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A C E Answers | Investigation 1 Applications 1. a. Answers will vary. Possible answers: Each grade’s goal is $60 more than the previous grade’s goal. The ratio does not tell you how many of these groups there are, so there are many possibilities. 5. Possible answers: eighths, twelfths and sixteenths (multiples of 4) 6. halves, fourths, twelfths The sixth-grade goal is 11 times the 7. The seventh-grade goal is twice the fifth-grade goal. 2 fifth-grade goal. 8. b. Answers will vary. Possible answers: 9. The teachers’ goal is 3 of the eighth4 10. a. Shown are 3 , 6 , 12 . grade goal. For every $75 the teachers plan to collect, the eighth graders plan to collect $100. The teachers’ goal is $75 less than the eighth graders’ goal. 2. 1 4 3 4 2 3 4 8 16 b. Another equivalent fraction would be 11. a. 15 . 20 5 is 5 the same as 1. b. Sally is correct. Any two segments are 2 of a whole. She is concentrating 24 or 3 32 4 5 on a fraction as a part of a whole. However, if you took any two segments and lined them up to start with 0, you would arrive at a location of 2 on the 3. a. This is true. If the teacher made groups of 2 boys and 4 girls, there would be six of these groups with no children left out of a group. b. Answers will vary. Possible answers: There are twice as many girls as boys. There are 12 more girls than boys. 4. There could be 3 boys and 2 girls. There could be 6 boys and 4 girls, 9 boys and 6 girls, etc. If the class is going to be close in size to the one in ACE Exercise 3, there could be 21 boys and 14 girls. In each of these possibilities, you can think about making groups of 3 boys and 2 girls. 5 number line. c. 1 would now be marked with 2 , 2 5 10 5 4 3 6 4 8 with , with , with , and 10 5 10 5 10 1 with 10 . These are equivalent 10 fractions. For every one fifth there are two tenths, so for two fifths there are four tenths, etc. d. Possible answers: For every one half, there would be 5 tenths. For every one whole, there would be 10 tenths. 12. Correct. (See Figure 1 for possible picture of number line and fraction strips.) Figure 1 Comparing Bits and Pieces 1 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1 A C E Answers | Investigation 1 13. Correct. (See Figure 2 for possible picture 18. Possible answer: You could draw a fraction of number line and fraction strips.) 14. Incorrect. (See Figure 3 for possible picture of number line and fraction strips.) strip and divide it into five equal parts. Shade three of these parts to represent 3 . Then divide each of the five parts into 5 15. Incorrect. (See Figure 4 for possible picture of number line and fraction strips.) 16. (See Figure 5.) two equal parts. You would then have ten equal parts, and six of the parts would be shaded. Therefore, 3 is the same as 6 , so 5 is equivalent to 17. (See Figure 6.) 10 3. 5 Figure 2 Figure 3 Figure 4 Figure 5 Comparing Bits and Pieces 2 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1 A C E Answers | Investigation 1 19. a. 3 , 6 , 12 4 8 26. 37 1 , 10 1 120 3 16 b. 2.1 GB 20. The diagram below shows that the 8 3 12 b. (See Figure 7.) of a dispenser is almost empty. (See Figure 9.) 21. 1 ; other estimates are acceptable 4 5 8 c. of a dispenser is almost empty. (See Figure 10.) other estimates are acceptable 28. 155 or 1 23. a. about two thirds 2 755 3 5 29. The MathCast: 45 or 3 of the podcast has b. about 80 cups c. about one third of a dispenser is almost full. (See Figure 8.) distance between these fractions is 1 . 22. 3 ; 8 5 6 27. a. 120 12 60 4 been downloaded. 1 3 The Fraction Podcast: 20 or 2 of the 30 d. about 40 cups 3 podcast has been downloaded. 24. A 25. J Figure 6 Figure 7 Figure 8 Comparing Bits and Pieces Figure 9 3 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Figure 10 Investigation 1 A C E Answers | Investigation 1 30. Answers will vary. Possible answer: The b. Answers will vary. Possible answers: MathCast is twice the size of the Fraction Podcast. 31. Answers will vary. Possible answer: The downloaded part of The MathCast is more than twice the downloaded part of the Fraction Podcast. It is possible that some students will take the directions to mean to compare the fractions from part (a). In this case, the downloaded fraction of The MathCast is only a little bit larger than the downloaded fraction of the Fraction Podcast. 32. Assuming a constant download rate, the MathCast takes 88 seconds from beginning to end. The Fraction Podcast takes 3 minutes. 33. a. Answers will vary. Possible answers: Dan 8 miles, Karim 4 miles; Dan 3 miles, Karim 11 miles, etc. 2 Karim 4 miles, Shawn 3 miles; Karim 8 miles, Shawn 6 miles; Karim 1 mile, Shawn 3 mile, etc. 4 c. Dan ran further than Karim, who ran further than Shawn. So Dan ran furthest. 34. a. Answers will vary. Possible answers: Kate could have scored 6 points, Sue 4 points. Kate could have scored 12 points, Sue 8 points, etc. Fractional numbers of points are not possible. The ratio of Kate’s points to Sue’s points is always 3 to 2. b. Lisa could have made only free throws, which are worth 1 point. c. Kate scored the most points because she scored more than Sue, who scored the same number as Lisa. d. Lisa made the most baskets because she made more than Sue, who made the same number as Kate. Connections 44. a. Miguel is correct. If a number is 35. Yes, because 450 can be divided evenly into groups of 5, 9, and 10 with no remainders. 36. Yes, because 12 × 4 = 48. 37. No, not evenly. 150 ÷ 4 = 37.5 38. Yes, because 3 × 17 = 51. 39. C 40. J 41. Mr. Chan: one third or 1 3 Mr. Will: one fourth or 1 4 Ms. Luke: one fourth or 1 4 42. Orange juice was the most popular in Mr. Chan’s class because 1 is greater than 1 . 3 4 43. a. Mr. Will: about 7 cans of orange juice 24 Ms. Luke: about 8 cans of orange juice is equivalent to b. Mr. Chan: 30 cans of juice 6 , 12 but you cannot measure 13 . (Note to teacher: Actually Mr. Will: about 28 cans of juice Ms. Luke: about 32 cans of juice Comparing Bits and Pieces divisible by 2, you can separate it into two equal halves. b. Manny is also correct. If a number is divisible by 3, you can separate it into 3 groups of equal size, or into thirds. c. Lupe is correct. If a number is divisible by n, you can separate it into n groups of equal size, or into nths. 45. a. Possible answer: You can measure with a twelfths strip all fractions with denominators that are factors of twelve (halves, thirds, fourths, sixths, and twelfths). You can also measure with a twelfths strip some fractions that have denominators that are multiples of twelve. For example, you can measure with a twelfths strip 12 , which 24 you can measure any fraction with a twelfths strip but you will not get a whole number numerator. This answer should not be excluded, but it is not expected.) 4 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1 A C E Answers | Investigation 1 b. Possible answer: If you start with a c. Assuming the two numbers in the ratio fraction strip folded into 2, 3, 4, or 6 parts of equal size, you can repartition the strip to make a twelfths strip. You can repartition strips that are factors of 12 to make a twelfths strip. are whole numbers, they will always have a common factor of 1. No other common factors are guaranteed. For example, the ratio 25 : 30 is equivalent to 5 : 6. The only common factor of 5 and 6 is 1. 46. a. Possible answer: You can measure with a tenths strip all fractions with denominators that are factors of ten (halves, fifths, and tenths). You can also measure with a tenths strip some fractions that have denominators that are multiples of ten. For example, you can measure with a tenths strip 12 , which is equivalent to 6 , but you 20 cannot measure 49. a. The common factors of 25 and 250 are 1, 5 and 25. b. The common factors of 30 and 300 are 1, 2, 3, 5, 6, 10, 15 and 30. c. Assuming all of the numbers in the ratios are whole numbers, the first numbers in two equivalent ratios will always have the common factor of 1. 10 11 . 24 (Note to teacher: Other common factors will depend on the “simplest form” of the ratio. The simplest form of a ratio is the equivalent ratio with the smallest whole numbers. In the case of the ratio 25 : 30, the simplest form is 5 : 6. The first number in the simplest form of the ratio (here 5) will be a common factor of the first numbers in any other equivalent ratios. Actually you can measure any fraction with a tenths strip but you will not get a whole number numerator. This answer should not be excluded, but it is not expected.) b. Possible answer: If you start with a fraction strip folded into 2 or 5 (factors of 10) parts, you can repartition the strip to make a tenths strip. 50. about 1 7 47. a. 4 beetles 51. about 5 7 b. 12 beetles 52. a. (See Figure 11.) c. 3 1 fraction strips long 4 b. 100 km, 60 km, about 67 km. Possible explanation: Divide each of the numbers by 3 and that will represent the distance that is 1 the total distance. 48. a. 1 and 5 are the common factors of 25 and 30. b. 1, 2, 5, 10, 25 and 50 are the common 3 factors of 250 and 300. Figure 11 300 km 180 km 200 km Comparing Bits and Pieces 5 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1 A C E Answers | Investigation 1 53. a. Brett (See Figure 12.) 54. a. Since 12.63 : 100, scaling up would produce 1,263 : 10,000. This means it would take the sprinter 1,263 seconds, or 21 minutes, 3 seconds. Jim (See Figure 13.) b. Brett – 3 kilometers (See Figure 14.) Jim – 6 kilometers (See Figure 15.) c. Brett 4 (See Figure 16.) b. Note: The following is used as time, not 5 a ratio. Jim 4 or 2 (See Figure 17.) 10 5 37:30 – 21:03 = 16:27 For every kilometer Brett runs, Jim needs to run two kilometers. The difference between the longdistance runner’s actual time and the sprinter’s hypothetical time is 16 minutes and 27 seconds. Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 Comparing Bits and Pieces 6 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1 A C E Answers | Investigation 1 55. C 60. 9 61. 1 56. 20 , 15 , 12 , 10 , 6 , 4 60 60 60 60 4 60 60 62. 2 3 63. 1 3 64. 2 5 57. 12 58. 3 59. 24 Extensions 65. Possible answers: 69. Possible answers: close to 1 : 10 or 12 2 22 close to 1 : 43 or 22 2 85 close to but greater than 1: 23 close to but greater than 1: 43 22 42 66. Possible answers: 70. Possible answers: close to 1 : 21 or 22 2 43 close to 1 : 17 or 43 2 33 close to but greater than 1: 44 6 67. Possible answers: 71. 11 2 close to 1 : 8 or 9 72. 12 17 close to but greater than 3 1: 18 17 73. 2 1 4 74. 3 1 68. Possible answers: close to 1 2 : 22 43 or 17 35 close to but greater than 1: 17 43 2 17 43 87 2 22 45 close to but greater than 1: 22 21 Comparing Bits and Pieces 7 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1 A C E Answers | Investigation 1 b. Yes, six people can have half if “half” 75. (See Figure 18.) means half of one pizza, making 6 halves. 76. (See Figure 19.) 77. (See Figure 20.) c. Yes, twelve people can have half if “half” means half of one half of a pizza or one fourth of a pizza. 78. (See Figure 21.) 79. (See Figure 22.) 82. Check students’ work to see if the 80. (See Figure 23.) 81. a. Yes, two people can have half if “half” means half of the three complete pizzas or 11 pizzas each. 2 thermometers are drawn to be the same length as the sixth- and seventh-grade thermometers. The thermometers should be partitioned and shaded to show that 3 4 of the goal has been met. Figure 18 Figure 19 Figure 20 Figure 21 Figure 22 Figure 23 Comparing Bits and Pieces 8 Copyright © Pearson Education, Inc., or its affiliates. All Rights Reserved. Investigation 1