Objective 1 To introduce multiplication with mixed numbers. materials Teaching the Lesson Key Activities Students review conversions from mixed numbers to fractions and from fractions to mixed numbers. Then they multiply mixed numbers by applying the conversions and by using the partial-products method. Key Concepts and Skills • Convert between fractions and mixed numbers. [Number and Numeration Goal 5] • Multiply mixed numbers. [Operations and Computation Goal 5] • Use the partial-products algorithm to multiply whole numbers, fractions, and mixed numbers. Math Journal 2, pp. 272–274 Study Link 8 7 Teaching Aid Master (Math Masters, 414; optional) slates See Advance Preparation [Operations and Computation Goal 5] Ongoing Assessment: Informing Instruction See page 661. Ongoing Assessment: Recognizing Student Achievement Use journal page 273. [Operations and Computation Goal 5] 2 Ongoing Learning & Practice Students practice using unit fractions to find a fraction of a number. Students practice and maintain skills through Math Boxes and Study Link activities. 3 materials Math Journal 2, pp. 275 and 276 Study Link Master (Math Masters, p. 237) materials Differentiation Options READINESS EXTRA PRACTICE Students compare and order improper fractions. Students practice converting between fractions, decimals, and percents by playing Frac-Tac-Toe. Additional Information Advance Preparation For Part 1, draw several blank “What’s My Rule?” rule boxes and tables on the board to use with the Study Link 8 7 Follow-up. Student Reference Book, pp. 309–311 Game Masters (Math Masters, pp. 472–484) number cards 0–10 (4 of each from the Everything Math Deck, if available); counters; slates calculator (optional) Technology Assessment Management System Journal page 273 See the iTLG. Lesson 8 8 659 Getting Started Mental Math and Reflexes Math Message Have students write each mixed number as a fraction. Suggestions: Complete journal page 272. 2 5 13 3 7 34 39 9 4 46 67 7 7 87 810 10 4 29 55 5 5 29 38 8 Study Link 8 7 Follow-Up Have partners compare answers and resolve differences. Ask volunteers to write the incomplete version of their “What’s My Rule?” table for the class to solve. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 2, p. 272) Ask students why the hexagons in the last row of the example on the journal page are divided into sixths. Sample answer: To add the 3 in 3 56, you need a common denominator. A simple way is to think of each whole as 66. Ask volunteers to share their solution strategies for Problems 1–8. Multiplying with WHOLE-CLASS ACTIVITY Mixed Numbers Ask students how they would use the partial-products method to calculate 6 435. Discuss student responses as you summarize the following strategy: Student Page Date Time LESSON Review Converting Fractions to Mixed Numbers 8 8 1. Think of 435 as 4 35. Math Message You know that fractions larger than 1 can be written in several ways. Whole Rule Example: hexagon If a is worth 1, what is worth? 5 5 5 The mixed-number name is 3 6 (3 6 means 3 6). 23 The fraction name is 6. Think sixths: 5 5 23 3 6, 3 6, and 6 are different names for the same number. Write the following mixed numbers as fractions. 13 5 5 3 3 1. 2 5 2 3. 1 3 3 9 8 7 2. 4 8 6 4. 3 4 18 4 , or 9 2 Write the following fractions as mixed or whole numbers. 1 2 3 7 5. 3 18 7. 4 6 6. 1 2 4 4 , or 4 1 2 9 8. 3 6 3 Add. 7 7 9. 2 8 3 11. 3 5 2 8 3 3 5 3 3 10. 1 4 1 12. 6 2 3 14 1 8 3 272 Math Journal 2, p. 272 660 Calculate the partial products and add. Unit 8 Fractions and Ratios 6 435 6 (4 35) 2. Write the problem as the sum of partial products. (6 4) (6 35) 3. Calculate the partial products. 24 158 4. Convert 158 to a mixed number. 24 335 5. Add. 2735 Ongoing Assessment: Informing Instruction Watch for students who have difficulty organizing partial products when multiplying two mixed numbers. Show them the diagram below, and have them write the partial products in a column. 3 4 3 23 6 6 2∗3 2 2∗ 2 3 3 4 2 34 2 3 2 3 2 3 2 3 ∗3 ∗ 3 43 6 3 6 4 6 12 2 112 3 4 1 2 922, or 10 Ask students how they might use improper fractions to solve the problem. Again, discuss student responses as you summarize the following strategy: Convert whole numbers and mixed numbers to improper fractions. 1. Think of 6 as 61 and 435 as 253. 6 435 2. Rewrite the problem as fraction multiplication. 61 253 623 3. Use a fraction multiplication algorithm. 15 4. Multiply. 138 5 5. Simplify the answer by converting 138 to a mixed number. 5 2735 Ask students to suggest advantages and disadvantages for each method. Expect a variety of responses. Sample answers: The partial-products method lets you work with smaller numbers but has more calculations. The improper-fraction method lets you multiply fractions where one of the denominators will be one, but you have a larger number to divide to simplify the answer. Ask students to work through two or three additional examples, using either of the above strategies or others of their own choosing. After each problem, ask volunteers to share their solution strategies. Suggestions: ● 4 35 225 ● 214 23 112 ● 223 3 8 Student Page Date Time LESSON Multiplying Fractions and Mixed Numbers 8 8 Using Partial Products Example 2: Example 1: 1 1 1 1 2 3 2 2 (2 3) (2 2) 1 2 1 2 3 4 5 .(3 4) 5 2 6 22 4 3 5 .5 2 2 1 1 1 4 1 3 2 2 3 1 3 1 2 1 1 5 2 1 2 5 .20 1 0 3 1 10 1 6 5 5 6 Converting Mixed Numbers to Fractions Example 3: 1 1 Example 4: 7 5 2 3 2 2 3 2 1 2 13 2 3 4 5 4 5 35 5 6 5 6 6 3 26 20 1 20 1 10 Solve the following fraction and mixed-number multiplication problems. 7 77 43 , or 7 , or 1 1 3 1 10 10 8 1. 3 2 2 5 2. 10 4 2 3. The back face of a calculator has an area of about 11 16 64 in2. 4. The area of this sheet of notebook paper is about 5 ACMECALC INC. Model# JETSciCalc Serial# 143 58 " 84 1 10 2 " in2. 7 28 " 8" 5. The area of this computer disk is about 5 13 12 in2. 3 58 6. The area of this 5 36 " 1 32 " flag is about 8165, or 825 yd2. 7. Is the flag’s area greater or less than that of your desk? 1 2 3 yd 3 3 5 yd Answers vary. 273 Math Journal 2, p. 273 Lesson 8 8 661 Student Page Date Time LESSON Track Records on the Moon and the Planets 8 8 Multiplying Fractions and PARTNER ACTIVITY Mixed Numbers Every moon and planet in our solar system pulls objects toward it with a force called gravity. 2 In a recent Olympic games, the winning high jump was 7 feet 8 inches, or 73 feet. The winning pole vault was 19 feet. Suppose that the Olympics were held on Earth’s Moon, or on Jupiter, Mars, or Venus. What height might we expect for a winning high jump or a winning pole vault? (Math Journal 2, pp. 273 and 274) Assign both journal pages. Encourage students to consider the numbers in each problem and then to use the method that is most efficient for that problem. Circulate and assist. 1. On the Moon, one could jump about 6 times as high as on Earth. What would be the height of the winning … 46 high jump? About feet 114 pole vault? About feet 3 2. On Jupiter, one could jump about 8 as high as on Earth. What would be the height of the winning … 69 7 high jump? About 24 , or 2 8 feet pole vault? About 3. On Mars, one could jump about 57 , 8 or 718 feet 2 2 3 times as high as on Earth. What would be the height of the winning … 152 184 4 , , or 20 9 feet high jump? About 9 pole vault? About 3 or 50 23 feet Ongoing Assessment: Recognizing Student Achievement 1 4. On Venus, one could jump about 1 7 times as high as on Earth. What would be the height of the winning … 152 184 16 , , or 8 21 feet high jump? About 21 pole vault? About 7 or 21 57 feet Journal Page 273 Problem 5 Use journal page 273, Problem 5 to assess students’ understanding of multiplication with mixed numbers. Have students complete an Exit Slip (Math Masters, 414) for the following: Explain how you solved Problem 5 on journal page 273. Students are making adequate progress if they correctly reference using partial products, improper fractions, or a method of their own. 5. Is Jupiter’s pull of gravity stronger or weaker than Earth’s? Explain your reasoning. Sample answer: Because you can’t jump as high on Jupiter as you can on Earth, the gravity pulling you back on Jupiter must be stronger. Try This 6. The winning pole-vault height given above was rounded to the nearest whole 1 number. The actual winning height was 19 feet 4 inch. If you used this actual [Operations and Computation Goal 5] measurement, about how many feet high would the winning jump be … 17 11418 7 128 on the Moon? on Jupiter? 50 1138 213412 on Mars? on Venus? 274 Math Journal 2, p. 274 2 Ongoing Learning & Practice Using Unit Fractions to Find INDEPENDENT ACTIVITY a Fraction of a Number (Math Journal 2, p. 275) Students practice using unit fractions to find the fraction of a number. Math Boxes 8 8 Student Page Date Time LESSON (Math Journal 2, p. 276) Finding Fractions of a Number 8 8 One way to find a fraction of a number is to use a unit fraction. A unit fraction is a fraction with 1 in the numerator. You can also use a diagram to help you understand the problem. 7 1 8 of 32 is 4. So Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-6. The skill in Problem 5 previews Unit 9 content. 32 Example: What is 8 of 32? 7 8 of 32 is 7 4 28. ? Solve. 1 1. 5 of 75 1 4. 8 of 120 15 15 2 2. 5 of 75 3 5. 8 of 120 30 45 4 3. 5 of 75 5 6. 8 of 120 60 75 Solve Problems 7–18. They come from a math book that was published in 1904. 1 2 First think of 3 of each of these numbers, and then state 3 of each. 7. 9 10. 3 6 2 8. 6 11. 21 4 14 9. 12 12. 30 8 20 1 3 First think of 4 of each of these numbers, and then state 4 of each. 13. 32 16. 24 24 18 14. 40 17. 20 30 15 15. 12 18. 28 9 21 19. Lydia has 7 pages of a 12-page song memorized. Has she memorized more 2 than 3 of the song? No 1 20. A CD that normally sells for $15 is on sale for 3 off. What is the sale price? $10 1 21. Christine bought a coat for 4 off the regular price. She saved $20. What did she pay for the coat? $60 22. Seri bought 12 avocados on sale for $8.28. What is the unit price, the cost for 1 avocado? $0.69 275 Math Journal 2, p. 275 662 INDEPENDENT ACTIVITY Unit 8 Fractions and Ratios Writing/Reasoning Have students write a response to the following: Explain how to use the division rule for finding equivalent fractions to solve Problem 4b. Sample answer: The division rule states that you can rename a fraction by dividing the numerator and the denominator by the same number. I divided the numerator and the denominator by 2 to rename the 4 4 2 2 fraction 50; 502 25 Student Page Study Link 8 8 INDEPENDENT ACTIVITY (Math Masters, p. 237) Date Time LESSON Math Boxes 8 8 1. a. Write a 7-digit numeral that has 5 in the ten-thousands place, 6 in the tens place, 9 in the ones place, 7 in the hundreds place, 3 in the hundredths place, and 2 in all the other places. b. Write this numeral in expanded notation. Home Connection Students practice multiplying fractions and mixed numbers. They find the areas of rectangles, triangles, and parallelograms. 52,769.23 50,000 2,000 700 60 9 0.2 0.03 4 1 3. Ellen played her guitar 2 hours on 3 Saturday and 11 hours on Sunday. How 4 2. Write 3 equivalent fractions for each number. Sample 4 6 8 14 21 28 6 9 12 3 b. 10 15 20 5 10 15 20 5 c. 16 24 32 8 2 4 6 20 d. 3 6 9 30 5 1 10 25 e. 2 10 20 50 , , , , , 2 a. 7 3 Differentiation Options answers: much longer did she play on Saturday? 2 13 114 h 1 hours h 1 12 Answer: , , , , , Number model: 59 READINESS Ordering Improper Fractions SMALL-GROUP ACTIVITY 5 2 4 b. 50 6 c. 20 scalene triangle. 2 8 a. 20 15–30 Min To review converting between fractions and mixed numbers and finding common denominators, have students order a set of improper fractions. Write the following fractions on the board: 7 4 7 11 , , , and . Ask students to suggest how to order the fractions 2 1 3 6 from least to greatest. Expect that students will suggest the same strategies they used with proper fractions, such as putting the numbers in order and then comparing them to a reference. Use their responses to discuss and demonstrate the following methods: 71 5. Use your Geometry Template to draw a 4. Complete. 25 3 How does the scalene triangle differ from other triangles on the Geometry Template? 10 1 2 d. 18 9 None of the sides are the same length. 108 109 134 276 Math Journal 2, p. 276 Rename each improper fraction as an equivalent fraction with a common denominator. Ask students what common denominator to use. Sample answer: Use 6 because the other denominators are all factors of 6. Have volunteers rename the fractions and write the equivalent fractions on the board underneath the first list of fractions. 261, 264, 164, 161 Write each fraction as a mixed number. Ask volunteers to write the mixed numbers on the board underneath the second list. 312, 4, 213, 156 Ask students to order the 3 lists on their slates. 161, 73, 72, 41; 5 11 14 21 24 1 1 , , , ; and 1, 2, 3, 4 Discuss any difficulties or curiosities 6 6 6 6 3 2 6 that students encountered. Study Link Master Name Date STUDY LINK 88 1. Multiply. 46 , 5 3 º 2 24 or 11121 85 , 5 1 c. 4 º 24 31234 a. 4 6 or 6 4 296 , 1 3 e. 3 º 1 60 12 EXTRA PRACTICE Playing Frac-Tac-Toe PARTNER ACTIVITY 2. Students play a favorite version of Frac-Tac-Toe to practice converting between fractions, decimals, and percents. or 5 5 b. 8 º 2 5 10 , 40 1 4 or 76–78 7 7 24 175 , 1 1 d. 2 º 3 24 or 364 , 4 2 f. 2 º 3 40 or 9110 3 41145 8 5 8 Find the area of each figure below. Area of a Rectangle Area of a Triangle Abºh A 2 º b º h 15–30 Min Area of a Parallelogram 1 a. (Student Reference Book, pp. 309–311; Math Masters, pp. 472–484) Time Multiplying Fractions and Mixed Numbers b. Abºh c. 5 6 1 2 3 yd ft 3 2 4 ft 1 2 2 ft 2 3 3 yd Area 3. 859 4 ft yd2 Area 512 ft2 2112 Area ft2 The dimensions of a large doghouse are 2 1 times the dimensions of 2 a small doghouse. a. If the width of the small doghouse is 2 feet, what is the width of the large doghouse? 5 b. feet If the length of the small doghouse is 2 1 feet, what is the length of the 4 large doghouse? 5 8 feet 5 2 ft 1 2 4 ft Math Masters, p. 237 Lesson 8 8 663

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