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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
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Investigation 1
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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
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Investigation 1
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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
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Figure 10
Investigation 1
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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
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Investigation 1
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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
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Investigation 1
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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
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Investigation 1
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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
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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
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Investigation 1