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Objective To develop addition concepts related to mixed numbers. 1 materials Teaching the Lesson Key Activities Students review fraction addition. They add mixed numbers in which the fractional parts have like or unlike denominators and rename the sums in simplest form. Key Concepts and Skills • Find equivalent fractions in simplest form. Math Journal 2, pp. 251 and 252 Study Link 8 1 Teaching Aid Master (Math Masters, p. 414; optional) slates Class Data Pad (optional) [Number and Numeration Goal 5] • Convert between and simplify fractions and mixed numbers. [Number and Numeration Goal 5] • Add fractions and mixed numbers. [Operations and Computation Goal 4] Ongoing Assessment: Informing Instruction See page 626. Ongoing Assessment: Recognizing Student Achievement Use journal page 252. [Operations and Computation Goal 4] 2 Ongoing Learning & Practice Students practice and maintain skills through Math Boxes and Study Link activities. 3 Students explore an alternate method for adding mixed numbers. Math Journal 2, p. 253 Study Link Master (Math Masters, p. 223) materials Differentiation Options READINESS materials EXTRA PRACTICE Students play Fraction Capture to practice comparing fractions and finding equivalent fractions. Math Journal 1, p. 198 Math Journal 2, p. 252 Game Master (Math Masters, p. 460) 2 six-sided dice Technology Assessment Management System Journal page 252 See the iTLG. 624 Unit 8 Fractions and Ratios Getting Started Mental Math and Reflexes Math Message Have students rename each fraction as a whole number or mixed number and each mixed number as an improper fraction. Solve Problems 1–9 at the top of journal page 251. 3 1 3 1 5 22 2 3 1 1 2 2 13 5 1 8 8 1 33 48 8 17 2 3 5 5 37 2 7 5 5 62 2 12 5 5 1 45 11 4 4 Study Link 8 1 Follow-Up Have partners compare answers and resolve differences. Ask volunteers to share their explanations for Problems 7, 14, and 21. 1 Teaching the Lesson Math Message Follow-Up WHOLE-CLASS ACTIVITY (Math Journal 2, p. 251) Ask volunteers to share their strategies for renaming the sums in Problems 3–6 to a whole or mixed number. Encourage students to use their understanding of multiplication facts to recognize when the simplest form will be a whole number. If the sum is an improper fraction and the numerator is not a multiple of the denominator, the simplest form will be a mixed number. If the numerator is a multiple of the denominator, the simplest form will be a whole number. Ask volunteers to explain how to recognize an improper fraction. If the numerator is greater than or equal to the denominator, the fraction is an improper fraction. To support English language learners, write and label examples of improper fractions on the board or Class Data Pad. Survey students for what methods they used to find the common denominators for Problems 7–9. Expect a mixture of the methods discussed in Lesson 8-1. Summarize the methods on the board or Class Data Pad for student reference throughout the lesson. Student Page Date Time LESSON 8 2 Math Message Add. Write the sums in simplest form. 3 1 1. 5 5 3 4. 7 Adding Mixed Numbers with Adding Fractions 5 7 4 5 117 3 1 2. 8 8 5 6 2 2 8. 3 5 1 2 7. 6 3 WHOLE-CLASS ACTIVITY Fractions Having Like Denominators Explain that one way to find the sum of mixed numbers is to treat the fraction and whole number parts separately. Write the problem on the next page onto the board or a transparency: 7 5. 10 1 2 2 2 2 3. 3 3 3 125 7 10 1115 2 5 7 6. 9 9 113 5 5 9. 6 8 11214 Adding Mixed Numbers Add. Write each sum as a whole number or mixed number. 3 5 1 10. 1 2 1 11. 12. 1 4 2 1 1 5 1 2 3 245 2 6 3 4 Fill in the missing numbers. 12 13. 5 6 7 5 16. 4 5 3 5 7 2 3 8 14. 7 5 8 11 17. 12 13 6 3 5 5 6 5 15. 2 3 4 13 18. 9 10 10 1 4 3 10 Add. Write each sum as a mixed number in simplest form. 2 3 3 19. 2 5 3 913 6 7 4 20. 21. 4 9 3 8 9 4 2 7 6 737 10 13 251 Math Journal 2, p. 251 Lesson 8 2 625 1 38 3 58 4 1 88 or 82 Add the whole-number parts. Then add the fraction parts. 7 28 Ask students to solve the following problem: 5 38 Discuss students’ solution strategies. Make sure the following strategy is presented: 7 28 1. Add the whole-number parts. 5 38 2. Add the fraction parts. 12 58 12 3. Rename 58 in simplest form. 12 8 8 4 4 4 8 8 1 8 18 12 4 12 4 4 1 Since 8 18, then 58 5 1 8 68 or 62. Model renaming the sum with a picture. 12 4 1 8 8 Pose a few more addition problems in which the addends are mixed numbers with like denominators. Suggestions: 4 3 ● 14 24 44, or 42 1 ● 810 510 145 ● 37 47 8 ● 623 23 9 3 7 3 2 9 1 3 Ongoing Assessment: Informing Instruction Watch for students who have difficulty renaming mixed-number sums such as 12 58. Discuss the meanings of numerator and denominator, and have students rename the fractional parts in the mixed numbers. Suggestions: 475 843 274 626 Unit 8 Fractions and Ratios 7 5 4 3 7 4 5 5 3 3 4 4 2 5 1 3 3 4 1 1 1 2 5 1 3 3 4 Adding Mixed Numbers with WHOLE-CLASS ACTIVITY Fractions Having Unlike Denominators Write the following problem on the board, and ask students to find the sum: 3 34 7 28 After a few minutes, ask students to share solution strategies. Make sure the following method is discussed: 3 7 1. Find a common denominator for 4 and 8 8, 16, 24, 32, ... 2. Rename the fraction parts of the mixed numbers so they have the same denominator. In this case, the least common denominator, 8, is the easiest to use. 34 3 38 6 7 28 7 28 13 58 3. Add. 13 8 5 5 5 58 5 8 8 5 1 8 68 4. Rename the sum. Pose a few more problems that involve finding common denominators to add mixed numbers. Suggestions: 1 3 7 2 5 3 4 1 1 5 3 13 7 1 1 ● 22 48 68 ● 53 16 76, or 72 ● 35 24 620 ● 66 35 1030 ● 18 86 1024 1 Student Page Adding Mixed Numbers Date PARTNER ACTIVITY (Math Journal 2, pp. 251 and 252) Time LESSON Adding Mixed Numbers 8 2 continued To add mixed numbers in which the fractions do not have the same denominator, you must first rename one or both fractions so that both fractions have a common denominator. Have students complete journal pages 251 and 252. Circulate and assist. 3 5 2 3 Example: 2 4 ? 3 5 Write the problem in vertical form, and rename the fractions. 3 5 9 15 2 2 Ongoing Assessment: Recognizing Student Achievement Journal Page 252 Problem 4 [Operations and Computation Goal 4] 2 4 3 ∑ 4 10 15 6 19 Add. Rename the sum. 19 6 15 6 15 15 15 4 15 4 15 4 15 6 1 7 Add. Write each sum as a mixed number in simplest form. Show your work. 7190 5172 1 1 1 2 1. 2 3 2. 5 2 3 Use journal page 252, Problem 4 to assess students’ facility with adding mixed numbers. Have students complete an Exit Slip (Math Masters, 414) for the following: Explain how you found the answer to Problem 4 on journal page 252. Students are making adequate progress if their responses demonstrate an understanding of renaming fractions to have common denominators and to be in simplest form. 2 3 Find a common denominator. The QCD of and is 5 3 15. 4 1 4 3. 6 2 3 9 1 5 5. 7 2 4 6 2 879 10112 5 1 3 4. 1 4 2 4 5 3 6. 3 3 6 4 614 7172 252 Math Journal 2, p. 252 Lesson 8 2 627 Student Page Date Time LESSON 2 Ongoing Learning & Practice Math Boxes 8 2 1. Add. 2. Use the patterns to fill in the missing 1 2 a. 4 4 3 1 b. 8 4 1 c. 2 1 8 numbers. 3 4 5 8 2 1 d. 3 6 8 a. 1, 2, 4, 5 8 5 6 4 2 2 e. 6 6 6 , or 32 , 41 c. 4, 34, 64, 94 , 124 62 d. 20, 34, 48, 2 3 68 e. 100, 152, 204, 3 3. The school band practiced 2 hours on 4 2 Saturday and 3 hours on Sunday. Was 3 16 , b. 5, 14, 23, , Math Boxes 8 2 (Math Journal 2, p. 253) 76 256 , 308 Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 8-4. The skill in Problem 6 previews Unit 9 content. parentheses. a. 18 11 3 10 More than 6 hours b. 18 11 3 4 Sample answer: Explain. 234 323 5 192 182 5 1172 5 1122 152 5 1 152 6 152 c. 14 7 5 1 13 ( ) ( ) ) ( ( ( Writing/Reasoning Have students write a response to the following: Explain your strategy for finding the values of the variables in Problem 5. Sample answer: I looked for the common factor of the numerators or denominators that were complete. Then I used the multiplication rule or the division rule to multiply or divide to find the values for the variables. ) d. 14 7 5 1 1 ) e. 14 7 5 1 3 71 5. Solve. 230 4. Make each sentence true by inserting the band’s total practice time more or less than 6 hours? 222 Solution 6. Circle the congruent line segments. x 5 a. 18 9 x 10 a. 40 8 b. y 25 y 125 b. 6 w c. 14 49 w 21 c. 28 7 d. z 9 z 36 d. 44 4 e. 77 v v7 INDEPENDENT ACTIVITY 155 108 109 253 Math Journal 2, p. 253 NOTE Alternately, students may use a table or chart to find an equivalent fraction with the given denominator or numerator. Study Link 8 2 INDEPENDENT ACTIVITY (Math Masters, p. 223) Home Connection Students practice adding mixed numbers and renaming improper fractions as mixed numbers in simplest form. 3 Differentiation Options Study Link Master Name Date STUDY LINK Time 82 Rename each mixed number in simplest form. 1 45 6 5 1. 3 3. 9 5. 4 16 2. 8 2 25 7 5 4. 1 6. 5 63 10 6 Add. Write each sum as a whole number or mixed number in simplest form. 1 4 3 4 1 3 2 3 7. 3 2 9. 9 4 6 1 5 4 5 5 7 6 7 2 9 5 9 8. 4 3 10. 3 8 3 8 3 12. 4 5 8 4 14 1 15 11. 8 15–30 Min 2 1 52 6 4 Adding Mixed Numbers (Math Journal 2, p. 252) 2 103 5 3 61 63 70 2 SMALL-GROUP ACTIVITY READINESS Adding Mixed Numbers 127 To explore mixed-number addition, have students use an oppositechange algorithm. Have students change one of the addends to a whole number. Pose the following problem: 2 5 13 76 7 54 99 2 1 2 1 1. Change 1 and 3 to a whole number by adding 3: 13 3 2 Add. 13. 5 8 3 4 2 14. 1 2 2 3 7 6 9 7 12 4 15. 6 3 3 3 116 1 84 98 16. 3 4 4 5 5 2 1 11 820 Practice 17. 3,540 6 590 18. 1,770 3 590 19. 7,080 / 12 590 20. (590 5) 2 1,475 Math Masters, p. 223 628 Unit 8 Fractions and Ratios 1 5 1 2 5 2 3 2. Subtract 3 from 76: 3 6; 76 – 6 76. 3 3 1 3. Add the new addends: 2 76 96, or 92. This strategy is most efficient when the sum of the fraction parts is greater than 1. Have students use this method to solve the following problems. Discuss how to recognize that the fraction parts are greater than 1: Game Master ● 8 1 3 Name 32 710 1110 3 ● 2 4 8 9 612 1 6 912 , 59 73 139 ● 410 36 830 5 Time 1 2 4 3 Fraction Capture Gameboard 1 92 1 2 2 ● 8 or Date 1 2 1 2 1 2 1 2 1 2 1 2 1 2 19 1 3 1 3 1 3 Before students begin journal page 252, have them identify problems for which this algorithm might apply. Problems 4–6 1 3 1 4 15–30 Min 1 4 1 4 1 4 1 5 1 5 1 5 1 5 1 4 (Math Journal 1, p. 198; Math Masters, p. 460) Students practice comparing fractions and finding equivalent fractions by playing Fraction Capture. Players roll dice, form fractions, and claim corresponding sections of squares. The rules are on Math Journal 1, page 198, and the gameboard is on Math Masters, page 460. 1 6 1 6 1 6 1 6 1 6 1 6 1 5 1 5 1 6 1 6 1 6 1 5 1 6 1 4 1 5 1 5 1 5 1 5 1 6 1 4 1 4 1 5 1 6 1 6 1 4 1 4 1 4 1 5 1 5 1 5 1 5 1 5 1 5 1 3 1 4 1 4 1 4 PARTNER ACTIVITY 1 3 1 3 1 3 1 3 1 4 Playing Fraction Capture 1 3 1 3 1 4 EXTRA PRACTICE 1 3 1 6 1 5 1 5 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 1 6 Math Masters, p. 460 Lesson 8 2 629