CMP3_G6_LBR_ACE1 - Mrs. Southward



Answers | Investigation 1

Applications

1. 1; Possible explanation: [pic] is 1, so [pic] is

more than 1.

2. [pic]; Possible explanation: Since [pic] is close to [pic].

3. 1; Possible explanation: [pic] is only [pic] less

than 1 whole.

4. [pic]; Possible explanation: [pic] of 100 is 50,

so [pic] is closer to [pic]

5. [pic]; Possible explanation: [pic] is 0.50 and [pic]

is 0.75, so 0.67 is close to [pic]

6. 0; Possible explanation: 0.0009999 is

a very small amount. It does not have

any tenths in it, and [pic] is equivalent to

5 tenths.

7. 1; Possible explanation: [pic] is a little

less than 1 and [pic] is a little less than [pic]

Together, a little less than 1 and a little less

than [pic] is a little less than [pic] or closer to 1

than to 2.

8. 2; Possible explanation: [pic] is closest to 2,

and [pic] is a small number that will make the

sum closer to, but not greater than, 2.

9. 3; Possible explanation: [pic] is a little

more than 1.3. 1.3 and 1.3 is 2.6, which is

greater than [pic], and closest to 3.

10. 0; Possible explanation: It would take

two [pic]’s to equal exactly [pic]. Since [pic] is less

than [pic], [pic] and [pic] is less than [pic] and closer

to 0.

11. 2; three tenths and eight tenths have a

sum greater than 1, so the total sum here

is greater than 2.

12. 2; [pic] is equivalent to 1.4 which, when

added to 0.375, is 1.775, which is close

to 2.

13. 0.5; Possible explanation: [pic] is a bit more

than half, and [pic] is a small fraction, not

big enough to push [pic] close to 1.

14. 0.5; Possible explanation: [pic] is exactly

in the middle of 0 and [pic] [pic] is a small

fraction so the sum is closer to [pic], but not

greater than [pic]

15. 0; Possible explanation: [pic]. Also, [pic] is

exactly halfway between 0 and [pic], so

[pic].Therefore the sum is closer to 0

than to [pic]

16. a. This is incorrect. [pic], so the sum here

is less than [pic].

b. Correct. Each fraction is equal to [pic], so

the sum is 1.

c. Correct. [pic], so the sum of [pic] and [pic]

is also greater.

d. Correct. [pic] and [pic], so the sum

is greater than [pic]

17. Possible answer: [pic] and [pic]

18. Possible answer: [pic] and [pic]

19. Possible answer: [pic] and [pic]

20. Possible answer: [pic] and [pic]

21. a. The greatest possible sum is 1.05 + [pic]

b. The least possible sum is [pic]

22. a. No; In the price list, the whole numbers

add up to 9: 2 + 1 + 1 + 1 + 3 + 1 = 9.

The cheese is $1.95, which brings the

total to $9.95. There’s clearly enough

in the cost of other items to put the

total over $10. (The exact cost of the

groceries is $12.42.)

b. Possible answers: Milk, cheese,

avocado ($4.92); eggs, cheese, honey,

bread ($4.91)

c. cereal, honey, and avocado ($4.94)

23. [pic] is closest to [pic]. Possible explanation: [pic]

The other [pic] makes [pic] just a little more than

[pic], but not close to [pic], or 1 whole.

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Answers | Investigation 1

24. Soo has enough molding. is less than

1 whole and is more than 1 whole or

. Since , when the is added to

the it will be greater than 1 yard, and

the total will be greater than 2 yards.

25. He had about 8 quarts.

26. No; If you add to , you will get .

But is a little less than , so there is not

enough.

27. a. Marigolds: ; Lantana: ;

Impatiens: ; Petunias: ; Lilies: ;

Begonias: ; Tulips: ; Daisies: ;

Irises:

b. [pic]

c. [pic], or [pic]

d. Incorrect. Possible explanation: The

number sentence for the situation is

[pic]. If you work out the

subtraction problem on the left of the

equal sign and the addition problem on

the right, the answers are not the same.

e. Possible combinations that total[pic], the

fraction planted with impatiens:

Marigolds + Petunias + Lantana:

[pic]

Lilies + Petunias: [pic], or [pic]

Marigolds + Begonias + Tulips +

Daisies: [pic], or [pic]

28. a. [pic] of the page is used for ads.

b. [pic] of the page remains.

29. (three -page ads, or )

plus (four -page ads, or )

plus (ten -page ads, or )

= pages.

30. [pic]pages

31. [pic]of the lasagna is eaten,

leaving [pic] of the lasagna uneaten.

32. [pic] of a small bag of chips

33. [pic]

34. [pic]

35. [pic]

36. [pic] or [pic]

37. [pic] is greater: [pic]

[pic]; [pic]

38. [pic] is greater: [pic]

[pic]; [pic]

39. [pic]

40. [pic]

41. [pic]

42. [pic]

43. [pic]

44. [pic]

45. [pic]

46. [pic]or[pic]

47. [pic]

48. [pic]

49. [pic]

50. [pic]

In all of the problems, you add unit fractions

where one fraction is half the size of the

other. The fraction in each part with the

lesser denominator is twice the value of the

unit fraction with the greater denominator.

You can think of the unit fraction with the

lesser denominator as two unit fractions

with the greater denominator. This gives

a sum with a 3 in the numerator over the

greater denominator.

51. No. If [pic] of all the pizza were eaten,

this would be less than one whole pizza.

If there are eight sections in each pizza,

then people are eating eighths. And all

together they ate [pic] or [pic] pizzas.

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Answers | Investigation 1

Connections

52. Divide the rectangle into thirds. Each third

of the rectangle represents 50% of the

original whole. So, 100% is represented

by the blue area.

53. 15 beans. There are 9 beans shown in the

picture representing three fifths. So each

fifth must contain 3 beans. Since the whole

is five fifths, there must be 15 beans on the

counter.

54. a. [pic] is 0.5 and 50%

[pic] is about 0.33 and 33%

[pic] is 0.25 and 25%

[pic] is about 0.66 or 66% (Some students

will argue for 0.67 and 67%, if the

convention of rounding up is used.)

[pic] is 0.75 and 75%

[pic] is about 0.16 and 16% (Some

students will argue for 0.17 and 17%, if

the convention of rounding up is used.)

[pic] is 0.2 or 20%

[pic] is 0.125 or 12.5%

b. (See Figure 1.)

55. D

56. a and d

a. This set of fractions can be renamed as

hundredths because each denominator,

2, 4, and 5, are factors of 100.

b. This set of fractions cannot be renamed

as hundredths. The denominator 10 is a

factor of 100, but 11 and 12 are not.

c. This set of fractions cannot be renamed

as hundredths. The denominators 6 and

8 are not factors of 100.

d. This set of fractions can be renamed

as hundredths because all the

denominators, 5, 10, and 20, are factors

of 100.

57. a. The least possible sum is [pic]. This is the

sum of the least numbers in each of

these intervals.

b. The greatest possible sum is 2.

That is the sum of the greatest numbers

in each of these intervals.

58. N = 2

59. N = 3

60. N = 8

61. N = 24

Figure 1

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Answers | Investigation 1

62. [pic]

63. [pic]

64. [pic]

65. 18.156 < 18.17

66. 4.0074 > 4.0008

67. G

68. Answers will vary. Possible answers include:

Foley, Burg, Walker, Theule, and Lapp.

[pic]

69. a. [pic] of a whole; 3 squares = [pic] of a

whole; 9 squares = 1 whole; so 12

squares = [pic] of a whole.

b. [pic] of a whole; 3 squares = [pic] or [pic] of a

whole; 9 squares = [pic] or [pic] of a whole;

so 7 squares = [pic] of a whole.

70. a. Note: The purple shaded region

represents 1. The green shaded region

represents the [pic] portion of [pic]. The

blue shaded region represents [pic].

Or,

b. Note: The Xs indicate the regions

subtracted. The purple and green

shaded regions each represent 1. The

blue shaded region represents [pic]. The

remaining portion is [pic].

Or,

71. a. Since [pic] and [pic], the sum

of these two numbers is the same no

matter which forms we use.

b. Answers will vary. Possible answers

include: [pic] and [pic].

c. Any problem with common

denominators should be easy to solve.

This includes Maribel’s problem as well

as this one: [pic].

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Answers | Investigation 1

Extensions

72. Answers will vary. Possible answers include:

[pic]

73. Answers will vary. Possible answers include:

[pic]

74. It is always possible to find a fraction

between any two fractions on the

number line. One way to know this is

that we can rewrite each given fraction

as an equivalent fraction, using common

denominators, and then choose a new

fraction with a numerator between the two

new numerators. For example, given [pic] and

[pic] we can rewrite as [pic] and [pic], and then

we know [pic] is between these.

Another method would be to take the

number that is halfway between the two

fractions, which is the average of the two

fractions. For example, to find the fraction

between [pic] and [pic] we can take the average.

[pic]

Now [pic],

so [pic] is between [pic] and [pic].

75. a. [pic] page costs $5, [pic] page costs $10,

[pic] page costs $20, [pic] page costs $40,

[pic] page costs $80, 1 full page costs $160.

b. 40 × 3 + 20 × 4 + 10 × 1 = $210

c. Yes. [pic] page costs $40 and [pic] page

costs $40, for a total of $80.

d. Answers will vary. Possible answers

include: Any combination adding

up to [pic] page. This includes two

[pic]-page ads, or one [pic]-page ad and

two [pic]-page ads.

76. a. 2 acres; 16 people can clear twice as

much in the same amount of time.

b. [pic] acre; one fourth of the people can

clear one fourth as much in the same

amount of time.

c. 24 people; three times as many people

are needed to clear three times the

area in the same amount of time.

d. 48 people; twice as many people are

needed to complete part (c) in half of

the time.

77. a. Answers will vary. Possible answers

include:[pic].

b. Answers will vary. Possible answers

include: [pic].

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Let’s Be Rational Investigation 1

Let’s Be Rational Investigation 1

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Let’s Be Rational Investigation 1

Let’s Be Rational Investigation 1

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Let’s Be Rational Investigation 1

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