Objectives To review multiplying fractions and mixed numbers; and to find reciprocals. 1 materials Teaching the Lesson Key Activities Students review multiplying fractions and mixed numbers. They also practice finding the reciprocal of a number. Key Concepts and Skills Щ— Math Journal 2, pp. 205 and 206 Щ— Student Reference Book, pp. 88вЂ“90 (optional) Щ— calculator вЂў Use an algorithm to multiply fractions and mixed numbers. [Operations and Computation Goal 4] вЂў Write a special case to illustrate a general pattern. [Patterns, Functions, and Algebra Goal 1] вЂў Define and find the reciprocal of a number. [Patterns, Functions, and Algebra Goal 4] Key Vocabulary reciprocal Ongoing Assessment: Recognizing Student Achievement Use journal page 206. [Patterns, Functions, and Algebra Goal 4] 2 materials Ongoing Learning & Practice Students practice finding and comparing products of whole numbers and fractions by playing Fraction/Whole Number Top-It. Щ— Math Journal 2, p. 207 Щ— Student Reference Book, pp. 319 or 320 Students practice and maintain skills through Math Boxes and Study Link activities. Щ— Game Master (Math Masters, p. 478) Щ— Study Link Master (Math Masters, p. 180) Щ— 4 each of number cards 1вЂ“10 (from the Everything Math Deck, if available) Щ— Geometry Template Щ— calculator (optional) 3 materials Differentiation Options READINESS Students convert mixed numbers to improper fractions. ENRICHMENT Students explore sums and products of reciprocals. EXTRA PRACTICE Students find reciprocals of whole numbers, fractions, and decimals. ELL SUPPORT Students add terms to their Math Word Banks. Щ— Teaching Masters (Math Masters, pp. 181 and 182) Щ— Differentiation Handbook Technology Assessment Management System Journal page 206, Problems 1вЂ“10 See the iTLG. 530 Unit 6 Number Systems and Algebra Concepts Getting Started Mental Math and Reflexes Math Message Students indicate thumbs-up if the fractions are equivalent or thumbs-down if they are not equivalent. Complete Problems 1вЂ“3 on journal page 205. Suggestions: 5 22 1.5 3 бЋЏбЋЏ and бЋЏбЋЏ thumbs up 5 10 7 70 бЋЏбЋЏ and бЋЏбЋЏ thumbs down 10 1,000 7 15 38 3 бЋЏбЋЏ and 7 бЋЏбЋЏ thumbs down 7 5 19 76 бЋЏбЋЏ and бЋЏбЋЏ thumbs up 20 80 1бЋЏ6бЋЏ and бЋЏ12бЋЏ thumbs up 1бЋЏ8бЋЏ and бЋЏ16бЋЏ thumbs down 1 Teaching the Lesson б¤ Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 2, p. 205) Go over the answers to Problems 1вЂ“3. Draw attention to the exception to the general pattern in Problem 2вЂ”any nonzero number divided by itself equals 1. Zero is excluded. Division by 0 0 is undefined, so бЋЏ0бЋЏ is undefined. 5 3 3.14 Write the special cases бЋЏ1бЋЏ, бЋЏ16бЋЏ П¬ 1, and бЋЏ1бЋЏ on the board. Ask students to give a general pattern in words for the three special cases. The quotient of a number divided by 1 is that number. ELL Adjusting the Activity To support English language learners, summarize studentsвЂ™ general patterns in words by writing the following on the board: A U D I T O R Y x бЋЏбЋЏ П x 1 x бЋЏбЋЏ П 1 x xвЂ«ШЎвЂ¬1Пx бњ K I N E S T H E T I C бњ T A C T I L E Student Page Date бњ V I S U A L Time LESSON Applying Properties of Multiplication 61 бњ Math Message 88 90 103 1. Write a general pattern in words for the group of three special cases. б¤ Reviewing Multiplication 19 вЂ« ШЎвЂ¬1 П 19 2 бЋЏбЋЏ 7 PARTNER ACTIVITY of Fractions For any number n, n П« 1 П n. General pattern: 2. Write a general pattern in words for the group of three special cases. (Math Journal 2, p. 205) Multiplication of fractions and mixed numbers was covered in Lessons 4-6 and 4-7. Pose several problems from the Student Reference Book, pages 88вЂ“90 to check studentsвЂ™ understanding. Then assign Problems 4вЂ“18 on journal page 205. 2 7 вЂ« ШЎвЂ¬1 П бЋЏбЋЏ 0.084 вЂ« ШЎвЂ¬1 П 0.084 58 бЋЏбЋЏ 58 П1 3 бЋЏбЋЏ 8 3 бЋЏбЋЏ 8 П1 7.02 бЋЏбЋЏ 7.02 П1 For any number n, General pattern: n n П 1. 3. Multiply. Write your answers in simplest form. a. 19 19 4 вЂ« ШЎвЂ¬бЋЏбЋЏ П 4 b. 2 бЋЏбЋЏ 3 2 бЋЏбЋЏ 3 6 6 вЂ« ШЎвЂ¬бЋЏбЋЏ П c. 0.5 2 2 0.5 вЂ« ШЎвЂ¬бЋЏбЋЏ П Multiply. Write your answers in simplest form. When you and your partner have finished solving the problems, compare your answers. 1 бЋЏбЋЏ 4 5. 6 вЂ« ШЎвЂ¬бЋЏбЋЏ П 11 8. 2 бЋЏ бЋЏ вЂ« ШЎвЂ¬1бЋЏ бЋЏ П 11. 3 бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П 14. 4 бЋЏбЋЏ 1 17. 5 1бЋЏбЋЏ 6 3 5 4. бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П 10 6 3 4 7. 2 бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П 4 1 1 2 10. бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П 4 5 7 3 13. бЋЏбЋЏ вЂ« ШЎвЂ¬2бЋЏбЋЏ П 5 10 7 16. бЋЏбЋЏ 8 вЂ«ШЎвЂ¬ 8 бЋЏбЋЏ 7 П 1 бЋЏбЋЏ 10 41 бЋЏбЋЏ 50 1 1 2 3 3 5 2 3 3 8 3 4 4 1 4бЋЏ3бЋЏ 17 бЋЏ бЋЏ 2 32 1 4 вЂ« ШЎвЂ¬бЋЏбЋЏ П вЂ«ШЎвЂ¬ 6 бЋЏбЋЏ 11 П 1 1 1 18. Write three special cases for the general pattern x вЂ« ШЎвЂ¬бЋЏбЋЏ П 1. x 7вЂ«ШЎвЂ¬ 1 бЋЏбЋЏ 7 П1 5 бЋЏбЋЏ 9 1 вЂ«ШЎвЂ¬бЋЏ П1 бЋЏ5бЋЏ 9 3 7 6. 7 вЂ« ШЎвЂ¬бЋЏбЋЏ П 9. 7 бЋЏбЋЏ 3 1 3 вЂ« ШЎвЂ¬бЋЏбЋЏ П 5 6 3 7 бЋЏбЋЏ 9 2 3 12. 1бЋЏ бЋЏ вЂ« ШЎвЂ¬4 бЋЏ бЋЏ П 15. 1 бЋЏбЋЏ 100 100 1 вЂ« ШЎвЂ¬бЋЏбЋЏ П 5 8бЋЏ9бЋЏ 1 Sample answers: 2.5 вЂ«ШЎвЂ¬ 1 бЋЏбЋЏ 2.5 П1 205 Math Journal 2, p. 205 Lesson 6 1 бњ 531 Student Page Date б¤ Defining the Reciprocal Time LESSON Reciprocals 61 бњ Reciprocal Property a b b a If a and b are any numbers except 0, then бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ П 1. aвЂ«ШЎвЂ¬ a бЋЏбЋЏ b 1 бЋЏбЋЏ a and b бЋЏбЋЏ a 93 are called reciprocals of each other. 1 a П 1, so a and бЋЏбЋЏ are reciprocals of each other. а¬™а¬™ а¬™ а¬™ а¬™ а¬™а¬™ а¬™ а¬™а¬™ Find the reciprocal of each number. Multiply to check your answers. 1 бЋЏбЋЏ 6 4 бЋЏбЋЏ 3 1. 6 3 3. бЋЏбЋЏ 4 8 бЋЏбЋЏ, or 3 2 бЋЏбЋЏ 17 3 бЋЏбЋЏ 14 3 5. бЋЏбЋЏ 8 1 7. 8 бЋЏбЋЏ 2 2 9. 4 бЋЏбЋЏ 3 11. 0.1 13. 0.75 15. 0.375 10 вЂ“ or 1.3 вЂ“ or 2.6 4 бЋЏбЋЏ, 3 8 бЋЏбЋЏ, 3 Solve mentally. 5 7 5 19. бЋЏ7бЋЏ вЂ« ШЎвЂ¬бЋЏ5бЋЏ вЂ« ШЎвЂ¬4 бЋЏ7бЋЏ вЂ«ШЎвЂ¬ 3 бЋЏбЋЏ 10 вЂ«ШЎвЂ¬ 7 бЋЏбЋЏ 10 вЂ«ШЎвЂ¬ 3 1бЋЏ7бЋЏ 1 4. бЋЏбЋЏ 3 13 6. бЋЏбЋЏ 16 5 8. 3 бЋЏбЋЏ 6 1 10. 6 бЋЏбЋЏ 4 12. 0.4 14. 2.5 16. 5.6 1 3бЋЏ2бЋЏ 5 4бЋЏ7бЋЏ 1 1 17. 3 бЋЏ2бЋЏ вЂ« ШЎвЂ¬4 вЂ« ШЎвЂ¬бЋЏ4бЋЏ 1 21. 3бЋЏ3бЋЏ 2 2бЋЏ3бЋЏ 2. 17 1 1 бЋЏбЋЏ 17 3 бЋЏбЋЏ 1 16 3 бЋЏбЋЏ, or 1бЋЏбЋЏ 13 13 6 бЋЏбЋЏ 23 4 бЋЏбЋЏ 25 5 бЋЏбЋЏ, or 2.5 2 2 бЋЏбЋЏ, or 0.4 5 5 бЋЏбЋЏ, or 0.1786 28 1 2 18. бЋЏ6бЋЏ вЂ« ШЎвЂ¬бЋЏ5бЋЏ вЂ« ШЎвЂ¬6 1 1 20. 2 вЂ« ШЎвЂ¬8 бЋЏ2бЋЏ вЂ« ШЎвЂ¬бЋЏ2бЋЏ 22. 3.875 вЂ« ШЎвЂ¬2.5 вЂ« ШЎвЂ¬0.4 2 бЋЏбЋЏ 5 1 8бЋЏ2бЋЏ 3.875 WHOLE-CLASS DISCUSSION of a Number (Math Journal 2, p. 205) Bring the class together to discuss Problems 4вЂ“18. Students may have discovered a visual pattern: If a fraction is multiplied by the same fraction turned upside down, then the product is 1. Pairs of numbers whose product is 1 are called reciprocals. To support English language learners, write reciprocal on the board and list some examples. Ask students for additional examples to add to the 1 list. (Do not erase the board.) For example, 2 is the reciprocal of бЋЏ2бЋЏ 1 1 and бЋЏ2бЋЏ is the reciprocal of 2 because бЋЏ2бЋЏ вЂ« ШЎвЂ¬2 П 1. The numbers 1 and вЂ“1 are their own reciprocals (1 вЂ« ШЎвЂ¬1 П 1 and вЂ“1 вЂ«вЂ“ ШЎвЂ¬1 П 1). Zero has no reciprocal because the product of 0 and any number is 0. Every other number has a reciprocal that is not equal to itself. Point out that reciprocals need not be in fraction form. For example, 0.8 and 1.25 are reciprocals because 0.8 вЂ« ШЎвЂ¬1.25 П 1. 206 Math Journal 2, p. 206 Adjusting the Activity Students can multiply 0.8 and 1.25 on a calculator to check that the numbers are reciprocals. They could also convert the numbers to fractions 8 1 4 5 бЋЏбЋЏ вЂ« ШЎвЂ¬1бЋЏбЋЏ П бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ. and multiply: 10 4 5 4 A U D I T O R Y бњ бњ K I N E S T H E T I C T A C T I L E б¤ Finding Reciprocals бњ V I S U A L WHOLE-CLASS ACTIVITY One way to test whether two numbers are reciprocals is to find their product. If the product is 1, then the numbers are reciprocals. Write the following number pairs on the board and ask the class to determine which pairs are reciprocals. As you discuss the answers, add to your list of reciprocals on the board. 1 4 and бЋЏ4бЋЏ Yes 7 бЋЏбЋЏ 4 4 and бЋЏ7бЋЏ Yes 1 3бЋЏ3бЋЏ and 3 бЋЏбЋЏ 10 Yes 2 бЋЏбЋЏ 3 2 бЋЏбЋЏ 5 1 and бЋЏ3бЋЏ No and 5 No 2.5 and 0.4 Yes Write the following numbers on the board and ask students to find their reciprocals. 5 бЋЏбЋЏ 6 7 бЋЏбЋЏ 5 1 бЋЏбЋЏ 9 532 Unit 6 Number Systems and Algebra Concepts 6 бЋЏбЋЏ , 5 5 бЋЏбЋЏ 7 9 1 or 1бЋЏ5бЋЏ 1 8 бЋЏ8бЋЏ 5 8 бЋЏбЋЏ 21 2бЋЏ8бЋЏ 8 3.125 0.32, or бЋЏ2бЋЏ5 NOTE Be sure students understand that the Links to the Future In Lesson 6-11 and Lesson 9-5, students will multiply both sides of an equation by the reciprocal of the fraction or decimal coefficient of the variable term. Recognizing and applying multiplicative inverses is a Grade 6 Goal. б¤ Practicing Finding Reciprocals reciprocal of a number may be represented in many ways, not only as a fraction turned upside down. PARTNER ACTIVITY (Math Journal 2, p. 206) Students complete the problems on journal page 206. Ongoing Assessment: Recognizing Student Achievement а¬™ Journal Page 206 Problems 1вЂ“10 Use journal page 206, Problems 1вЂ“10 to assess studentsвЂ™ ability to name and identify the reciprocal of a number. Students are making adequate progress if they are able to solve Problems 1вЂ“10. Some students may be able to find reciprocals for Problems 11вЂ“16 without a calculator and use reciprocals in Problems 17вЂ“22 to mentally find the products of fractions and mixed numbers. [Patterns, Functions, and Algebra Goal 4] Adjusting the Activity Consider having some students use the F D key on a calculator for Problems 11вЂ“16 to convert between decimals and fractions. (Not all calculators have a F D key.) A U D I T O R Y бњ K I N E S T H E T I C бњ T A C T I L E б¤ Using a Calculator to бњ V I S U A L WHOLE-CLASS DISCUSSION Find Reciprocals Some calculators have keys that can be used to find reciprocals. 5 For example, to find the reciprocal of бЋЏ6бЋЏ using a calculator that has 6 the key, enter 5 6 . The display shows бЋЏ5бЋЏ. If students are using calculators that do not have the key, remind them that a fraction is a division problem and the 1 reciprocal of a number can be expressed as бЋЏaбЋЏ. For example, to 3 find the reciprocal of бЋЏ8бЋЏ, key in: TI-15: 1 П¬ 3 n 8 fx-55: 1 П¬ 3 8 d ; . Have students use calculators to find reciprocals. Suggestions: 1 20 бЋЏбЋЏ бЋЏ бЋЏ 20 1 3 8 7бЋЏ8бЋЏ бЋЏ5бЋЏ9 1 3.14 бЋЏбЋЏ бЋЏбЋЏ 3.14 1 1 вЂ“9 вЂ“0.1 а·†, or вЂ“ бЋЏ9бЋЏ NOTE Keystrokes vary for different calculators. Check the calculatorвЂ™s instruction manual for specific keystrokes. If students are using calculators with the x вЂ“1 key, they must 1 understand that x вЂ“1 П бЋЏxбЋЏ. Students have 1 вЂ“1 learned that 10 П бЋЏ10бЋЏ from writing numbers in expanded notation. Lesson 6 1 бњ 533 Student Page Date Time LESSON 2 Ongoing Learning & Practice Math Boxes 61 бњ 1. Rename each mixed number as a fraction. 7 a. 3 бЋЏбЋЏ П 8 53 бЋЏбЋЏ 9 53 бЋЏбЋЏ 6 51 бЋЏбЋЏ 7 b. c. d. 2 e. 14 бЋЏбЋЏ 3 2. Multiply. 31 бЋЏбЋЏ 8 П 1 b. бЋЏбЋЏ 2 7 бЋЏбЋЏ П 6 1 2 бЋЏбЋЏ 3 вЂ«ШЎвЂ¬ 5 6 1 2 c. бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ вЂ« ШЎвЂ¬8 П 8 9 9 7 d. П 8 бЋЏбЋЏ П 6 бЋЏбЋЏ 44 бЋЏбЋЏ 3 П 14 1 , or 1бЋЏ6бЋЏ 1 a. 3 бЋЏбЋЏ вЂ« ШЎвЂ¬4 П 2 8 5 бЋЏбЋЏ 9 1 б¤ Playing Fraction/Whole Number 2 бЋЏбЋЏ 9 1 5 Top-It 1 2 П бЋЏбЋЏ вЂ« ШЎвЂ¬бЋЏбЋЏ вЂ« ШЎвЂ¬10 71 72 88 89 3. Circle the number sentence that describes y 3 11 5 15 B. (2 вЂ« ШЎвЂ¬x) П© 5 П y C. y ПЄ 2 П (5 ПЄ x) D. y ПЄ 8 П x 0 5 10 25 1 b. бЋЏбЋЏ 8 П 3 d. 1бЋЏбЋЏ П 4 3 e. бЋЏбЋЏ П 100 1 бЋЏбЋЏ) 10 b. (9 вЂ«ШЎвЂ¬ П© (7 вЂ«ШЎвЂ¬ 0.976 ) 1 бЋЏ бЋЏ) 100 Distribute four each of number cards 1вЂ“10 (from the Everything Math Deck, if available) to each partnership. П© (6 вЂ«ШЎвЂ¬ 1 бЋЏбЋЏ) 1,000 6. a. Use your Geometry Template to draw a spinner with colored sectors so the chances of landing on these colors are as follows: 80% 12.5% 87.5% 175% 3% 7 c. бЋЏбЋЏ П 8 0 53.04 5. Write a percent for each fraction. 4 a. бЋЏбЋЏ П 5 ПЄ2 a. (5 вЂ« ШЎвЂ¬10 ) П© (3 вЂ« ШЎвЂ¬10 ) П© (4 вЂ« ШЎвЂ¬10 1 x (Student Reference Book, pp. 319 or 320; Math Masters, p. 478) 4. Write each number in standard notation. the numbers in the table. A. y П x П© 10 PARTNER ACTIVITY green 3 10 red: бЋЏбЋЏ Students use cards to form whole numbers and fractions. They find the products of the numbers they form and then compare those products. Students can use Math Masters, page 478 to record their products and comparisons for each round of play. blue blue: 0.33 green: 20% red b. On this spinner, what is the chance of not landing on red, blue, or green? б¤ Math Boxes 6 1 бњ 17% 146 59 60 207 INDEPENDENT ACTIVITY (Math Journal 2, p. 207) Math Journal 2, p. 207 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 6-3. The skills in Problems 5 and 6 preview Unit 7 content. Writing/Reasoning Have students write a response to the following: Explain how to mentally solve Problems 2b 1 and 2d. Sample answer: For 2b: Multiply (бЋЏ2бЋЏ вЂ« ШЎвЂ¬2) and 1 1 1 1 1 1 1 (бЋЏ2бЋЏ вЂ« ШЎвЂ¬бЋЏ3бЋЏ). Add the products: 1 П© бЋЏ6бЋЏ П 1бЋЏ6бЋЏ. For 2d: бЋЏ5бЋЏ вЂ« ШЎвЂ¬бЋЏ2бЋЏ П бЋЏ10бЋЏ. Apply 1 the reciprocal property: бЋЏ10бЋЏ вЂ« ШЎвЂ¬10 П 1. б¤ Study Link 6 1 бњ INDEPENDENT ACTIVITY (Math Masters, p. 180) Study Link Master Name Date STUDY LINK Home Connection Students find pairs of equivalent fractions, find the reciprocals of numbers, and multiply fractions and mixed numbers to solve number stories. Time Practice with Fractions 61 бњ Put a check mark next to each pair of equivalent fractions. 2 бЋЏбЋЏ 3 1. вњ“ вњ“ 3. 5. 5 6 and бЋЏбЋЏ 24 бЋЏбЋЏ 30 and бЋЏбЋЏ 56 бЋЏбЋЏ 8 and бЋЏбЋЏ вњ“ 2. 73 90 93 3 4 28 16 1бЋЏбЋЏ and бЋЏбЋЏ 4 5 4. 7 бЋЏбЋЏ 3 49 7 6. 2бЋЏбЋЏ and бЋЏбЋЏ 3 7 and бЋЏбЋЏ 3 8 19 4 Find the reciprocal of each number. Multiply to check your answers. 1 1 бЋЏбЋЏ 2 5 бЋЏбЋЏ 19 7. 19 8. бЋЏбЋЏ бЋЏ2бЋЏ 2 5 7 бЋЏбЋЏ 5 1 2 6 9. 3 бЋЏбЋЏ 10. бЋЏбЋЏ 7 , or 2 6 6 Multiply. Write your answers in simplest form. Show your work. 3 бЋЏбЋЏ 2 1 1 7 4 11. бЋЏбЋЏ Вє 1бЋЏбЋЏ П 12. 3 бЋЏбЋЏ Вє бЋЏбЋЏ П 3 8 7 22 1 Solve the number stories. 13. How much does a box containing 5 horseshoes 1 weigh if each horseshoe weighs about 2бЋЏбЋЏ pounds? 12бЋЏ12бЋЏ lb One and one-half dozen golf tees are laid in a straight 1 line, end to end. If each tee is 2бЋЏбЋЏ inches long, how 8 long is the line of tees? 38бЋЏ14бЋЏ in. 2 14. 15. 1 A standard-size brick is 8 inches long and 2бЋЏбЋЏ inches 4 3 high and has a depth of 3 бЋЏбЋЏ inches. What is the volume 4 of a standard-size brick? Practice 16. 107 П© (ПЄ82) П© 56 П 18. ПЄ85 П© 66 П© (ПЄ48) П 81 67бЋЏ2бЋЏ in.3 17. 4 П© (12 П© ПЄ18) П 19. 7 П© (ПЄ11 П© ПЄ22) П 1 Math Masters, p. 180 534 Unit 6 Number Systems and Algebra Concepts Teaching Master Name 3 Differentiation Options 61 бњ 1. б¤ Converting Mixed Numbers to 5вЂ“15 Min Improper Fractions Products and Sums of Reciprocals Read Statement 1. Then find each reciprocal or product to help you decide whether the statement is true or false. 24 d. The product of 4 and 6 is e. The reciprocal of the product of 4 and 6 is f. g. . 1 бЋЏбЋЏ 24 . Repeat Problems 1aвЂ“1e using a different pair of positive numbers. Do you think Statement 1 is true or false for all positive numbers? Explain. Sample answer: True. For any positive numbers 1 1 1 бЋЏ a and b, бЋЏaбЋЏ вЂ« ШЎвЂ¬бЋЏbбЋЏ П бЋЏ (a вЂ« ШЎвЂ¬b) . To provide experience converting mixed numbers to improper fractions, have students use a shortcut. 2. Read Statement 2. Then find each reciprocal or sum to help you decide whether the statement is true or false. Statement 2 The sum of the reciprocals of two positive numbers is equal to the reciprocal of their sum. 1 1 бЋЏбЋЏ бЋЏбЋЏ 5 . 10 . a. The reciprocal of 5 is b. The reciprocal of 10 is 3 бЋЏбЋЏ 10 . c. The sum of the reciprocals from 2a and 2b is Example 1: П© 1 вЂ“ 2 3 (2 вЂ« ШЎвЂ¬3) П© 1 6П©1 7 П бЋЏ2бЋЏ П бЋЏ2бЋЏ П бЋЏ2бЋЏ в€— The sum of 5 and 10 is e. The reciprocal of the sum of 5 and 10 is f. Example 2: 15 d. g. . 1 бЋЏбЋЏ 15 . Repeat Problems 2aвЂ“2e using a different pair of positive numbers. Do you think Statement 2 is true or false for all numbers having reciprocals? Explain. Sample answer: False. A common denominator of a and b a П© b. П© (16 вЂ« ШЎвЂ¬5бЋЏ )П©3 3 ПбЋЏ вЂ” 16 16 5 Time Statement 1 The product of the reciprocals of two positive numbers is equal to the reciprocal of their product. 1 1 бЋЏбЋЏ бЋЏбЋЏ 4 . 6 . a. The reciprocal of 4 is b. The reciprocal of 6 is 1 бЋЏбЋЏ 24 . c. The product of the reciprocals from 1a and 1b is SMALL-GROUP ACTIVITY READINESS Date LESSON 80 П© 3 83 П бЋЏ16бЋЏ П бЋЏ16бЋЏ Math Masters, p. 181 в€— Have students use the shortcut to convert the following mixed numbers to improper fractions. 3 19 4 бЋЏ4бЋЏ бЋЏ4бЋЏ 4 64 2 23 7 бЋЏ3бЋЏ бЋЏ3бЋЏ 7 79 12 бЋЏ5бЋЏ бЋЏ5бЋЏ 9 бЋЏ8бЋЏ бЋЏ8бЋЏ 8 53 6бЋЏ7бЋЏ бЋЏ7бЋЏ 5бЋЏ9бЋЏ бЋЏ9бЋЏ 6 48 ENRICHMENT б¤ Exploring Products and INDEPENDENT ACTIVITY 15вЂ“30 Min Sums of Reciprocals (Math Masters, p. 181) To explore the properties of reciprocals, students use specific cases to decide whether general statements about reciprocals are true or false. Lesson 6 1 бњ 535 INDEPENDENT ACTIVITY EXTRA PRACTICE б¤ Finding Reciprocals 15вЂ“30 Min (Math Masters, p. 182) Students find reciprocals of whole numbers, fractions, and decimals. INDEPENDENT ACTIVITY ELL SUPPORT б¤ Building a Math Word Bank 5вЂ“15 Min (Differentiation Handbook) To provide language support for multiplication of fractions, have students use the Word Bank template in the Differentiation Handbook. Ask students to write any terms with which they are unfamiliar, draw pictures relating to each term, and write other related words. See the Differentiation Handbook for more information. Teaching Master Name Date LESSON Time Finding Reciprocals 61 бњ Solve. 3. 1 бЋЏбЋЏ 5 1 бЋЏбЋЏ 17 5. 10 бЋЏбЋЏ 6 1. 7. Вє5П1 2. Вє 17 П 1 4. 2 4 Вє 0.6 П 1 6. 1 бЋЏnбЋЏ 1 2 Вє бЋЏбЋЏ П 1 Вє 0.25 П 1 ВєnП1 Explain how you solved Problem 5. Sample answer: I renamed the decimal 6 as a fraction (бЋЏ1бЋЏ0 ) and then found 10 its reciprocal (бЋЏ6бЋЏ). For each number, fill in the circle next to the reciprocal. (There may be more than one correct answer.) 5 8. бЋЏбЋЏ 6 56 1 5 1бЋЏбЋЏ 1.2 6 бЋЏбЋЏ 5 12. 9. 2 7 7 бЋЏбЋЏ 3 7 бЋЏбЋЏ 12 7 бЋЏбЋЏ 9 1бЋЏбЋЏ 2.7 10. 3 11. 1.25 9 бЋЏбЋЏ 3 3 бЋЏбЋЏ 9 1 бЋЏбЋЏ 3 5 бЋЏбЋЏ 4 1.3 12 бЋЏбЋЏ 5 5.21 0.8 Explain how you solved Problem 10. Sample answer: I renamed 3 as бЋЏ31бЋЏ and found its reciprocal, бЋЏ13бЋЏ. бЋЏ39бЋЏ is equivalent to 1 3 бЋЏ3бЋЏ, so бЋЏ9бЋЏ is also a reciprocal of 3. Math Masters, p. 182 536 Unit 6 Number Systems and Algebra Concepts