Answers | Investigation 3

Applications

1. 73 and 1

32 are equivalent. It takes 3 thirds to

make 1 whole, so 7 thirds is equivalent to 2 wholes and one third.

2. B 3. (See Figure 1.) 4. Answers will vary. (See Figure 2.) Pay

attention to whether students understand the meaning of greater and less in the context of negative numbers (i.e., do they know that –2 > –3, or do they think –3 > –2?)

5. 53

6. 274

7. 889

−

8. 307

−

9. 245 or 1

25

10. 461 or 2

31

11. 253−

12. 484− or 1

24−

13. 122 or 1

22−

14. Answers will vary. Possible answers include 3, –3, 100, –100, 3

42 and 3

42− .

15. a. They moved 14 yards. |7| + |–2| + |4| + |–1| = 7 + 2 + 4 + 1 = 14

b. No. It is because they only got 8 yards forward. Gain 7 and lost 2 makes a gain of 5. Then 5 gain 4 makes a gain of 9. Finally, 9 lost 1 makes a total of 8 yards gained forward.

16. 15°F > 5°F

17. –8°F < 6°F 18. –10°F > –25°F 19. –5°F < –3°F 20. The Blue team is the winner, –50 points is

the highest point value. You might remind students that numbers on the right of the number line are greater than numbers to the left. –50 is the number furthest to the right of all of them.

21. a. Ordering #1 is ordered from least to greatest.

b. Ordering #2 is ordered least to greatest by absolute value.

c. Ordering #3 is ordered from greatest to least.

Figure1

Figure 2

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Comparing Bits and Pieces Investigation 3

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Answers | Investigation 3

22. Ordering #1 (least to greatest): –9, –2, 1, 5, 7; Ordering #2 (least to greatest by absolute value): 1, –2, 5, 7, –9; Ordering #3 (greatest to least): 7, 5, 1, –2, –9; One other ordering that could be done is to reverse ordering #2 (greatest to least by absolute value): –9, 7, 6, –2, 1

23. Jeremiah is correct, because the winner is the person closest to zero. They should be concerned about the absolute values of the numbers.

Note: The term par, which is included in the phrase that Rosemary rewrote, is the number of strokes an expert golf player would take to complete a particular hole in the golf course.

24. a. Order from lowest to highest: Death Valley, Indio, New Orleans, Denver, Wenzuan

b. Order from least to greatest distance from sea level: New Orleans, Indio, Death Valley, Denver, Wenzuan

c. I used absolute value in part b because the order of the cities is in terms of distance, not numerical value. Therefore, it doesn’t matter whether the city’s elevation level is above or below seal level. The order starts from the city closest to sea level, 0 feet, to the city farthest away from sea level.

25. The person owes $15.00, or has a debt of $15.00

26. The elevation is 20 feet below sea level. 27. The quarterback lost 8 yards. 28. The lemonade stand lost ten dollars. 29. 8 3

10 8>

30. 2 43 9>

31. 3 55 12>

32. 1 23 3<

33. 3 34 5>

34. 3 72 6> −

35. 8 612 9

− <

36. 9 1010 11<

37. 3 712 12

− >−

38. 5 56 8

− <−

39. 3 67 14

− = −

40. 4 75 8

− >− For Exercises 41–44, answers may vary:

41. 316

42. 324

43. 312

−

44. 14

45. between 12 and 1; nearer benchmark: 12

46. between 1 and 121 ; nearer benchmark: 1

21

47. between 1 and 121 ; nearer benchmark: 1

48. between 0 and 12 ; nearer benchmark: 0

49. between 121 and 2; nearer benchmark: 2

50. between 121 and 2; nearer benchmark: 2

51. F 52. D 53. Each person gets 120 or 5

100 = 0.05 of a

pan. (5 servings) 54. Each person gets or = 0.025 of a

pan. ( 122 servings)

55. Each person gets or = 0.0125 of a pan. ( servings)

56. 810

57. 410

58. 4100

−

59. 84100

−

60. 75100

0.75=

61. 14100

0.14=

62. 52100

0.52− = −

63. 710

0.7− = −

64. Possible answer: 0.60 is greater than 0.45 because it has a larger value in the tenths place. Also, we can compare these as fractions: 45 60

100 100< .

2

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Comparing Bits and Pieces Investigation 3

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180

114100114

140

12 2100

Answers | Investigation 3

65. Possible answer: 0.6 and 0.60 are equivalent because they both cover the same area of a hundredths grid. Also, we can compare these as fractions: 6 60

10 100= .

66. 40 4100 10

= =0.4

67. 62.5 625100 1000

= =0.625

68. 6 106100 1001 = =1.06

69. Possible answers: 410 , 40100

, 25 , 2050 . The

decimal value 0.40 means 40100

.

If you divide 100 into 10 equal parts or tenths, each tenth will have 10 hundredths in it and 4 of those tenths will have 40 hundredths in it, so 410 is equivalent to 40

100.

70. The step is 0.01. (See Figure 3.) 71. The step is 0.001. (See Figure 4.)

72. The step is 0.1. (See Figure 5.) 73. 1

3−

74. 12

−

75. 18

−

76. 16

−

77. 0.205 < 0.21 78. 0.1 = 0.1000 79. –0.04 > –0.050 80. –1.03 < –0.03

81. 510

0.6<

82. 35

0.3− < −

83. –0.127, –0.12, 0.2, 0.33, 4510

84. 4510- , –0.005, 3

1000, 0.34

85. G 86. Use Pilar’s approximation for 19 .

a. 29

2 0.1111 0.2222= × =

b. 119

11 0.1111 1.2222= × =

c. 69

6 0.1111 0.6666= × =

d. 23 is equivalent to 69 =, or 0.6666.

87. When Belinda entered 21 ÷ 28 into her calculator, the display read 0.75. This is the decimal equivalent for 34 . Since 2128 is

equivalent to 34 , the decimal equivalent is

also 0.75. 88. Possible answer: I would tell the new

student to divide the numerator by the denominator on the calculator. I would tell her to round to the nearest hundredth. To show her this makes sense, I would write this decimal as a fraction with a power of ten in the denominator and show her that the original fraction and this fraction are nearly the same amount. I could do this by finding a number to multiply the numerator and the denominator of the original fraction by that gives a denominator close to a power of ten.

Figure 3

Figure 4

Figure 5

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Comparing Bits and Pieces Investigation 3

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Answers | Investigation 3

Connections

89. 133 miles

90. If four days at this rate, the Drama Club can clean 203 miles. So it is correct that they

can clean at least 193 miles, but not that

they will clean exactly 193 miles.

91. They are both correct. 284 = 7

92. a. 810 or 45 ; 0.8

b. Possible answer: If every student were to receive one piece of every pizza, each pizza would have to be divided into ten equal pieces, with each piece being 110 of a pizza. Each student would

receive eight pieces, giving each 810 of

a pizza. The picture below shows one student’s share of the pizza. (See Figure 6.)

93. a. Each part represents 110 , 0.1, or one

tenth of a centimeter. (See Figure 7.) b. Each part represents 1

100, 0.01, or one

hundredth of a centimeter. c. Each part represents 1

1,000, 0.001, one

thousandth of a centimeter. 94. 1 2 4 5 6 7 8 9, , , , , , ,

5 10 20 25 30 35 40 45

95. 8 16 24 32 40 48 56 64, , , , , , , ,3 6 9 12 15 18 21 24

72 80 88 96 104 112 120 128, , , , , , , 27 30 33 36 39 42 45 48

96. Written as improper fractions, 5 10 15 20 25 30 35 40, , , , , , , ,3 6 9 12 15 18 21 24

45 50 55 60 65 70 75 80, , , , , , , 27 30 33 36 39 42 45 48

97. 8 410 5

− =− , so there are no fractions between them.

98. 45 is the largest part of a whole. This is a difficult comparison. One way to compare these fractions is based on knowing that if we multiply or divide the numerator and denominator of a fraction by a common factor, we get an equivalent fraction. In comparing 1723 to 5168 , multiply its numerator and denominator by 3. The equivalent fraction is 5169 . Now that they have the

same numerator, we see that 5168 is larger (because sixty-eighths are bigger than sixty-ninths). Next, compare 5168 to 45 by dividing its numerator and denominator by 17. 5168 reduces to the equivalent

fraction, 3 .4 45 is greater than 34 so it’s the

largest part of the whole.

Figure 6

Figure 7

Figure 8

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Comparing Bits and Pieces Investigation 3

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Answers | Investigation 3

Connections

99. (See Figure 8, previous page.)

100. 3. Possible explanation: If you break 12 into fourths, or 4 groups of equal size, one group or 14 of 12 will be 3. (See Figure 9.)

101. 6. Possible explanation: If you partition 8 into fourths or 4 groups of equal size, one group or 14 of 8 is 2. Three of those groups

or 34 of 8 is 6.

102. 23 . Possible explanation: If you partition 3

wholes into 9 parts of equal size, each part will be 13 of 1 whole and 2 of the nine parts

or 29 of 3 will be 2 of the one-third size

parts or 23 of one of the wholes.

103. Possible explanation: If you partition 18 into ninths or 9 groups of equal size, one group or 19 of 18 is 2. Two of those groups

or 29 of 18 is 4. (See Figure 10.)

104. 34 . Possible explanation: If you have 3 wholes and partition each whole into fourths, you can take 14 from each of the 3 wholes. You will have 3 one-fourth-size parts, which is 34 of one whole.

105. 1

42 or 94 or 2.25. Possible explanation:

If you have 3 wholes and partition each whole into fourths, you can take 3 fourth- size pieces or 34 from each of the 3 wholes.

You will have 9 one-fourth-size parts or 94

which is equivalent to 2 14

.

is the same as

Figure 9

Figure 10

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Comparing Bits and Pieces Investigation 3

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