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 Math Journal 2, pp. 205 and 206 Student Reference Book, pp. 88–90 (optional) calculator • Use an algorithm to multiply fractions and mixed numbers. [Operations and Computation Goal 4] • Write a special case to illustrate a general pattern. [Patterns, Functions, and Algebra Goal 1] • 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. Math Journal 2, p. 207 Student Reference Book, pp. 319 or 320 Students practice and maintain skills through Math Boxes and Study Link activities. Game Master (Math Masters, p. 478) Study Link Master (Math Masters, p. 180) 4 each of number cards 1–10 (from the Everything Math Deck, if available) Geometry Template 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. Teaching Masters (Math Masters, pp. 181 and 182) Differentiation Handbook Technology Assessment Management System Journal page 206, Problems 1–10 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 1–3 on journal page 205. Suggestions: 5 22 1.5 3 and thumbs up 5 10 7 70 and thumbs down 10 1,000 7 15 38 3 and 7 thumbs down 7 5 19 76 and thumbs up 20 80 16 and 12 thumbs up 18 and 16 thumbs down 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS DISCUSSION (Math Journal 2, p. 205) Go over the answers to Problems 1–3. Draw attention to the exception to the general pattern in Problem 2—any nonzero number divided by itself equals 1. Zero is excluded. Division by 0 0 is undefined, so 0 is undefined. 5 3 3.14 Write the special cases 1, 16 1, and 1 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 students’ general patterns in words by writing the following on the board: A U D I T O R Y x x 1 x 1 x x1x K I N E S T H E T I C T A C T I L E Student Page Date V I S U A L Time LESSON Applying Properties of Multiplication 61 Math Message 88 90 103 1. Write a general pattern in words for the group of three special cases. Reviewing Multiplication 19 1 19 2 7 PARTNER ACTIVITY of Fractions For any number n, n 1 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 88–90 to check students’ understanding. Then assign Problems 4–18 on journal page 205. 2 7 1 0.084 1 0.084 58 58 1 3 8 3 8 1 7.02 7.02 1 For any number n, General pattern: n n 1. 3. Multiply. Write your answers in simplest form. a. 19 19 4 4 b. 2 3 2 3 6 6 c. 0.5 2 2 0.5 Multiply. Write your answers in simplest form. When you and your partner have finished solving the problems, compare your answers. 1 4 5. 6 11 8. 2 1 11. 3 14. 4 1 17. 5 1 6 3 5 4. 10 6 3 4 7. 2 4 1 1 2 10. 4 5 7 3 13. 2 5 10 7 16. 8 8 7 1 10 41 50 1 1 2 3 3 5 2 3 3 8 3 4 4 1 43 17 2 32 1 4 6 11 1 1 1 18. Write three special cases for the general pattern x 1. x 7 1 7 1 5 9 1 1 5 9 3 7 6. 7 9. 7 3 1 3 5 6 3 7 9 2 3 12. 1 4 15. 1 100 100 1 5 89 1 Sample answers: 2.5 1 2.5 1 205 Math Journal 2, p. 205 Lesson 6 1 531 Student Page Date Defining the Reciprocal Time LESSON Reciprocals 61 Reciprocal Property a b b a If a and b are any numbers except 0, then 1. a a b 1 a and b a 93 are called reciprocals of each other. 1 a 1, so a and are reciprocals of each other. Find the reciprocal of each number. Multiply to check your answers. 1 6 4 3 1. 6 3 3. 4 8 , or 3 2 17 3 14 3 5. 8 1 7. 8 2 2 9. 4 3 11. 0.1 13. 0.75 15. 0.375 10 – or 1.3 – or 2.6 4 , 3 8 , 3 Solve mentally. 5 7 5 19. 7 5 4 7 3 10 7 10 3 17 1 4. 3 13 6. 16 5 8. 3 6 1 10. 6 4 12. 0.4 14. 2.5 16. 5.6 1 32 5 47 1 1 17. 3 2 4 4 1 21. 33 2 23 2. 17 1 1 17 3 1 16 3 , or 1 13 13 6 23 4 25 5 , or 2.5 2 2 , or 0.4 5 5 , or 0.1786 28 1 2 18. 6 5 6 1 1 20. 2 8 2 2 22. 3.875 2.5 0.4 2 5 1 82 3.875 WHOLE-CLASS DISCUSSION of a Number (Math Journal 2, p. 205) Bring the class together to discuss Problems 4–18. 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 2 1 1 and 2 is the reciprocal of 2 because 2 2 1. The numbers 1 and –1 are their own reciprocals (1 1 1 and –1 –1 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 1.25 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 1 . and multiply: 10 4 5 4 A U D I T O R Y K I N E S T H E T I C T A C T I L E Finding Reciprocals 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 4 Yes 7 4 4 and 7 Yes 1 33 and 3 10 Yes 2 3 2 5 1 and 3 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 6 7 5 1 9 532 Unit 6 Number Systems and Algebra Concepts 6 , 5 5 7 9 1 or 15 1 8 8 5 8 21 28 8 3.125 0.32, or 25 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. 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 Journal Page 206 Problems 1–10 Use journal page 206, Problems 1–10 to assess students’ ability to name and identify the reciprocal of a number. Students are making adequate progress if they are able to solve Problems 1–10. Some students may be able to find reciprocals for Problems 11–16 without a calculator and use reciprocals in Problems 17–22 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 11–16 to convert between decimals and fractions. (Not all calculators have a F D key.) A U D I T O R Y K I N E S T H E T I C T A C T I L E Using a Calculator to 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 6 using a calculator that has 6 the key, enter 5 6 . The display shows 5. 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 a. For example, to 3 find the reciprocal of 8, key in: TI-15: 1 3 n 8 fx-55: 1 3 8 d ; . Have students use calculators to find reciprocals. Suggestions: 1 20 20 1 3 8 78 59 1 3.14 3.14 1 1 –9 –0.1 , or – 9 NOTE Keystrokes vary for different calculators. Check the calculator’s instruction manual for specific keystrokes. If students are using calculators with the x –1 key, they must 1 understand that x –1 x. Students have 1 –1 learned that 10 10 from writing numbers in expanded notation. Lesson 6 1 533 Student Page Date Time LESSON 2 Ongoing Learning & Practice Math Boxes 61 1. Rename each mixed number as a fraction. 7 a. 3 8 53 9 53 6 51 7 b. c. d. 2 e. 14 3 2. Multiply. 31 8 1 b. 2 7 6 1 2 3 5 6 1 2 c. 8 8 9 9 7 d. 8 6 44 3 14 1 , or 16 1 a. 3 4 2 8 5 9 1 Playing Fraction/Whole Number 2 9 1 5 Top-It 1 2 10 71 72 88 89 3. Circle the number sentence that describes y 3 11 5 15 B. (2 x) 5 y C. y 2 (5 x) D. y 8 x 0 5 10 25 1 b. 8 3 d. 1 4 3 e. 100 1 ) 10 b. (9 (7 0.976 ) 1 ) 100 Distribute four each of number cards 1–10 (from the Everything Math Deck, if available) to each partnership. (6 1 ) 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. 8 0 53.04 5. Write a percent for each fraction. 4 a. 5 2 a. (5 10 ) (3 10 ) (4 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 x 10 PARTNER ACTIVITY green 3 10 red: 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? Math Boxes 6 1 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 (2 2) and 1 1 1 1 1 1 1 (2 3). Add the products: 1 6 16. For 2d: 5 2 10. Apply 1 the reciprocal property: 10 10 1. Study Link 6 1 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 Put a check mark next to each pair of equivalent fractions. 2 3 1. ✓ ✓ 3. 5. 5 6 and 24 30 and 56 8 and ✓ 2. 73 90 93 3 4 28 16 1 and 4 5 4. 7 3 49 7 6. 2 and 3 7 and 3 8 19 4 Find the reciprocal of each number. Multiply to check your answers. 1 1 2 5 19 7. 19 8. 2 2 5 7 5 1 2 6 9. 3 10. 7 , or 2 6 6 Multiply. Write your answers in simplest form. Show your work. 3 2 1 1 7 4 11. º 1 12. 3 º 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 2 pounds? 1212 lb One and one-half dozen golf tees are laid in a straight 1 line, end to end. If each tee is 2 inches long, how 8 long is the line of tees? 3814 in. 2 14. 15. 1 A standard-size brick is 8 inches long and 2 inches 4 3 high and has a depth of 3 inches. What is the volume 4 of a standard-size brick? Practice 16. 107 (82) 56 18. 85 66 (48) 81 672 in.3 17. 4 (12 18) 19. 7 (11 22) 1 Math Masters, p. 180 534 Unit 6 Number Systems and Algebra Concepts Teaching Master Name 3 Differentiation Options 61 1. Converting Mixed Numbers to 5–15 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 24 . Repeat Problems 1a–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 a and b, a b (a 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 5 . 10 . a. The reciprocal of 5 is b. The reciprocal of 10 is 3 10 . c. The sum of the reciprocals from 2a and 2b is Example 1: 1 – 2 3 (2 3) 1 61 7 2 2 2 ∗ 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 15 . Repeat Problems 2a–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 b. (16 5 )3 3 — 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 4 . 6 . a. The reciprocal of 4 is b. The reciprocal of 6 is 1 24 . c. The product of the reciprocals from 1a and 1b is SMALL-GROUP ACTIVITY READINESS Date LESSON 80 3 83 16 16 Math Masters, p. 181 ∗ Have students use the shortcut to convert the following mixed numbers to improper fractions. 3 19 4 4 4 4 64 2 23 7 3 3 7 79 12 5 5 9 8 8 8 53 67 7 59 9 6 48 ENRICHMENT Exploring Products and INDEPENDENT ACTIVITY 15–30 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 1 535 INDEPENDENT ACTIVITY EXTRA PRACTICE Finding Reciprocals 15–30 Min (Math Masters, p. 182) Students find reciprocals of whole numbers, fractions, and decimals. INDEPENDENT ACTIVITY ELL SUPPORT Building a Math Word Bank 5–15 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 Solve. 3. 1 5 1 17 5. 10 6 1. 7. º51 2. º 17 1 4. 2 4 º 0.6 1 6. 1 n 1 2 º 1 º 0.25 1 ºn1 Explain how you solved Problem 5. Sample answer: I renamed the decimal 6 as a fraction (10 ) and then found 10 its reciprocal (6). For each number, fill in the circle next to the reciprocal. (There may be more than one correct answer.) 5 8. 6 56 1 5 1 1.2 6 5 12. 9. 2 7 7 3 7 12 7 9 1 2.7 10. 3 11. 1.25 9 3 3 9 1 3 5 4 1.3 12 5 5.21 0.8 Explain how you solved Problem 10. Sample answer: I renamed 3 as 31 and found its reciprocal, 13. 39 is equivalent to 1 3 3, so 9 is also a reciprocal of 3. Math Masters, p. 182 536 Unit 6 Number Systems and Algebra Concepts

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