Answers Investigation 3

Answers | Investigation 3

Applications

1. and 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?)

9. or

10. or

11.

12. or

13. or

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

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

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.

30.

31.

32.

33.

34.

35.

36.

37.

38.

39.

40.

For Exercises 41–44, answers may vary:

41.

42.

43.

44.

45. between and 1; nearer benchmark:

46. between 1 and ; nearer benchmark:

47. between 1 and ; nearer benchmark: 1

48. between 0 and ; nearer benchmark: 0

49. between and 2; nearer benchmark: 2

50. between and 2; nearer benchmark: 2

51. F

52. D

53. Each person gets or = 0.05 of a
pan. (5 servings)

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

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

56.

57.

58.

59.

60.

61.

62.

63.

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: .

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: .

66.

67.

68.

69. Possible answers: , , , . The
decimal value 0.40 means .
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 is equivalent to .

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.

74.

75.

76.

77. 0.205 0.21

78. 0.1 = 0.1000

79. –0.04 –0.050

80. –1.03 –0.03

81.

82.

83. –0.127, –0.12, 0.2, 0.33,

84. , –0.005, , 0.34

85. G

86. Use Pilar’s approximation for .

d. is equivalent to =, or 0.6666.

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

Answers | Investigation 3

Connections

89. miles

90. If four days at this rate, the Drama Club
can clean miles. So it is correct that they
can clean at least miles, but not that
they will clean exactly miles.

91. They are both correct. = 7

92. a. or ; 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 of a pizza. Each student would
receive eight pieces, giving each of
a pizza. The picture below shows one
student’s share of the pizza.
(See Figure 6.)

93. a. Each part represents , 0.1, or one
tenth of a centimeter. (See Figure 7.)

b. Each part represents , 0.01, or one
hundredth of a centimeter.

c. Each part represents , 0.001, one
thousandth of a centimeter.

94.

95.

96. Written as improper fractions,

97. , so there are no fractions
between them.

98. 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 to , multiply its numerator
and denominator by 3. The equivalent
fraction is . Now that they have the
same numerator, we see that is larger
(because sixty-eighths are bigger than
sixty-ninths). Next, compare to by
dividing its numerator and denominator
by 17. reduces to the equivalent
fraction, is greater than so it’s the
largest part of the whole.

Figure 6

Figure 7

Figure 8

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 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 of 8 is 2. Three of those groups
or of 8 is 6.

102. . Possible explanation: If you partition 3
wholes into 9 parts of equal size, each part
will be of 1 whole and 2 of the nine parts
or of 3 will be 2 of the one-third size
parts or of one of the wholes.

103. Possible explanation: If you partition 18
into ninths or 9 groups of equal size, one
group or of 18 is 2. Two of those groups
or of 18 is 4. (See Figure 10.)

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

105. or or 2.25. Possible explanation:
If you have 3 wholes and partition each
whole into fourths, you can take 3 fourth-
size pieces or from each of the 3 wholes.
You will have 9 one-fourth-size parts or
which is equivalent to .

is the same as

Figure 9

Figure 10