the height to model 0.8 and shade 16 hundredths across to model units Find the number of squares shaded. Each square represents one thousandth of the full rectangle. There are 128 out of 1,000 squares, or 128 thousandths, shaded. So, = and Apply 10 units Use decimal models to show each product See students work for models. 10 units the Results 11. MAKE A CONJECTURE Find the product of 0.04 and 0.08 without using a model = Explain why = 0.2. Since 4 rows of 5 will be shaded this represents 0.20 or MAKE A CONJECTURE How does the number of decimal places in the product relate to the number of decimal places in the factors? 14. Analyze each product See margin. First Second Product Factor Factor a. Explain why the first product is less than = 0.54 b. Explain why the second product is equal to = 0.60 c. Explain why the third product is greater than = 0.90 Lesson 39 4_043_C1L1_ indd 39 1/21/09 9:00:59 AM 3 ASSESS Activity 2 Make sure students understand why they need a 10-by-100 grid that shows thousandths. Reinforce to students that the model is not to scale. Ask: How many decimal places are in the factors? 3 How many decimal places will be in the product? 3 What is the place value of a decimal with three places? thousandths Use Exercises 1 and 8 to determine whether students understand how to use models to multiply decimals. From Concrete to Abstract Use Exercise 13 to bridge the gap between using models to find the product of decimals and using rules to determine where to place the decimal point in the product. Extending the Concept Have students discuss how to estimate the product of Sample answer: 0.84 is approximately 1 and 0.9 is approximately = 1 Additional Answers 13. Sample answer: The total number of decimals in the factors is the same as the number of places the decimal is placed from right to left in the product. 14a. Sample answer: The first product is less than the multiplicand because the multiplier is less than one. You don t want all of what you started with, only a part of a whole, so the product is less than the multiplicand. 14b. Sample answer: The second product is equal to the multiplicand because of the Identity Property. Any number times one is equal to that number itself. 14c. Sample answer: The third product is greater than the multiplicand because the multiplier is more than one. So, the product is more than what you started with. Lesson 39

14 Multi-Part Lesson A B C D E E Objective Students will multiply decimals by decimals. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Also addresses MA.6.A.1.3, MA.6.A FOCUS Activity Write the following on the board. Ask students to evaluate each expression and describe the pattern in the products ; 70; 7; 0.7; 0.07; Sample answer: Each product is one tenth of the previous product. 2 TEACH Scaffolding Questions Display the table of download prices. Downloads Multiply Decimals by Decimals Price MP3 $0.99 Unlimited play MP3 $1.49 Music video $2.89 Ask: How could you estimate the cost of 3 MP3 downloads? Sample answer: Round $0.99 to $1, and multiply 1 by 3. So 3 downloads cost about $3. How would you estimate the cost of 6 music video downloads? Sample answer: Round $2.89 to $3 and multiply 3 by 6. So 6 video downloads cost about $18. Multi-Part Lesson Multipy Decimals A B C D E Main Idea Multiply decimals by decimals. MA.6.A.1.1 Explain and justify procedures for multiplying and dividing fractions and decimals. MA.6.A.1.2 Multiply and divide fractions and decimals efficiently. Also addresses MA.6.A.1.3, MA.6.A Chapter 1 Multiply and Divide Decimals How would you find the estimated cost of 8 unlimited play MP3 downloads? Sample answer: Round $1.49 to $1 and multiply 1 by 8. So 8 unlimited play downloads costs about $8. Is your estimate for 8 unlimited play downloads greater or less than the actual cost? Explain. Sample answer: less, since the actual cost was rounded down by Decimals PLANETS The table shows the weight of a 1-pound object on each planet. 1. A 0.5-pound object weighs one half as much as a 1-pound object. What would this object weigh on Jupiter? 1.15 lb 2. What would an object that weighs 5 pounds on Earth weigh on Jupiter? 11.5 lb Planet Weight (pounds) Mercury 0.3 Venus 0.9 Mars 0.3 Earth 1 Jupiter 2.3 Saturn 1 Uranus 0.8 Neptune MAKE A CONJECTURE How can you use the results of Exercises 1 and 2 to find ? Find 5 23 and place the decimal point two places from the right. When multiplying a decimal by a decimal, multiply as with whole numbers. To place the decimal point, find the sum of the number of decimal places in each factor. The product has the same number of decimal places. Find Estimate or one decimal place 6.7 one decimal place two decimal places The product is Compared to the estimate, the product is reasonable. Find Estimate or one decimal place 0.05 two decimal places three decimal places The product is or Once you place the decimal point, you can drop the zero at the right. a b c _043_C1L1_ indd 40 1/21/09 9:01:03 Internet Research Show students a video on multiplying decimals. You can find videos on education video-sharing Web sites. Have students work in small groups to take notes on what they learned. Tell them they will make their own video in a later lesson. 40 Chapter 1 Multiply and Divide Decimals

15 Annex a Zero Find three decimal places 1.4 one decimal place Annex a zero to make four decimal places. d e f Formative Assessment Use the Check Your Progress exercises after each Example to determine students understanding of concepts. Find Find Find Real-World Link A car that can travel 21 miles on one gallon of gasoline will cost about $925 per year more, in gasoline costs alone, than a car that can travel 31 miles on one gallon of gasoline. indicates multi-step problem Examples 1 3 (pp ) Example 4 (p. 41) PRACTICE Formative Assessment Use Exercises 1 10 to check for understanding. Then use the table on the next page to customize assignments for your students. CARS A certain car can travel miles with one gallon of gasoline. The gasoline tank can hold 11.5 gallons. How many miles can this car travel on a full tank of gas? Justify your answer. Estimate or two decimal places 11.5 one decimal place The product has three decimal places. The car could travel miles. Since , the answer is reasonable. g g; = 9,375, and there are 3 total decimal places, so the decimal point is placed 3 places from the right. g. NUTRITION FACTS A nutrition label indicates that one serving of apple crisp oatmeal has 2.5 grams of fat. How many grams of fat are there in 3.75 servings? Justify your answer. Multiply MEASUREMENT A mile is equal to approximately kilometers. How many kilometers is 2.5 miles? Justify your answer km; and = 3.75; Lesson 41 Focus on Mathematical Content 4_043_C1L1_ indd 41 1/28/09 3:54:34 PM In adding decimals, the decimal points must be aligned. In multiplying decimals, the decimal points are not aligned. The decimals are multiplied as if they were whole numbers. The position of the decimal point in the product is based on the sum of the number of decimal places in each factor. MONEY Jennifer makes $8.79 an hour walking dogs. She walks dogs for 6.75 hours per week. How much money does she earn each week? $59.33 Justify your answer. Since $ is about 9 7, or 63, the answer is reasonable. IWB INTERACTIVE WHITEBOARD READY AL Alternate Teaching Strategy If students have trouble multiplying decimals by decimals, Then use one of these reteach options: 1 Quick Review Math Handbook, pp Use grid paper to line up the numbers and keep track of decimal placement. Some students may benefit from removing the decimals. Their work might resemble the following. Have students highlight the squares in each factor that are after the decimal. Then have them count the highlighted squares, and use that number to place the decimal X Lesson 41

16 Odd/Even Assignments Exercises are structured so that students practice the same concepts whether they are assigned odd or even problems. Common Error Students may have trouble remembering the order of operations in Exercises Remind them to multiply before adding and subtracting. Examples 1 3 (pp ) Example 4 (p. 41) Multiply. = Step-by-Step Solutions begin on page R1. Extra Practice begins on page EP ANIMALS A giraffe can run up to feet per second. How far could a giraffe run in 1.8 seconds? Justify your answer ft; = MEASUREMENT Katelyn has a vegetable garden that measures feet in length and 5.8 feet in width. Find the area of the garden. Justify your answer ft 2 ; = 102! Exercise Alert! FIND THE DATA In Exercise 36, students should use the Data File on pp. 2 5 to write a real-world problem. Focus Question Follow-Up Ask: What do you know about multiplying a whole number by a decimal greater than 1? Sample answer: The product is greater than the number. What do you know about multiplying a whole number by a decimal between 0 and 1? Sample answer: The product is less than the number. Do you think those same rules apply to dividing by a decimal? Sample answer: I think they will be the opposite. AL OL ELL Intrapersonal Have students work through at least one problem in this lesson on their own. Ask them to describe the steps they used to solve the problem in a short paragraph. Encourage students to include an explanation of why their answers are reasonable when compared to their original estimates. Additional Answer in 2 ; Since = and 3( ) = 9, then the area of the fi gure is or in 2. Algebra To show multiplication with variables, the variables and/or numbers are placed next to each other. ab means a b 6.023a means a. Multiply. B Chapter 1 Multiply and Divide Decimals Differentiated Homework Options Level Assignment Two-Day Option AL Approaching Level 11 24, 39 42, odd, 47, even, 39 42, 46, OL On Level odd, 28, 29, 31, 33 37, 39 42, WALKING Junnie walked for 2.5 hours at a speed of 3.2 miles per hour. Maurice walked for 1.8 hours at a speed of 4.1 miles per hour. (Hint: distance equals speed times time.) a. Who walked farther? Junnie b. How much farther did that person walk? 0.62 mile farther ALGEBRA Evaluate each expression if a = 1.3, b = 0.042, and c = ab + c a - c c + b abc MEASUREMENT Find the area of the figure at the right. Justify your procedure. See margin. 34. ALGEBRA Which of the three numbers 9.2, 9.5, or 9.7 is the correct solution of 2.65t = ? Explain your answer. 9.7; = Sample answer: If a = 1.5 and b = 0.2, then ab = 0.3, which is less than 1; if 35 GROCERY SHOPPING Pears cost $0.98 per a = 1.5 and b = 0.8, then ab = 1.2, which is not less than Sample answer: If ab = 0.63, then a could be 0.9 and b could be 0.7, which are both less than 1; if ab = 0.5, then a could be 0.5, which is less than 1, but b would be 1.0, which is not less than in. 3.1 in. 6.1 in. pound and apples cost $1.05 per pound. Mr. Bonilla bought 3.75 pounds of pears and 2.1 pounds of apples. How much did he pay for the pears and apples? Explain your answer. $5.88; Each price is about $1. He bought about 6 pounds of fruit. 6 1 = 6 $ FIND THE DATA Refer to the Data File on pages 2 5. Choose some data and write a real-world problem in which you would multiply decimals. See students work. For each statement below, find two decimals a and b that make the statement true. Then find two decimals a and b that make the statement false. Explain your reasoning. 37. If a > 1 and b < 1, then ab < If ab < 1, then a < 1 and b < _043_C1L1_ indd 42 1/28/09 3:54: , 47, , 46, in. BL Beyond Level Chapter 1 Multiply and Divide Decimals

17 ; Sample answer: rounds to 4 8 = 32, so the answer must be about always; Sample answer: = 0.15; = greater than 0.4; It is being multiplied by a decimal greater than 1. Practice Multiply. (Lesson C) The distance around Earth at the equator is 24, mi about 24, miles. The distance around Earth through the North Pole and South Pole is about 24, miles. (Lesson A) a b. Sample answers given. 24, mi a. Estimate how many miles you would travel if you circled the equator 3 times. about 75,000 mi b. Estimate how many more miles you would travel if you circled the equator 10 times rather than 10 times around the poles. about 1,000 mi Add or subtract. (Lesson 0-3) 54. 3_ _ 10 4_ 5 C 39. OPEN ENDED Write a multiplication problem in which the product is between 0.05 and Sample answer: NUMBER SENSE Place the decimal point in the answer to make it correct. Explain your reasoning = REASONING Determine whether the following statement is always, sometimes, or never true. Give examples to justify your answer. The product of two decimals less than 1 is less than either of the factors. 42. NUMBER SENSE Is the product of greater than or less than 0.4? Explain your reasoning. CHALLENGE Evaluate each expression (3-0.5) (7-2.8) ( ) Describe two methods for determining where to place the decimal point in the product of two decimals. See Ch. 1 Answer Appendix. MA.6.A.1.1, MA.6.A What is the area of the rectangle? D 5.62 cm 1.4 cm A cm 2 C cm 2 B cm 2 D cm _ 9 + 7_ _ 8-2_ SHORT RESPONSE A soccer ball and 12 golf balls weigh a total of 1 kilogram. Each golf ball weighs about kilogram. What is the weight of the soccer ball? Explain your reasoning kilogram; The golf balls total mass is or kilogram. 3_ _ _ 11 3_ 11 Lesson 43 4 ASSESS Ask students to write a short paragraph explaining how the previous lesson on multiplying decimals by whole numbers helped them in understanding how to multiply decimals by decimals. Formative Assessment Check for student understanding of concepts in Multi-Part Lesson. CRM Chapter Quiz 1, p. 64 Are students continuing to struggle with multiplying decimals by decimals? If Yes If No AL Reteach Worksheet CRM (p. 24) OL Skills Practice Worksheet CRM (p. 25) BL Enrich Worksheet CRM (p. 29) BL Differentiated Instruction Option 1 (p. 27c) Practice Diagnose Student Errors Survey student responses for each item. Class trends may indicate common errors and misconceptions. 47. A. found perimeter, not area B. made mistake in computation C. made mistake in computation D. correct _043_C1L1_ indd 43 10/1/10 3:21:42 PM Ask students to compare and contrast multiplying whole numbers and decimals. Sample answer: Multiplying whole numbers and decimals is similar because you can multiply decimals as if they were whole numbers. Multiply each digit of one factor by the ones, tens, hundreds, and so on of the other factor. Multiplying decimals is different from multiplying whole numbers because you need to place the decimal point in the product. You can count the number of decimal places in each factor and use this number for the number of decimal places in the product, annexing zeros to the left if needed. Practice exercises help students solidify their knowledge using exercises in a format similar to the state test. Lesson 43