Objective To provide experience finding the product of a whole number and a fraction. 1 materials Teaching the Lesson Key Activities Students use area models and a fraction multiplication algorithm to find the products of whole numbers and fractions. Key Concepts and Skills Math Journal 2, pp. 268–270 Study Link 8 6 slates Class Data Pad (optional) • Use given denominators to rename numbers as fractions. [Number and Numeration Goal 5] • Find fractions of a set. [Number and Numeration Goal 5] • Use an area model and a fraction multiplication algorithm to find fraction-by-wholenumber products. [Operations and Computation Goal 5] 2 materials Ongoing Learning & Practice Students practice using order of operations by playing Name That Number. Students practice and maintain skills through Math Boxes and Study Link activities. Ongoing Assessment: Recognizing Student Achievement Use Math Boxes, Problem 1. [Number and Numeration Goal 5] 3 materials Differentiation Options READINESS Students rename whole numbers as fractions and find common denominators. ENRICHMENT Students explore the use of the commutative property to simplify finding the product of fractions. Math Journal 2, p. 271 Student Reference Book, p. 325 Study Link Master (Math Masters, p. 235) number cards 0–9 (4 of each from the Everything Math Deck, if available) calculator (optional) EXTRA PRACTICE Students practice multiplying fractions and whole numbers. Teaching Master (Math Masters, p. 236) 5-Minute Math, pp. 23 and 185 Class Data Pad (optional) slates Technology Assessment Management System Math Boxes, Problem 1 See the iTLG. 654 Unit 8 Fractions and Ratios Getting Started Mental Math and Reflexes Math Message Have students solve fraction-of problems. Remind them to think of when multiplying fractions. Complete journal page 268. 1 1 1 of 5 2 10 1 1 1 of 8 2 16 2 1 1 of 14 14 2 3 1 1 14 3 14 5 2 of 12 6 9 3 6 2 1 4 3 2 3 1 of 3 4 2 4 4 16 of 5 5 25 8 2 16 9 3 27 Study Link 8 6 Follow-Up Have partners compare answers and resolve differences. Ask volunteers to share their solution strategies for Problems 10 and 11. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 2, p. 268) Ask students whether they remember problems of this type from earlier grades. Point out that in the fifth grade they still solve the problems, but they also write the number models for calculations that solve the problems. Review the problems by having volunteers rewrite each problem in the form ab dc on the board and then write the product. 1. 1 6 2. a. b. c. 3. a. b. of 1 16 11 16 3 4 2 3 2 2 1 4 1 8 4. a. 1 of 1 34 1 34 b. of 1 23 11 23 c. of 1 22 11 22 1 5. 5 6 1 2 1 8 1 2 of 14 of 12 of 18 1 2 1 8 1 2 14 18 12 11 6 18 11 6 of 12 56 112 660 10 6 of 16 14 116 14 4 of 16 1 8 16 1 16 8 Student Page Date Time LESSON A Blast from the Past 8 7 1. From Kindergarten Everyday Mathematics: 2 This slice of pizza is what fraction of the whole pizza? 1 6 2. From First Grade Everyday Mathematics: Using an Area Model to Write a fraction in each part of the diagrams below. Then color the figures as directed. WHOLE-CLASS ACTIVITY Represent the Product of a Fraction and a Whole Number a. b. 1 4 1 4 1 4 1 4 c. 1 3 3 4 1 3 1 3 2 3 Color . 1 1 2 2 2 2 Color . Color . 3. From Second Grade Everyday Mathematics: a. b. (Math Journal 2, p. 269) Ask students to solve the following problem on their slates: 1 2 3 2? Ask volunteers to show on the board or Class Data Pad how an area model might be used to represent this problem. The basic idea is that there are several wholes, each of which is divided into fractional parts. Summarize students’ presentations using the steps on the next page: 1 Color 4 of the beads. Color 8 of the beads. 4. From Third Grade Everyday Mathematics: 1 1 a. of 2 4 1 8 1 1 b. of 8 2 1 16 1 1 c. of 2 8 1 16 5. From Fourth Grade Everyday Mathematics: 5 6 Cross out of the dimes. 268 Math Journal 2, p. 268 Lesson 8 7 655 Student Page Date Time LESSON Area Models 8 7 Draw an area model for each product. Then write the product as a fraction or as a mixed number. 2 3 Example: 2 4 1 1. 4 3 3 4 , 3 or 113 , or 113 2. Note that the denominator of the fraction is 3. Divide both rectangles into thirds. 3 4 1 2. 3 4 3 3. 2 5 5 6 , or 115 9 , or 118 3 4. 3 8 8 1. Draw a number of rectangles equal to the whole number. In this example, the whole is 2. 3. Note that the numerator of the fraction is 2. Shade 23 of each rectangle. 269 Math Journal 2, p. 269 In each rectangle, there are 3 parts; 2 of them are shaded. In the 2 rectangles, there are 4 shaded thirds altogether. So 23 2 43 or 113. Assign the journal page. When most students have finished, bring the class together to discuss answers. Using an Algorithm to Multiply a Fraction and a Whole Number Student Page Date (Math Journal 2, p. 270) Time LESSON Using the Fraction Multiplication Algorithm 8 7 An Algorithm for Fraction Multiplication a b c d ac bd The denominator of the product is the product of the denominators, and the numerator of the product is the product of the numerators. 2 3 Example: 2 2 3 2 2 2 3 1 22 31 4 1 , or 1 3 3 2 1 Think of 2 as . Apply the algorithm. Calculate the numerator and denominator. 18 1 , or 4 4 2 15 1 , or 1 3 2 5 10 21 , 8 24 , 4 6 5 7 2. 3 8 3. 4. 10 5. Use the given rule to complete the table. in ( ) Rule 3 5 º 1 2 2 4 5 3 4 3 5 2 or 258 or 445 Assign the journal page. Circulate and assist. 6. What is the rule for the table below? out ( ) 3 10 6 1 , or 1 5 5 12 25 9 20 9 4 , or 1 5 5 in ( ) out ( ) Rule º 1 2 270 Math Journal 2, p. 270 656 Refer students to the top of journal page 270, and ask how this algorithm could be used to multiply a fraction and a whole number such as 23 2. Rewrite the whole number as a fraction. Remind students that any number can be thought of as a fraction with a denominator of 1. Ask a volunteer to demonstrate using the algorithm to solve 3 2. 22 2 2 4 1 , or 1 3 1 3 3 31 Use the fraction multiplication algorithm to calculate the following products. 3 1. 6 4 PARTNER ACTIVITY Unit 8 Fractions and Ratios 2 3 3 4 2 6 3 8 7 8 7 16 3 11 2 When students have completed the page, ask what patterns they notice about the numerators and denominators when multiplying fractions by whole numbers. The denominators in the products are always the same as the denominator of the fraction factor. The numerator is the product of the whole number and the numerator of the fraction factor. Student Page Emphasize that, when rewriting the whole number as a fraction, the denominator is always 1. Ask students what true statement they can make about multiplying by 1. Any number times 1 is itself. Accordingly, the patterns for multiplying fractions by whole ac numbers can be represented as ab c b . Date Time LESSON 8 7 Math Boxes 2. Write true or false for each number sentence. 1. Complete. 1 a. 5 2 b. 3 5 c. 8 4 d. 7 4 20 6 15 15 0 false 1 5 1 1 1 5 c. 2 6 3 3 2 6 40 d. 16 (4 8 2) / 2 3 32 1 2,160 25 24 42 b. (2 10 ) (1 10 ) (6 10 ) 10 true a. 5 (6 3) (5 6) (5 3) 30 2 9 24 6 108 109 56 e. 10 1 billion 6 true false false 222 223 3. On the grid, draw each animal whose 6 location is given below. 2 Ongoing Learning & Practice Lake 5 a. A bird in C2. 4 b. A fish in D6. 3 c. A turtle in E3. 2 d. A snake in F1. Playing Name That Number 1 PARTNER ACTIVITY e. A frog in F4. 208 A 4. Draw an isosceles triangle. (Student Reference Book, p. 325) B C D E F 5. The shapes below represent geometric solids. Name the solids. Write a definition of an isosceles triangle. An isosceles triangle has two sides and two angles that are equal. Students play Name That Number to practice writing number sentences using order of operations. Encourage them to find number sentences that use all five numbers. Students can use numbers as exponents or to form fractions. a. cone b. triangular prism 147–149 144 271 Math Journal 2, p. 271 Math Boxes 8 7 INDEPENDENT ACTIVITY (Math Journal 2, p. 271) Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-5. The skills in Problems 4 and 5 preview Unit 9 content. Ongoing Assessment: Recognizing Student Achievement Math Boxes Problem 1 Use Math Boxes, Problem 1 to assess students’ understanding of converting fractions to decimals and percents. Have students write a response to the following: Convert the fractions in Math Boxes, Problem 1 to the decimal and the percent equivalents, and explain your solution strategy. Students are making adequate progress if their conversions are correct and their writing demonstrates an understanding of the role of the numerator and the denominator. Some students might refer to the fact that the fractions are equivalent; therefore, they have the same decimal and percent equivalents. NOTE Students may use calculators. If they do, remind them to explain the part of the conversion process the calculator is performing. [Number and Numeration Goal 5] Study Link Master Name STUDY LINK 87 Date Use the fraction multiplication algorithm to calculate the following products. 45 3 , or 15 5 1. 3 º9 1 3. 8 º5 5 8 70 , º 14 6 or 1123 5 5. 6 7. 3 2. 8 in () º4 2 3 4 5 8 9 INDEPENDENT ACTIVITY (Math Masters, p. 235) 5 4 7 3 8. 1 º 4 9. 4. 20 º 3 6. 27 º 2 4 9 36 1 or 4 2 8 60 , or 15 4 54 , or 6 9 , 73 out () 8 3 16 5 , or 223 , or 3 15 32 , or 35 9 9 20 , or 5 4 28 , or 91 3 3 What is the rule for the table below? Rule Home Connection Students solve problems to find a fraction of a whole number and a fraction of a fraction. They solve “What’s My Rule?” problems and make a function table for fraction multiplication. º 12 Use the given rule to complete the table. Rule Study Link 8 7 Time Multiplying Fractions and Whole Numbers in () out () 2 1 2 3 3 4 5 6 5 24 2 3 1 6 Make and complete your own “What’s My Rule?” table on the back of this page. Answers vary. Math Masters, p. 235 Lesson 8 7 657 3 Differentiation Options READINESS Writing Whole Numbers SMALL-GROUP ACTIVITY 15–30 Min as Fractions To reinforce students’ understanding of whole numbers written as fractions, guide them through the following activity: Remind students that any number can be thought of as a fraction with a denominator of 1. Write the examples on the board or Class Data Pad. 236 0.5 Examples: 3 31, 236 1 , and 0.5 1 Ask students why this is true. The denominator represents how many parts it takes to make a whole. If it takes only 1 part, then the numerator represents wholes. When applying a multiplication algorithm to problems of the form ab n, where one factor is a fraction and the other factor is a whole number, think of the whole number as n1. Ask students to write each number as a fraction on their slates. Then repeat the numbers, and ask students to rename each as a fraction with a denominator of 2. 5 51; 120 3.5 7 3.5 1 ; 2 0 20 2 5 11 ; 2 7 71; 124 1 11; 22 100% 11; 22 140 280 140 1 ; 2 0.5 1 0.5 1 ; 2 23 81; 126 ENRICHMENT Teaching Master Name Date LESSON 87 Time An Algorithm for Fraction Multiplication c d aºc bºd º The denominator of the product is the product of the factor denominators, and the numerator of the product is the product of the factor numerators. aºc bºd cºa dºb The commutative property lets us write as . Study the examples. 6 112 112 8 14 2 7 º 16 ; , or Example 1: 7 º 1 8 º 21 8 21 168 168 8 21 3 6 16 2 1 2 7 7 º 16 2º1 Example 2: 7 º 1 º º 8 1. 8 º 21 21 8 21 1 3 1º3 3 1 2 Example 3: 78 º 16 21 1 1º2 1º3 2 3 Describe the similarities and differences between Examples 2 and 3. Use what you have discovered to solve the following problems. Show your work. 14 3. 60 2 2 º 1 15 21 EXTRA PRACTICE 5-Minute Math SMALL-GROUP ACTIVITY 5–15 Min 3 Both examples have the same factors and products. Example 3 has fewer steps than Example 2 because the fractions are reduced without rearranging them first. 36 4. 88 3 3 º 3 16 72 25 5. 54 Math Masters, p. 236 658 To extend students’ understanding of fraction multiplication and lowest terms, have students explore the process of reducing factors in fraction multiplication problems. When students have completed the Math Masters page, discuss any difficulties or curiosities they encountered. Describe the similarities and differences between Examples 1 and 2. Both examples have the same factors and products. Example 1 is renamed in simplest form after multiplying. Example 2 is renamed in simplest form before multiplying. 2. 15–30 Min (Math Masters, p. 236) Simplifying Fraction Factors a b Simplifying Fraction Factors PARTNER ACTIVITY Unit 8 Fractions and Ratios 10 º 36 27 45 To offer students more experience with fractions and whole numbers, see 5-Minute Math, pages 23 and 185.

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