4-5 Subtracting Mixed Numbers 1

4-5 4-5 Subtracting Mixed Numbers 1. Plan

Lesson Preview What You’ll Learn Check Skills You’ll Need For help, go to Lesson 3-5. OBJECTIVE PowerPoint 1 To subtract mixed numbers Write each improper fraction as a mixed number in simplest form. Check Skills You’ll Need OBJECTIVE 3 8 23 To subtract mixed 11 22 3 2 1.2 2.3 3 3. 45 Writing Fractions as numbers with 2 5 20 15 24 Mixed Numbers renaming 4.21 5.21 6. 22 Lesson 3-5: Example 3. Extra 8 2 6 2 10 5 Practice p. 644. ...And Why 7. Explain the steps you used to write 24 as a mixed number in To subtract two weights, 10 as in Example 1 simplest form. Divide 24 by 10. The quotient 2 is the integer of the Lesson Resources mixed number. The remainder 4 is the numerator, and 10 is the 4 2 denominator: 210 . Reduce to 25 . Teaching Resources OBJECTIVE Practice, Reteaching, Enrichment Interactive lesson includes instant self-check, tutorials, and activities. 1 Subtracting Mixed Numbers Reaching All Students Practice Workbook 4-5 To subtract mixed numbers, ﬁrst you may need to write the fractions with a Spanish Practice Workbook 4-5 Guided Problem Solving 4-5 common denominator. Then subtract the whole numbers and the fraction Hands-On Activities 14 parts separately.

Presentation Assistant Plus! 1 EXAMPLE Real-World Problem Solving Transparencies • Check Skills You’ll Need 4-5 3 Lions At birth, one lion cub weighs 3 pounds. Another cub in the same • Problem of the Day 4-5 5 4 litter weighs 2 pounds. How much more does the heavier cub weigh? • Additional Examples 4-5 8 • Student Edition Answers 4-5 3 5 To calculate the difference in weights, ﬁnd 3 - 2 . • Lesson Quiz 4-5 4 8 3 6 d 3 6 PH Presentation Pro CD-ROM 4-5 3 3 The LCD is 8. Write as . 4 8 4 8 S 5 5 2 2 2 2 Male cubs grow to an adult 8 8 Computer Test Generator CD Subtract the whole numbers. weight of 600 pounds. 1 d 1 Then subtract the fractions. 8 Technology 1 The heavier cub weighs 1 pounds more than the other cub. Resource Pro® CD-ROM 8 Computer Test Generator CD 3 5 Check for Reasonableness Round each mixed number: 3 < 4; 2 < 3. PH Presentation Pro CD-ROM 4 8 - = 1 Then subtract: 4 3 1. The answer 18 is close to the estimate. So, the www.PHSchool.com answer is reasonable. Student Site 3 7 Teacher Web Code: aak-5500 • Check Understanding 1 a. A supporting wedge for a window is 216 inches wide and 28 inches long. • Self-grading Lesson Quiz 11 How much longer is the wedge than it is wide? 16 in. PH SuccessNet Teacher Center b. Reasoning In Example 1, could you use 32 as the common denominator? • Lesson Planner Explain your answer. Yes; answers may vary. Sample: 32 is a common • Resources multiple of 8 and 4, so it is also a common denominator.

Plus 190 Chapter 4 Adding and Subtracting Fractions

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S Ongoing Assessment and Intervention Before the Lesson During the Lesson After the Lesson Diagnose prerequisite skills using: Monitor progress using: Assess knowledge using: • Check Skills You’ll Need • Check Understanding • Lesson Quiz • Additional Examples • Computer Test Generator CD • Test Prep 190 OBJECTIVE 2 Subtracting Mixed Numbers by Renaming 2. Teach

Sometimes you need to rename whole numbers or fractions in order to 1 Math Background subtract from them. Here is how to rename 3 4 . Subtracting mixed numbers is Need Help? ϭ 1 1 3 5 2 1 1 similar to adding mixed numbers For help writing mixed 4 4 in that whole numbers and numbers as improper 5 fractions, go to Lesson 3-5. 5 2 1 fraction parts are dealt with 1 5 4 3 ϭ 2 separately and the results are 4 4 5 5 2 combined. The new step in 4 subtracting mixed numbers involves renaming. If the fraction 2 EXAMPLE Renaming Whole Numbers being subtracted is greater than 5 the one from which it is being - 2 Find 7 8. subtracted, the fraction from which you are subtracting must Write 7 as a mixed number. Use 8 for the denominator since you must 5 be renamed. This requires also subtract 8 . renaming the whole number part 8 d 8 8 of the mixed number. For 7 6 Rename 7 as 6 1 1 5 6 1 , or 6 . 8 8 8 1 2 S instance, in subtracting 4 5 2 2 5 , 1 6 5 5 you rename 4 5 as 3 5 . First you 2 2 2 2 8 8 subtract the fractions. Then you 3 d Subtract the whole numbers. subtract the whole numbers. 4 8 Then subtract the fractions. 6 2 4 3 5 2 2 5 5 15 2 1 1 3 Check Understanding 2 Find each difference. a. 5 - 3 1 b. 10 - 4 5 3 3 4 4 Teaching Notes

Inclusion 3 EXAMPLE Renaming Mixed Numbers You may need to review how to 2 find the LCD (Least Common Distance (miles) Biology In one hour, a bee can fly 53 miles and a moth can fly Denominator) of two fractions. 1 Moth 11 6 miles. How much farther can the moth fly in one hour? 1 1 EXAMPLE Error Prevention 116 1 2 1 2 1 To answer the question, find 11 - 5 . Since , , rename 11 . 6 3 6 3 6 Guide students to begin with the 2 fractions’ column when they 5 3 1 2 Estimate 11 2 5 < 11 2 6 5 5 subtract mixed numbers. In Bee Difference 6 3 Example 1 no renaming is 1 7 1 1 7 11 10 d Rename 11 as 10 151 10 . required; but in problems that 6 6 6 6 6 S do require renaming, subtracting d The LCD is 6. Write 2 as 4 . the whole numbers first can cause 2 4 3 6 2 5 2 5 3 6 errors. 3 d Subtract. 3. Answers may vary. 5 5 2 EXAMPLE Teaching Tip 6 Sample: If the Call attention to Check benchmark of the 1 d Simplify. first fraction is less 5 5 Understanding 2b. Review that 2 0 than the benchmark 1 10 is the same as 10 4 . Ask: Check for Reasonableness The answer 5 is close to the estimate of 5. 4 of the second 2 How do you rename 10? 9 4 fraction, then rename 1 PowerPoint before subtracting. The moth can fly 5 2 miles farther than the bee. Additional Examples Check Understanding 3 Number Sense How can you use benchmarks to tell whether you will have 1 to rename before subtracting? See above left. 1 A black bear is about 5 4 ft long. An Alaskan brown bear 1 is about 7 2 ft long. How much 4-5 Subtracting Mixed Numbers 191 longer is an Alaskan brown 1 bear than a black bear? 2 4 ft 2 1 Reaching All Students 2 Find 9 2 13. 7 3 3 A two-week-old panda bear Below Level Have students rename Advanced Learners Match up E, F, Inclusion 3 mixed numbers as improper fractions. G, and H to the digits 1, 2, 5, and 8 See note on page 191. weighed 4 pound. At age 2 = 3 2 5 to solve this mystery problem. Visual Learners one-month, the cub weighed 13 3 1 3 5 3 3 G F 9 2 10 pounds. How many pounds H 2 F 5 6 E ≠ 5, F ≠ 1, See note on page 192. 14 = 9 1 4 5 13 E G 10 did it gain? 111 lb 9 9 9 9 2 1 9 G ≠ 2, H ≠ 8 20 5 = 6 5 11 8 5 2 12 5 6 10 16 6 1 6 5 6 191 Visual Learners More Than One Way For the More Than One Way 1 feature, have students make a Suppose you caught two fish. The first one is 48 inches long. The second drawing to help them visualize 1 one is 2 inches long. How much longer is the first fish? the problem situation. Then work 4 through both solution methods Leon’s Method using colors to illustrate each step. For instance, have volunteers write 1 1 1 I need to subtract the lengths. Since , , I will rename 4 . the subtraction on the board or 8 4 8 an overhead transparency and use 1 9 1 1 9 4 3 d Rename 4 as 3 151 3 . a color to circle the denominators. 8 S 8 8 8 8 Then have students use another 1 2 1 2 2 2 2 4 d The LCD is 8. Write as . color to circle the numerators that 4 8 4 8 will be subtracted, and a third 7 1 d Find the difference. color for the number that has 8 been renamed. 7 The first fish is 1 8 inches longer than the second one. Closure Lauren’s Method • When do you have to rename when you are subtracting mixed I need to subtract the lengths. I will change both mixed numbers? when the fraction being subtracted is greater numbers to improper fractions with the same denominator. than the one from which it is 1 1 33 9 d 4 -=-2 Write as improper fractions. being subtracted 8 4 8 4 • Explain how you would rename =-33 18 d Rename as equivalent fractions 1 1 8 8 with a like denominator. 9 8. Sample: Think of 9 8 as ± ± 1 8 1 8. Change the 1 to 15 7 d Subtract. Write the difference = 1 8 8 , or 8 in simplest form. eighths, or 8 . Combine 8 1 9 8 ±±to get 8 . 7 8 8 8 The first fish is 18 inches longer than the second one.

Choose a Method 1 8 10 - 7 Find 3 9. Describe your method and explain your choice. 4 1 12 8 4 29 ; Sample: I renamed 103 as 99 and subtracted 79 ; the difference is 29 .

EXERCISES For more practice, see Extra Practice.

A Practice by Example Find each difference. Exercise 1 has been started for you. 3 6 3 2 7 5 1 3 4 4 8 Example 1 1. 12 12 2.7 2 6 1 3.2 2 1 1 4. 9 2 4 5 4 8 4 5 20 8 4 8 5 15 15 (page 190) S 3 3 2 10 2 10 3 8 8 1. 28 3 1 1 11 1 1 1 1 2 1 1 5. 21 - 11 10 6. 15 - 11 7. 12 - 4 8 8. 3 - 1 2 8 4 8 12 2 4 8 8 3 6 2 5 412 2 1 9. You spend 2 3 hours reading and 1 2 hours watching a movie. How much 1 more time did you spend reading than watching a movie? 1h6

192 Chapter 4 Adding and Subtracting Fractions

192 Example 2 Find each difference. Exercise 10 has been started for you. (page 191) 3. Practice 4 1 3 1 10. 4 3 1 11.23 3 12. 32 15 S 4 4 8 2 3 3 5 1 Assignment Guide 2 2 2 2 2 19 16 4 4 8 2 2 1 Objective 1 1 2 3 3 5 5 3 2 1 2 A B Core 1–9, 18, 21–22, Example 3 13. 10 - 3 14. 3 - 1 1 15. 4 - 1 2 16. 6 - 2 10 5 8 4 8 12 4 3 5 3 (page 191) 27–31 6 7 8 10 315 17. Science You and your partner are growing bean plants for a science 2 Objective 2 7 A B project. After one week, one plant is 7 inches tall and another Core 10–17, 19–20, 15 8 23–26, 32 plant is 5 16 inches tall. Find the difference in the heights of the 15 C B Extension 33–35, 36 two plants. 116 in. Test Prep 37–39 Mixed Review 40–49 B Find each difference. 1 22 2 2 1 5 2 1 3 1 - - - 1 - Auditory Learners 18. 9 5 4 19. 1 6 20. 3 1 3 21. 12 10 3 3 6 3 4 4 Exercises 13–16 Have students 7 1 2 5 6 1 4 2 1 3 3 verbalize to a partner the steps 22.5 2 2 3 23. 8 - 3 4 24.5 2 4 25. 9 - 6 2 9 9 3 11 11 5 5 5 8 4 8 they used to rename mixed numbers. 26. Data File, p. 169 Find the difference in the top speed of a peregrine Practice 4-5 Subtracting Mixed Numbers falcon and the top speed of a cheetah. 21 mi per min Write each difference in simplest form. 613 523 53 6 11 7 16 1 3 24 2 5 1.10 16 2 3 8 2.8 3 2 2 8 3. 9 2 3 5 213 423 41 3 3 16 1 2 30 1 2 27. Answers may vary. 27. Writing in Math Explain how you can use mental math to 4.5 16 2 2 8 5.8 6 2 3 5 6. 7 2 2 3 15 2 1 55 3 1 8 1 1 16 2 5 6 Sample: Mentally 1 - 3 7.2 4 2 18 8.4 8 2 2 16 9. 9 3 2 3 6 ﬁnd 10 .12 See left. 7 7 1 3 9 4 4 1 2 10 7 1 12 7 7 16 1 5 10.2 2 1 11.15 2 8 12. 6 2 2 rename 12 as 11 ; 10 5 12 2 16 8 131 4 3 211 4 4 1 11 3 2 1 20 2 3 12 3 13.27 13 14.5 1 15. 10 7 3 4 2 12 5 2 4 3 2 4 28. On Monday, the snowfall in the mountains was 15 inches. On Tuesday, subtract 10 ; simplify 31 6 1 1413 4 4 3 1 4 5 1 12 7 1 16 1 16.5 4 2 2 2 17.16 12 2 10 3 18. 23 8 2 9 16 2 1 the difference 1 as 1 . GPS the snowfall was 18 inches. What was the difference in snowfall? 3 3 7 7 62 2 1 1 10 1 3 12 1 1 5 4 2 19.35 2 32 20.25 2 17 21. 33 2 27 3 2 5 3 4 2 10 2 in. 513 4 3 33 4 3 5 24 3 3 16 1 1 4 22.24 8 2 18 6 23.12 8 2 8 16 24. 9 4 2 5 2 Olympics Use the table for Women’s Olympic Long Jump Winners Solve. 25. Robbie needs to buy fencing for his square vegetable garden 3 that measures 16 4 feet on a side. One side borders the back of Exercises 29–31. Winner, CountryYear Distance the garage. The fencing costs $4 per feet. Estimate how much the fencing will cost. about $204

1984 Anisoara Stanciu, 22 ft 10 in. 3 29. How much farther did 26. Paula has 2 yards of elastic. One project needs a piece 4 yard. Does she have enough for another project that needs 11 yards? Explain. Romania 3 1 No. Sample answer: she will have only 14 yd left Heike Drechsler jump 1 1 1 after the first project and 14 R 13. 6 in. 1988 Jackie Joyner-Kersee, 24 ft 3 in. in 1992 than in 2000? 2 27. Use a ruler or measuring tape to find the perimeter of your desk. U.S.A. Measure to the nearest half inch. Check students’ work. 1 width: length: perimeter: 30. Which two jumps were 1992 Heike Drechsler, 23 ft 5 in. Now find the perimeter of your teacher’s desk. 4 width: length: perimeter: closest in length? Germany Subtract to find the difference in the perimeters. Explain. See below right. 1 1996 Chioma Ajunwa, 23 ft 4 2 in. Nigeria Reteaching 4-5 Subtracting Mixed Numbers 31. Find the difference between 1 Some mixed numbers can be subtracted Sometimes you must rename the first fraction 2000 Heike Drechsler, 22 ft 11 in. mentally. before subtracting. 2 1 1 3 the longest and the shortest 4 Example 1: Find 5 2 2 . Example 2: Find 6 2 2 . Germany 3 6 2 4 winning jumps shown. 1 Subtract the whole numbers. 1 Write with a common denominator. 1 3 2 3 1 3 5 2 3 6 2 2 2 4 5 6 4 2 2 4 30. 1992 and 1996; the 1992 jump is in. 2 6 3 1 ft 5 in. 4 2 Then, subtract the fractions. 2 Rename 6 4. 5 5 4 2 2 4 2 2 1 4 1 3 1 3 longer than the 1996 jump. 3 2 6 5 6 2 6 5 6 5 2 3 Subtract the whole numbers. 5 3 4 Then, subtract the fractions. 3 Combine the two parts. Simplify, if necessary. 32. Gardening Carlos plants a spruce tree in a garden of a new school. The 1 1 3 1 2 5 3 2 1 3 3 1 6 2 2 2 4 5 3 4 2 1 1 height of the tree when he plants it is 3 feet. He measures the tree two 5 2 2 5 3 2 3 6 2 3 7 years later. It is 4 feet tall. How much has the tree grown? ft 8 8 Find each difference. 7 3 3 1 2 1 1.7 10 2 2 10 2.3 4 2 12 3. 6 3 2 2 6 2 1 1 55 24 42

7 3 1 1 1 1 4.9 8 2 7 4 5.8 2 2 3 4 6. 14 3 2 8 4 21 51 6 1 4-5 Subtracting Mixed Numbers 193 8 4 12 1 2 5 3 5 13 7.12 3 2 9 3 8.6 8 2 2 4 9. 7 7 2 4 14 2 7 11 23 38 214

2 5 7 1 2 2 Use the Guided Problem 10.10 3 2 7 6 11.5 16 2 12 12. 8 5 2 3 3 25 315 411 GPS Solving worksheet with 6 16 15 1 1 5 3 3 1 13.6 8 2 3 16 14.9 12 2 5 4 15. 12 4 2 6 8 Exercise 28. 1 2 5 316 33 68

2 1 5 1 1 4 16.7 5 2 2 4 17.15 12 2 8 3 18. 4 10 2 2 5 3 1 3 520 712 110

193 4. Assess C Algebra Solve each equation.

4 13 9 3 2 25 1 3 1 33. x =-9 8 34. x =-6 2 3 35. x +=3 4 1 PowerPoint 7 14 14 16 3 48 2 4 4 Lesson Quiz 4-5 Find each difference. 36. Stretch Your Thinking Fill in each with 745 7 1 5 one of the digits 4, 5, 6, 7, 8, or 9 to ﬁnd the 2 698 1. 7 8 2 4 4 3 8 — least possible whole number difference. 47 2 8 3 2. 8 5 2 6 10 1 5 Use each digit only once. 1 11 5 3. 9 3 2 5 12 3 12 1 5 17 4. 14 2 6 7 3 8 24 Test Prep

Reading Comprehension Read the passage and answer the questions below. Alternative Assessment Have students work in pairs. Refer Time of Day them to Exercises 13–16. For each Take It to the NET exercise, partners must verify with Online lessonJun. 21LatitudeDec. quiz at 21 The tilt of Earth’s axis affects the June 21 is sometimes referred to one another that they have found www.PHSchool.com length of daylight in a given region as the “longest day of the year.” 20°N 13 1 h 10 4 h a common denominator and 5 5 throughout the year. Latitude, the December 21 is the “shortest day have renamed the mixed number measure of the distance from the of the year.” The table shows the 40°N 14 1 h 9 1 h accurately before subtracting. 2 6 equator toward the poles, also number of daylight hours for Partners should then compare affects the length of daylight. some latitudes in the Northern 60°N 18 1 h 5 1 h their answers before moving to 2 2 In the Northern Hemisphere, Hemisphere. the next exercise.

Test Prep 37. On December 21, what is the difference between the number of 3 For additional practice with a daylight hours at 208 latitude and 608 latitude? 5h10 variety of test item formats: • Test-Prep, p. 213 38. For 208 latitude, what is the difference in daylight hours between the 2h2 • Test-Taking Strategies, p. 209 Take It to the NET shortest and longest days of the year? 5 • Test-Taking Strategies With Online lesson quiz at Transparencies www.PHSchool.com 39. For 608 latitude, what is the difference in daylight hours between the Web Code: aaa-0405 shortest and longest days of the year? 13 h

Mixed Review

Lesson 4-2 Find each sum or difference. Enrichment 4-5 Subtracting Mixed Numbers Critical Thinking 10 7 9 6 21 5 A large financial institution trading on the New York Stock Exchange + 2 + 1 - 2 3 40. 1 41. 1 42. listed its highest selling price in the last year at 80 8 points. The 15 2 3 1 15 15 10 10 24 24 difference between its highest and lowest prices was 26 2 points. Write and solve an equation to find the lowest selling price. 803 8 points 1. What is the highest selling price of the stock? 23 3 4 11 5 8 9 3 3 261 - + - 2 points 43. 44. 45. 2. What is the difference between its highest and lowest prices? 25 25 5 18 18 9 28 28 14 3 1 8 3. What is the least common denominator for 8 and 2 ? 4. Write the prices using the least common denominator. 803 264 8 8 a. Highest selling price b. Difference in selling price Lesson 3-4 Write each fraction in simplest form. subtraction 5. Which operation will you use to find the difference? 803 x 264 8 2 5 8 6. Write an equation to find the lowest selling price. 15 16 36 8 537 3 2 2 8 46. 47. 48. 49. 2 7. Solve the equation. What is the lowest selling price? 25 5 56 7 54 3 4 8. Write an equation that will find the lowest selling price using another operation. 264 x 803 8 1 5 8

3 9. The highest selling price of a stock was 75 4.The difference 1 between its highest and lowest prices was 18 8 . Write and solve Chapter 4 Adding and Subtracting Fractions an equation to find the lowest selling price. 194 575 753 181 x x 575 8;; 4 2 8 5 5 8

Complete the table.

Difference between Monday Wednesday Friday highest and lowest price.

1 5 3 17 10. Stock A 328 326 327 24 1 1 7 35 11. Stock B 24 52 58 8 5 1 9 75 12. Stock C 1036 1018 10812 8 1 1 1 57 13. Stock D 772 734 798 8

194 Reading Understanding Word Problems Reading Math Math Understanding For Use With Lesson 4-5 Word Problems Some problems contain too much information. You need to decide Students need to be able to read which information is necessary for solving the problem. You can use with comprehension to understand the problem-solving plan you learned in Lesson 1-6. Start by asking and solve problems. This page presents two basic questions: yourself, “What do I know?” and “What do I need to find out?” “What do I know?” and “What do I need to find out?” to help Real students identify the key EXAMPLE World information in word problems. Track Team Each member of the track team runs 15 miles each week. On A third related question is “What 7 1 1 information is not needed?” Monday, Celine runs 2 8 miles. She runs 3 4 miles on Tuesday and 2 2 miles on Wednesday. She runs 11 miles per hour. How many more miles does she need to run? Teaching Notes Discuss with students the need to Read and Understand Read for understanding. Summarize the problem. read and solve word problems. Because real-world problems What do I know? What do I need to find out? usually come in either written or verbal form, students must be able • Each team member runs 15 miles • How many miles must to extract relevant information, each week. Celine run on Thursday ignore extra information, and 7 1 • Celine has already run 2 8 , 3 4 , and Friday? figure out what the problem asks. 1 and 2 2 miles. Have a volunteer read the opening • Celine runs 11 miles per hour. paragraph. Ask: What are the steps in the Problem-Solving Plan What information is not needed? you learned in Chapter 1? Read • How fast Celine runs is not needed to solve the problem. and understand; plan and solve; look back and check.

1 EXAMPLE Teaching Tip Have a volunteer read through EXERCISES the example. Have students summarize the problem using For each word problem, answer the questions “What do I know?” and their own plain language. You may “What do I need to ﬁnd out?” Identify any information not needed to solve want to write a list as shown on the problem. 1–2. See margin. the student page to summarize the three types of information. 1. Dressmaking A dressmaker sends 250 dresses to several stores. The Remind students that these same number of dresses is sent to each store. The dressmaker charges questions are part of the ﬁrst step of the problem-solving plan. $59 for each dress. How much money does the dressmaker receive?

3 1 Exercises 3 4 2. Carey worked 4 hours on Monday. Ricky worked 2 hours on Monday Have students work independently and 2 hours on Tuesday.Who worked more hours on Monday? How on the Exercises. Then have them many more? form groups in which members share and evaluate their answers. Students should make adjustments in their work based upon their group discussions. Group members can work together to solve the problems. Reading Math Understanding Word Problems 195

1. I know 250 dresses were The fact that the same and 2 h on Tuesday. I sent, the same number number of dresses was need to ﬁnd out whose was sent to each store, sent to each store is not work time on Monday is and the dressmaker needed. greater and the charges $59 for each difference between the dress. I need to ﬁnd how 2. I know Carey worked two work times. The much money the 3 time Ricky worked on 3h4 on Monday. Ricky dressmaker receives. 1 Tuesday is not needed. worked 42 h on Monday 195