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 Щ— Math Journal 2, pp. 268–270 Щ— Study Link 8 6 бњ Щ— slates Щ— 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. Щ— Math Journal 2, p. 271 Щ— Student Reference Book, p. 325 Щ— Study Link Master (Math Masters, p. 235) Щ— number cards 0–9 (4 of each from the Everything Math Deck, if available) Щ— calculator (optional) EXTRA PRACTICE Students practice multiplying fractions and whole numbers. Щ— Teaching Master (Math Masters, p. 236) Щ— 5-Minute Math, pp. 23 and 185 Щ— Class Data Pad (optional) Щ— 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 бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ 5 2 10 1 1 1 бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ 8 2 16 2 1 1 бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ 14 14 2 3 1 1 бЋЏбЋЏ ‫ ء‬ᎏᎏ бЋЏбЋЏ 14 3 14 5 2 бЋЏбЋЏ of 12 6бЋЏбЋЏ 9 3 6 2 бЋЏбЋЏ ‫ ء‬ᎏᎏ 1 4 3 2 3 1 бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ 3 4 2 4 4 16 бЋЏбЋЏ of бЋЏбЋЏ бЋЏбЋЏ 5 5 25 8 2 16 бЋЏбЋЏ ‫ ء‬ᎏᎏ бЋЏбЋЏ 9 3 27 Study Link 8 6 Follow-Up бњ Have partners compare answers and resolve differences. Ask volunteers to share their solution strategies for Problems 10 and 11. 1 Teaching the Lesson б¤ 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 бЋЏabбЋЏ ‫ ء‬ᎏdcбЋЏ on the board and then write the product. 1. 1 бЋЏбЋЏ 6 2. a. b. c. 3. a. b. of 1 бЋЏ16бЋЏ ‫ ء‬ᎏ11бЋЏ П бЋЏ16бЋЏ 3 бЋЏбЋЏ 4 2 бЋЏбЋЏ 3 2 бЋЏбЋЏ 2 1 бЋЏбЋЏ 4 1 бЋЏбЋЏ 8 4. a. 1 of 1 бЋЏ34бЋЏ ‫ ء‬ᎏ1бЋЏ П бЋЏ34бЋЏ b. of 1 бЋЏ23бЋЏ ‫ ء‬ᎏ11бЋЏ П бЋЏ23бЋЏ c. of 1 бЋЏ22бЋЏ ‫ ء‬ᎏ11бЋЏ П бЋЏ22бЋЏ П 1 5. 5 бЋЏбЋЏ 6 1 бЋЏбЋЏ 2 1 бЋЏбЋЏ 8 1 бЋЏбЋЏ 2 of бЋЏ14бЋЏ of бЋЏ12бЋЏ of бЋЏ18бЋЏ 1 бЋЏбЋЏ 2 1 бЋЏбЋЏ 8 1 бЋЏбЋЏ 2 ‫ ء‬ᎏ14бЋЏ П бЋЏ18бЋЏ ‫ ء‬ᎏ12бЋЏ П бЋЏ11бЋЏ 6 ‫ ء‬ᎏ18бЋЏ П бЋЏ11бЋЏ 6 of 12 бЋЏ56бЋЏ ‫ ء‬ᎏ112бЋЏ П бЋЏ660бЋЏ П 10 6 of 16 бЋЏ14бЋЏ ‫ ء‬ᎏ116бЋЏ П бЋЏ14бЋЏ П4 of 16 1 бЋЏбЋЏ 8 ‫ء‬ 16 бЋЏбЋЏ 1 П 16 бЋЏбЋЏ 8 Student Page Date Time LESSON A Blast from the Past 8 7 бњ 1. From Kindergarten Everyday Mathematics: П2 This slice of pizza is what fraction of the whole pizza? 1 бЋЏбЋЏ 6 2. From First Grade Everyday Mathematics: б¤ 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 бЋЏбЋЏ. 1 1 2 2 2 2 Color бЋЏбЋЏ. Color бЋЏбЋЏ. 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 бЋЏбЋЏ 3 ‫ء‬2П? 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 бЋЏ4бЋЏ of the beads. Color бЋЏ8бЋЏ of the beads. 4. From Third Grade Everyday Mathematics: 1 1 a. бЋЏбЋЏ of бЋЏбЋЏ П 2 4 1 бЋЏбЋЏ 8 1 1 b. бЋЏбЋЏ of бЋЏбЋЏ П 8 2 1 бЋЏбЋЏ 16 1 1 c. бЋЏбЋЏ of бЋЏбЋЏ П 2 8 1 бЋЏбЋЏ 16 5. From Fourth Grade Everyday Mathematics: 5 6 Cross out бЋЏбЋЏ of the dimes. 268 Math Journal 2, p. 268 Lesson 8 7 бњ 655 Student Page Date Time LESSON Area Models 8 7 бњ Draw an area model for each product. Then write the product as a fraction or as a mixed number. 2 3 Example: бЋЏбЋЏ ‫ ء‬2 П 4 бЋЏбЋЏ 1 1. бЋЏбЋЏ ‫ ء‬4 П 3 3 4 бЋЏбЋЏ, 3 or 1бЋЏ13бЋЏ , or 1бЋЏ13бЋЏ 2. Note that the denominator of the fraction is 3. Divide both rectangles into thirds. 3 бЋЏбЋЏ 4 1 2. бЋЏбЋЏ ‫ ء‬3 П 4 бЋЏбЋЏ 3 3. 2 ‫ ء‬ᎏᎏ П 5 5 6 , or 1бЋЏ15бЋЏ 9 , or 1бЋЏ18бЋЏ бЋЏбЋЏ 3 4. бЋЏбЋЏ ‫ ء‬3 П 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 бЋЏ23бЋЏ 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 бЋЏ23бЋЏ ‫ ء‬2 П бЋЏ43бЋЏ or 1бЋЏ13бЋЏ. Assign the journal page. When most students have finished, bring the class together to discuss answers. б¤ 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 бњ An Algorithm for Fraction Multiplication a бЋЏбЋЏ b c d a‫ء‬c b‫ء‬d ‫ ء‬ᎏᎏ П бЋЏбЋЏ 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: бЋЏбЋЏ ‫ ء‬2 2 бЋЏбЋЏ 3 ‫ء‬2 2 2 3 1 2‫ء‬2 3‫ء‬1 4 1 бЋЏбЋЏ, or 1бЋЏбЋЏ 3 3 2 1 П бЋЏбЋЏ ‫ ء‬ᎏ бЋЏ Think of 2 as бЋЏбЋЏ. П бЋЏбЋЏ Apply the algorithm. П Calculate the numerator and denominator. 18 1 бЋЏбЋЏ, or 4бЋЏбЋЏ 4 2 15 1 бЋЏбЋЏ, or 1бЋЏбЋЏ 3 2 бЋЏбЋЏ ‫ ء‬5 П 10 21 бЋЏбЋЏ , 8 24 бЋЏбЋЏ , 4 6 ‫ ء‬ᎏᎏ П 5 7 2. бЋЏбЋЏ ‫ ء‬3 П 8 3. 4. 10 5. Use the given rule to complete the table. in ( ) Rule П 3 5 Вє бЋЏбЋЏ 1 бЋЏбЋЏ 2 2 4 бЋЏбЋЏ 5 3 бЋЏбЋЏ 4 3 5 2 or 2бЋЏ58бЋЏ or 4бЋЏ45бЋЏ Assign the journal page. Circulate and assist. 6. What is the rule for the table below? out ( ) 3 бЋЏбЋЏ 10 6 1 бЋЏбЋЏ, or 1бЋЏбЋЏ 5 5 12 бЋЏбЋЏ 25 9 бЋЏбЋЏ 20 9 4 бЋЏбЋЏ, or 1бЋЏбЋЏ 5 5 in ( ) out ( ) Rule П Вє 1 бЋЏбЋЏ 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 бЋЏ23бЋЏ ‫ ء‬2. 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 бЋЏ3бЋЏ ‫ ء‬2. 2‫ء‬2 2 2 4 1 бЋЏбЋЏ ‫ ء‬ᎏᎏ П бЋЏбЋЏ П бЋЏбЋЏ, or 1бЋЏбЋЏ 3 1 3 3 3‫ء‬1 Use the fraction multiplication algorithm to calculate the following products. 3 1. бЋЏбЋЏ ‫ ء‬6 П 4 PARTNER ACTIVITY Unit 8 Fractions and Ratios 2 бЋЏбЋЏ 3 3 бЋЏбЋЏ 4 2 бЋЏбЋЏ 6 3 бЋЏбЋЏ 8 7 бЋЏбЋЏ 8 7 бЋЏбЋЏ 16 3 1бЋЏ1бЋЏ 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 a‫ء‬c бЋЏ numbers can be represented as бЋЏabбЋЏ ‫ ء‬c П бЋЏ b . Date Time LESSON 8 7 ᜠଙ Math Boxes 2. Write true or false for each number sentence. 1. Complete. 1 a. бЋЏбЋЏ П 5 2 b. бЋЏбЋЏ П 3 5 c. бЋЏбЋЏ П 8 4 d. бЋЏбЋЏ П 7 4 20 6 15 15 0 false 1 5 1 1 1 5 c. бЋЏбЋЏ П© бЋЏбЋЏ П© бЋЏбЋЏ П бЋЏбЋЏ П© бЋЏбЋЏ П© бЋЏбЋЏ 2 6 3 3 2 6 40 d. 16 ПЄ (4 П© 8 ПЄ 2) / 2 П 3 32 П 1 П 2,160 25 П 24 42 b. (2 ‫ ء‬10 ) П© (1 ‫ ء‬10 ) П© (6 ‫ ء‬10 ) 10 П true a. 5 ‫( ء‬6 П© 3) П (5 ‫ ء‬6) П© (5 ‫ ء‬3) 30 2 9 24 6 П 108 109 56 e. 10 П 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. б¤ 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 б¤ Math Boxes 8 7 INDEPENDENT ACTIVITY бњ (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 ଙ 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 бњ Date Use the fraction multiplication algorithm to calculate the following products. 45 бЋЏбЋЏ 3 , or 15 5 1. бЋЏбЋЏ 3 Вє9П 1 3. бЋЏбЋЏ 8 Вє5П 5 бЋЏбЋЏ 8 70 бЋЏбЋЏ, Вє 14 П 6 or 11бЋЏ23бЋЏ 5 5. бЋЏбЋЏ 6 7. 3 2. бЋЏбЋЏ 8 in (б®Ђ) бќПб®ЂВє4 2 бЋЏбЋЏ 3 4 бЋЏбЋЏ 5 бњ 8 бЋЏбЋЏ 9 INDEPENDENT ACTIVITY (Math Masters, p. 235) 5 бЋЏбЋЏ 4 7 бЋЏбЋЏ 3 8. 1 бќ П б®Ђ Вє бЋЏбЋЏ 4 9. 4. 20 Вє бЋЏ3бЋЏ П 6. 27 Вє бЋЏ2бЋЏ П 4 9 36 1 бЋЏбЋЏ or 4 бЋЏбЋЏ 2 8 60 бЋЏбЋЏ, or 15 4 54 бЋЏбЋЏ, or 6 9 , 73 out (бќ) 8 бЋЏбЋЏ 3 16 бЋЏбЋЏ 5 , or 2бЋЏ23бЋЏ , or 3 бЋЏ15бЋЏ 32 бЋЏбЋЏ, or 3бЋЏ5бЋЏ 9 9 20 бЋЏбЋЏ, or 5 4 28 бЋЏбЋЏ, or 9бЋЏ1бЋЏ 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. Вє 12 П Use the given rule to complete the table. Rule б¤ Study Link 8 7 Time Multiplying Fractions and Whole Numbers in (б®Ђ) out (бќ) 2 1 бЋЏбЋЏ 2 3 3 бЋЏбЋЏ 4 5 бЋЏбЋЏ 6 5 бЋЏбЋЏ 24 2 бЋЏбЋЏ 3 1 бЋЏбЋЏ 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 бњ 657 3 Differentiation Options READINESS б¤ 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: бџ 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 бЋЏ бЋЏбЋЏ Examples: 3 П бЋЏ31бЋЏ, 236 П бЋЏ 1 , and 0.5 П 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. бџ When applying a multiplication algorithm to problems of the form бЋЏabбЋЏ ‫ ء‬n, where one factor is a fraction and the other factor is a whole number, think of the whole number as бЋЏn1бЋЏ. бџ 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 бЋЏ51бЋЏ; бЋЏ12бЋЏ0 3.5 7 бЋЏ бЋЏбЋЏ 3.5 бЋЏ 1 ; 2 0 20 2 ‫ ء‬5 бЋЏ11бЋЏ ; бЋЏ2бЋЏ 7 бЋЏ71бЋЏ; бЋЏ12бЋЏ4 1 бЋЏ11бЋЏ; бЋЏ22бЋЏ 100% бЋЏ11бЋЏ; бЋЏ22бЋЏ 140 280 бЋЏ бЋЏбЋЏ 140 бЋЏ 1 ; 2 0.5 1 бЋЏ бЋЏбЋЏ 0.5 бЋЏ 1 ; 2 23 бЋЏ81бЋЏ; бЋЏ12бЋЏ6 ENRICHMENT Teaching Master Name Date LESSON 87 бњ Time An Algorithm for Fraction Multiplication c d aВєc bВєd Вє бЋЏбЋЏ П бЋЏбЋЏ 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. aВєc bВєd cВєa dВєb The commutative property lets us write бЋЏбЋЏ as бЋЏбЋЏ. Study the examples. 6 112 112 8 14 2 7 Вє 16 ПбЋЏ бЋЏ П бЋЏбЋЏ; бЋЏбЋЏ П¬ бЋЏбЋЏ П бЋЏбЋЏ, or бЋЏбЋЏ Example 1: бЋЏ7бЋЏ Вє бЋЏ1бЋЏ 8 Вє 21 8 21 168 168 8 21 3 6 16 2 1 2 7 7 Вє 16 2Вє1 Example 2: бЋЏ7бЋЏ Вє бЋЏ1бЋЏ ПбЋЏ бЋЏ П бЋЏбЋЏ Вє бЋЏбЋЏ П бЋЏбЋЏ Вє бЋЏбЋЏ П бЋЏбЋЏ П бЋЏбЋЏ 8 1. 8 Вє 21 21 8 21 1 3 1Вє3 3 1 2 Example 3: 78 Вє 16 21 1 1Вє2 1Вє3 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. бЋЏбЋЏ 60 2 2 бЋЏ Вє бЋЏ1бЋЏ П бЋЏ 15 21 EXTRA PRACTICE б¤ 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. бЋЏбЋЏ 88 3 3 бЋЏ Вє бЋЏ3бЋЏ П бЋЏ 16 72 25 5. бЋЏбЋЏ 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 бЋЏбЋЏ b б¤ Simplifying Fraction Factors PARTNER ACTIVITY Unit 8 Fractions and Ratios 10 бЋЏ Вє бЋЏ36бЋЏ П бЋЏ 27 45 To offer students more experience with fractions and whole numbers, see 5-Minute Math, pages 23 and 185.
© Copyright 2024 Paperzz
