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Lesson 10 Use after Ready Instruction page 87 Add and Subtract Fractions Name: Find Equivalent Fractions Study the example showing how you can use models and multiplication to find equivalent fractions. Then solve problems 1–7. Example The model is divided into 4 equal parts. The shaded section shows the fraction 1 . 4 ·· You can divide the same whole into 2 times as many equal parts. There are 2 times as many parts shaded. You can divide the same whole into 3 times as many equal parts. There are 3 times as many parts shaded. The fractions 1 , 2 , and 3 are equivalent because they 4 ·· 8 12 ·· ·· 1 4 ·· 1 3 2 5 2 432 ····· 8 ·· 5 3 1 3 3 433 ····· 12 ·· each show the same shaded part of a whole. 1 Look at the model to the right. 3 of the whole is 5 ·· shaded. Divide the model into a different number of equal parts to find an equivalent fraction. Complete the equation. 3 5 5 ·· ···· 2 Write the missing numbers to describe the equivalent fraction you found in problem 1. 33 5 ······· ···· 5 3 There are times as many shaded parts. There are times as many equal parts. ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 Add and Subtract Fractions 103 Solve. 3 Shade the model to show 2 . Then divide the model to 3 ·· show 6 equal parts. 4 Look at the model in problem 3. Write the missing numbers to show the equivalent fraction you formed by dividing it into 6 equal parts. 23 5 ······· 33 ···· 5 Explain how you can multiply to find equivalent fractions. 6 Choose Yes or No to tell whether the fraction is equivalent to 2 . 5 ·· 4 a. Yes 10 ·· b. 5 Yes 8 ·· c. 6 Yes 15 ·· d. 6 Yes 20 ·· Yes e. 10 25 ·· u u u u u 23? 5 ····· 53? ···· u No u No u No u No u No 7 How did you determine whether a fraction was equivalent to 2 in problem 6? Explain. 5 ·· 104 Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 • Use after Ready Instruction page 89 Name: Add Fractions with Unlike Denominators Study the example showing one way to add fractions with unlike denominators. Then solve problems 1–4. Example What is 3 1 1 1 ? 4 ·· 6 ·· To add fractions, the size of the parts must be the same. Write each addend as an equivalent fraction with a common denominator. 3 4 1 1 1 6 9 12 1 1 2 12 Identify a common multiple of the denominators, 4 and 6: 12. Divide models into 12 equal parts. Write the equivalent fractions. 3 5 9 and 1 1 5 1 2 4 ·· 12 ·· 6 ·· 12 ·· Find the sum. 3 1 1 1 5 9 1 1 2 5 1 11 4 6 ·· 12 12 12 ·· ·· ·· ·· 1 The example uses 12 as the common multiple of 4 and 6. a. Name a different common multiple of 4 and 6. b. Using the common multiple from part a., how would the models be different? How would they be the same? c. Use the common multiple from part a. as the common denominator to write equivalent fractions for 3 and 1 1 . 4 ·· 6 ·· 3 5 4 ·· 1 1 5 1 6 ·· ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 Add and Subtract Fractions 105 Solve. 2 One way to find a common denominator is by multiplying the denominators of the two fractions together and using the product as the common denominator. Use this method to find a common denominator for each pair of fractions. Write the equivalent fractions. a. 1 3 5 1 5 ·· 20 ···· 1 3 5 1 20 ···· 4 ·· b. 2 1 5 4 5 2 ·· 5 ·· c. 3 5 1 5 8 ·· 6 ·· 3 Show how to add 2 1 1 4 using the number line 2 ·· 5 ·· below. 2 3 4 Write an equation to represent the problem. 4 Maya is packing her backpack for a hike. In one pocket she puts in a 1 -pound bag of trail mix, a water bottle 4 ·· weighing 2 1 pounds, and a flashlight weighing 5 ·· 1 pound. How much weight do these three items add 4 ·· to her backpack? Show your work. Solution: 106 Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 • Use after Ready Instruction page 91 Name: Subtract Fractions with Unlike Denominators Study the example problem showing one way to subtract fractions with unlike denominators. Then solve problems 1–5. Example Felicia lives 1 1 miles from school and 9 mile from the 5 ·· 10 ·· soccer field. How much closer does she live to the field than to school? You can show 1 1 2 9 using a number line. 5 ·· 10 ·· First find the common denominator. Identify a common multiple of 5 and 10: 10. Rewrite the fractions as needed. 1 1 5 1 2 5 ·· 10 ·· Divide the number line into tenths. Start at the point 1 2 and jump left 9 . 10 ·· 10 ·· Find the difference. 0 1 3 10 2 1 10 1 1 2 9 5 1 2 2 9 5 3 . 5 ·· 10 ·· 10 ·· 10 ·· 10 ·· Felicia lives 3 mile closer to the field than to school. 10 ·· 1 How would the model and answer in the example problem change if Felicia lives 7 mile from the soccer 10 ·· field? 2 Eric says he knows 9 is 1 less than 10 , or 1 mile. 10 ·· 10 10 ·· ·· So, he is going to subtract 1 mile, then add 1 back. 10 ·· Can he use this method to solve the example problem? Explain. ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 Add and Subtract Fractions 107 Solve. 3 Sometimes it is helpful to rewrite mixed numbers that include a fraction greater than 1. Use the number line to write the missing numbers. 2 2 16 0 1 a. 1 2 5 6 ·· 6 ···· b. 2 5 5 1 6 ·· 6 ···· 5 26 2 d. 3 1 5 6 ·· 36 3 c. 2 2 5 1 6 ·· 1 26 ···· ···· 4 What is 3 1 2 1 1 ? 3 2 ·· ·· Show your work. Solution: 5 Emil’s backpack weighs 6 3 pounds. He removes a 8 ·· 3 book that weighs pound. Then he removes a book 4 ·· 1 that weighs pound. How much does Emil’s backpack 2 ·· weigh now? Show your work. I can find the total weight of the books first and then subtract, or subtract the weight of each book separately. Solution: 108 Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 • Use after Ready Instruction page 93 Name: Add and Subtract Fractions with Unlike Denominators Solve. 1 Which statement and reasoning is true for finding a common denominator for the fractions 1 and 1 ? Circle 4 8 ·· ·· the letter of all that apply. A I can use 8 because 2 3 4 5 8. Can a pair of fractions have more than one common denominator? B I can use 12 because 2 3 4 5 8 and 4 1 8 5 12. C I can use 16 because 4 3 4 5 16 and 2 3 8 5 16. D I can use 24 because 6 3 4 5 24 and 3 3 8 5 24. 2 What is 3 1 1 3 ? 4 ·· 8 ·· I need to find a common denominator before I can add. Show your work. Solution: 3 Kado spent 1 2 hours painting a fence. Then he spent 3 ·· 4 of an hour walking his dog. How much longer did he 5 ·· spend painting than walking? A 2 hour C 1 2 hours B 13 hour D 1 13 hours 15 ·· 15 ·· 15 ·· I know he spent more time walking his dog because 1 ·23 is greater than ··45 . 15 ·· Orleans chose C as the correct answer. How did she get that answer? ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 Add and Subtract Fractions 109 Solve. 4 A piece of string is 5 5 inches long. How much should 8 ·· Lena cut off to make it 3 1 inches long? 2 ·· Show your work. Do I represent this problem with an addition or subtraction expression? Solution: 5 Erin and Ethan add the fractions 2 1 and 1 1 . Look at 3 2 ·· ·· their work below. 1 1 1 1 Erin 2 ·· 3 2 ·· How are the pictures for the two answers alike? 2 1 3 2 5 2 2 1 1 3 3 5 1 3 332 ····· 6 ·· 233 ····· 6 ·· 2 2 1 1 3 5 3 5 6 ·· 6 ·· 6 ·· 1 1 1 1 Ethan 2 ·· 3 2 ·· 2 1 3 4 5 2 4 1 1 3 6 5 1 6 334 ····· 12 ·· 236 ····· 12 ·· 2 4 1 1 6 5 3 10 12 ·· 12 ·· 12 ·· Both Erin’s and Ethan’s answers are correct. Use pictures to explain why this is true. 110 Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Practice Lesson 10 Add and Subtract Fractions Lesson 10 Unit 2 Use after Ready Instruction page 87 Add and Subtract Fractions Solve. Name: Find Equivalent Fractions M 3 Shade the model to show 2 . Then divide the model to 3 ·· M 4 Look at the model in problem 3. Write the missing show 6 equal parts. Study the example showing how you can use models and multiplication to find equivalent fractions. Then solve problems 1–7. numbers to show the equivalent fraction you formed by dividing it into 6 equal parts. Example 23 2 The model is divided into 4 equal parts. ······· The shaded section shows the fraction 1 . 4 ·· You can divide the same whole into 2 times as many equal parts. There are 2 times as many parts shaded. M 5 4 ···· 6 5 Explain how you can multiply to find equivalent fractions. 13252 4 3 2 ·· 8 ····· You can divide the same whole into 3 times as many equal parts. There are 3 times as many parts shaded. Answers will vary. Students should show an understanding that the numerator and denominator of a fraction is multiplied by the same number in order to find an equivalent fraction. 1335 3 4 3 3 ·· 12 ····· The fractions 1 , 2 , and 3 are equivalent because they 4 ·· 8 ·· 33 2 1 4 ·· 12 ·· each show the same shaded part of a whole. B C 1 Look at the model to the right. 3 of the whole is 5 ·· u 3 u u 3 u u 3 shaded. Divide the model into a different number of equal parts to find an equivalent fraction. Complete the equation. 35 6 5 ·· B ···· 10 2 Write the missing numbers to describe the equivalent fraction you found in problem 1. There are 2 times as many equal parts. There are 33 2 ······· times as many shaded parts. 5 3 2 2 6 Choose Yes or No to tell whether the fraction is equivalent to 2 . 5 ·· Yes a. 4 10 ·· b. 5 Yes 8 ·· Yes c. 6 15 ·· d. 6 Yes 20 ·· Yes e. 10 25 ·· 5 M 6 ···· 10 23? ····· 53? 5 ···· u No u 3 No u No u 3 No u No 7 How did you determine whether a fraction was equivalent to 2 in problem 6? Explain. 5 ·· Answers will vary. Students may indicate a preference for one method or the other (using models or multiplying) or explain they use some combination of both methods. ©Curriculum Associates, LLC Lesson 10 Add and Subtract Fractions Copying is not permitted. Lesson 10 • Use after Ready Instruction page 89 103 103 104 104 Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Name: Solve. Add Fractions with Unlike Denominators M Study the example showing one way to add fractions with unlike denominators. Then solve problems 1–4. 2 One way to find a common denominator is by multiplying the denominators of the two fractions together and using the product as the common denominator. Example Use this method to find a common denominator for each pair of fractions. Write the equivalent fractions. What is 3 1 1 1 ? 4 ·· 6 ·· To add fractions, the size of the parts must be the same. Write each addend as an equivalent fraction with a common denominator. a. 1 3 5 1 5 ·· 3 4 1 1 1 6 c. Write the equivalent fractions. M 9 12 1 1 4 ·· 2 ·· Divide models into 12 equal parts. 3 5 9 and 1 1 5 1 2 12 6 12 ·· ·· ·· Find the sum. 3 1 1 1 5 9 1 1 2 5 1 11 4 6 ·· 12 12 12 ·· ·· ·· ·· 1351 5 b. 2 1 5 2 ··· 10 Identify a common multiple of the denominators, 4 and 6: 12. 4 ·· 12 20 ···· M 2 12 35 8 ·· 18 48 ··· 15 20 ···· 8 5 ·· 45 10 ··· 15 48 ··· 8 6 ·· 3 Show how to add 2 1 1 4 using the number line 2 ·· 5 ·· below. 2 1 The example uses 12 as the common multiple of 4 and 6. 3 5 2 10 4 Write an equation to represent the problem. 211452 5 1 8 53 3 a. Name a different common multiple of 4 and 6. 2 ·· Possible answer: 24 C b. Using the common multiple from part a., how would the models be different? How would they be the same? Possible answer: There would be twice as many equal parts in each model, and each part would be smaller. The areas shaded would be the same. 5 ·· 10 ··· 10 ··· 10 ··· 4 Maya is packing her backpack for a hike. In one pocket she puts in a 1 -pound bag of trail mix, a water bottle 4 ·· weighing 2 1 pounds, and a flashlight weighing 5 ·· 1 pound. How much weight do these three items add 4 ·· to her backpack? Show your work. Students might use models, number lines, c. Use the common multiple from part a. as the equations, or some other method to find common denominator to write equivalent fractions 1 1 2 1 1 1. 5 ·· 4 ·· 4 ·· for 3 and 1 1 . 4 ·· 35 4 ·· 18 24 ··· ©Curriculum Associates, LLC 24 6 ·· 1151 6 ·· 4 24 ··· Copying is not permitted. Practice and Problem Solving Solution: Lesson 10 Add and Subtract Fractions 105 105 106 106 2 14 pounds or 2 7 pounds 20 ··· 10 ··· Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Unit 2 Number and Operations—Fractions ©Curriculum Associates, LLC Copying is not permitted. Practice Lesson 10 Add and Subtract Fractions Lesson 10 • Use after Ready Instruction page 91 Unit 2 Name: Solve. Subtract Fractions with Unlike Denominators M Study the example problem showing one way to subtract fractions with unlike denominators. Then solve problems 1–5. 3 Sometimes it is helpful to rewrite mixed numbers that include a fraction greater than 1. Use the number line to write the missing numbers. 2 2 16 Example Felicia lives 1 1 miles from school and 9 mile from the 5 ·· 10 ·· 0 soccer field. How much closer does she live to the field than to school? 1 a. 10 ·· b. 2 5 5 1 First find the common denominator. 6 ·· 3 6 7 d. 3 1 5 2 6 ·· ···· 6 6 ···· Identify a common multiple of 5 and 10: 10. Rewrite the fractions as needed. 1 1 5 1 2 5 ·· 4 What is 3 1 2 1 1 ? 3 2 ·· ·· M 10 ·· Show your work. Students might use models, number lines, Divide the number line into tenths. Start at the point 1 2 10 ·· and jump left 9 . 10 ·· Find the difference. equations, or some other method to find 3 1 2 1 1 . 0 1 3 10 3 ·· 2 1 10 10 ·· 5 Emil’s backpack weighs 6 3 pounds. He removes a 8 ·· book that weighs 3 pound. Then he removes a book 4 ·· 1 that weighs pound. How much does Emil’s backpack 2 ·· C 1 How would the model and answer in the example problem change if Felicia lives 7 mile from the soccer 10 ·· first and then subtract, or subtract the weight of each Show your work. book separately. Students might use models, equations, or some change from 3 mile to 5 , or 1 mile. 10 10 2 ··· ··· ·· other method to find the answer. They may first add the weights of the two books 1 3 1 1 2 and 2 Eric says he knows 9 is 1 less than 10 , or 1 mile. 10 ·· 10 10 ·· ·· So, he is going to subtract 1 mile, then add 1 back. 10 ·· 4 ·· 2 ·· subtract the combined weight from 6 3 pounds, 8 ·· or they may subtract the weight of each book Can he use this method to solve the example individually from the weight of the backpack. problem? Explain. Yes. Answers will vary, but students should show understanding that Eric’s method will result in the same answer, 3 . Solution: 10 ··· ©Curriculum Associates, LLC I can find the total weight of the books weigh now? field? Possible answer: The model would show 7 jumps instead of 9. The difference would M 2 ·· 15 6 Solution: ·· 112 9 51 2 2 9 5 3 . 5 ·· 10 10 ·· 10 ·· 10 ·· ·· Felicia lives 3 mile closer to the field than to school. B 36 8 2251 6 ·· ···· c. 11 1 26 2 8 125 6 ···· 6 ·· You can show 1 1 2 9 using a number line. 5 ·· 5 26 Lesson 10 Add and Subtract Fractions Copying is not permitted. Lesson 10 • Use after Ready Instruction page 93 107 107 108 108 5 1 pounds 8 ·· Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Name: Solve. Add and Subtract Fractions with Unlike Denominators Solve. M B 1 Which statement and reasoning is true for finding a common denominator for the fractions 1 and 1 ? Circle 4 8 ·· ·· the letter of all that apply. A I can use 8 because 2 3 4 5 8. B I can use 12 because 2 3 4 5 8 and 4 1 8 5 12. C I can use 16 because 4 3 4 5 16 and 2 3 8 5 16. 4 A piece of string is 5 5 inches long. How much should 8 ·· Lena cut off to make it 3 1 inches long? 2 ·· Show your work. Students might use models, equations, or some Can a pair of fractions have more than one common denominator? other method to find 5 5 2 3 1 . 8 ·· Solution: C D I can use 24 because 6 3 4 5 24 and 3 3 8 5 24. 2 ·· She should cut off 2 1 inches. 8 ·· 5 Erin and Ethan add the fractions 2 1 and 1 1 . Look at 3 2 ·· ·· their work below. B 2 What is 3 1 1 3 ? 4 ·· 8 ·· Show your work. Students might use models, equations, or some other method to find 3 1 1 3 . 4 ·· Solution: 8 ·· I need to find a common denominator before I can add. Erin 21111 3 ·· 1133513 233 ····· 21111 Ethan 8 ·· 3 ·· 334 ····· spend painting than walking? A B 2 hour 15 ·· 13 hour 15 ·· C 2 ·· D 12 ·· more time walking 15 ··· 236 ····· 12 ·· 12 ·· 21522 2152 4 11513 1151 6 3 ·· 2 ·· subtracted 10 from 1 12 instead of 12 from 1 10. 15 ··· 12 ·· Both Erin’s and Ethan’s answers are correct. Use pictures to explain why this is true. his dog because 1 ·23 is Orleans chose C as the correct answer. How did she get that answer? Answers will vary. Possible answer: She 15 ··· 113651 6 12 ·· 2 4 1 1 6 5 3 10 I know he spent greater than ··45 . 1 2 hours 15 ·· 1 13 hours 15 ·· 6 ·· 22113535 6 6 6 ·· ·· ·· 3 5 pounds 3 Kado spent 1 2 hours painting a fence. Then he spent 3 ·· 4 of an hour walking his dog. How much longer did he 5 ·· How are the pictures for the two answers alike? 2 ·· 2132522 332 6 ····· ·· 213452 4 M Do I represent this problem with an addition or subtraction expression? 6 ·· 6 ·· 3 ·· 2 ·· 12 ··· 12 ··· Answers will vary. Students should show an understanding that 3 5 and 3 10 15 ··· 6 ·· 12 ··· are equivalent. ©Curriculum Associates, LLC Copying is not permitted. Lesson 10 Add and Subtract Fractions Practice and Problem Solving ©Curriculum Associates, LLC Copying is not permitted. 109 109 110 110 Lesson 10 Add and Subtract Fractions ©Curriculum Associates, LLC Copying is not permitted. Unit 2 Number and Operations—Fractions 25
