Answers |
A C E
Investigation 1
Applications
1. 16.95 – 8.69 – 1.95 – 4.29 =
= 30.975, so they will pay $30.98.
You can round the numbers involved to 17,
9, 2, and 4. This way, you can estimate how
much Billie has left over as $2.
= 2.02; Billie will get $2.02 in change.
b. 2.95(6.75 + 5.2) =
Since the exact answer of $2.02 and the
estimate of $2 are close, you know that
your calculations are probably correct.
2. a. 1.8 ÷ 2 =
, or 1.8 × 0.5 =
.
= 35.2525, so they’ll have to pay
$35.25.
= 0.9 mile, so the music store is
0.9 mile from his school.
Since the exact answer of $35.25
is close to the estimate of $36, you
can assume that your calculations
are correct.
Since the exact answer of 0.9 mile
and the estimate of 1 mile are close,
you know that your calculations are
probably correct.
4. 250 ÷ 0.6 =
= 416. 6 , so 416 drinks can be dispensed
before a refill is needed.
= 0.54 mile, so Greg needs to walk
0.54 mile more.
Since the exact answer of 416 is fairly close
to 500, you know that your calculations are
probably correct.
Since the exact answer of 0.54 mile
and the estimate of 0.5 mile are close,
you know that your calculations are
probably correct.
5. You can use estimates to check your work.
If your estimate is close to your exact
calculation, you can assume that your
calculations are correct. Estimates will vary,
but sample estimates are given below.
.
You can round 1.8 to 2 and 0.36 to 0.5
and estimate that Greg has about 1.5
miles left to walk home.
a. 48.92 – (12.35 + 13.12 + 11.91) =
= 11.54 seconds
To break the record, Dwayne will
have to run his leg in less than 11.54
seconds.
Since the exact answer of 1.44 miles
and the estimate of 1.5 miles are close,
you know that your calculations are
probably correct.
Note: In this problem, an estimate is not
likely to be useful because the record is
calculated to the hundredth of a second.
You could, however, calculate
49–12–13–12 = 12 to see that the
answer is certainly “in the ballpark.”
.
You can round 10.5 to 11 and 2.95 to 3
and estimate that the Payne family has
to pay about $33 for the raspberries.
Decimal Ops
,
or 48.92 – 12.35 – 13.12 – 11.91 =
= 1.44 miles, so Greg needs to walk
1.44 miles more to reach home.
3. a. 10.5 × 2.95 =
.
You can round 0.6 to 0.5 to estimate that
about 500 drinks can be dispensed before
a refill is needed.
.
You can round 0.9 to 1 and 0.36 to 0.5
and estimate that Greg needs to walk
about 0.5 mile more to reach the store.
c. 1.8 – 0.36 =
.
You can round 2.95 to 3, 6.75 to 7, and
5.2 to 5. Your estimate would then
be that the Payne family buys about
12 pounds of berries and that they will
need to pay about $36.
You can round 1.8 to 2 and estimate
that the music store is about 2 ÷ 2, or
1 mile away from Greg’s school.
b. 0.9 – 0.36 =
Since the exact answer of $30.98
is close to the estimate of $33, you
can assume that your calculations are
correct.
1
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Investigation 1
.
A C E
Answers |
Investigation 1
b. 400 ÷ 47.65 =
;
estimate: 400 ÷ 50 = 8;
x = 0.138; y = 0.168; z = 0.115
You can check your work by comparing
the exact answers with the estimates.
Since the exact answers are close to the
estimates, you can assume that your
calculations are correct.
≈ 8.39 meters per second
c. 47.65 ÷ 4 =
;
estimate: 48 ÷ 4 = 12;
= 11.9125 seconds
d. Lou’s tree grew the most from
December to April.
Karen: 1.602 – 0.794 = x;
estimate: 1.6 – 0.8 = 0.8
Lou: 1.551 – 0.510 = y;
estimate: 1.6 – 0.5 = 1.1
Jeff: 1.61 – 0.788 = z;
estimate: 1.6 – 0.8 = 0.8
x = 0.808 ; y = 1.041 ; z = 0.822
You can check your work by comparing
the exact answers with the estimates.
Since the exact answers are close to the
estimates, you can assume that your
calculations are correct.
Note: At this time, students may not know
how to calculate the mean. They can use
their understanding of rates to help them
solve this problem.
6. You can use estimates to check your work.
If your estimate is close to your exact
calculation, you can assume that your
calculations are correct. Estimates will vary,
but sample estimates are given below.
a. 1.5 – 0.25 =
;
estimate: 1.5 – 0.3 = 1.2;
= 1.25 miles left
b. 0.25 + 0.375 =
;
estimate: 0.3 + 0.4 = 0.7;
8. 0.2 =
= 0.625 miles
1
5
9. 0.25 =
c. 0.287 + 0.02 =
estimate: 0.3 + 0 = 0.3;
10.
= 0.307 miles
11.
d. 0.85 + 0.005 =
;
estimate: 0.9 + 0 = 0.9;
12.
She did not clean 0.9 mile because
0.85 + 0.005 = 0.855. While the
estimate does imply that she cleaned
0.9 mile, the exact calculation shows
that she did not.
13.
14.
7. a. Karen’s tree was tallest tree in
15.
December. This part is a comparison
question, so it does not require a
number sentence or estimation.
16.
b. Jeff’s tree was tallest by the end
17.
of April. This part is a comparison
question, so it does not require a
number sentence or estimation.
1
4
2
0.4 =
5
1
0.5 =
2
3
0.6 =
5
2
0.67 ≈
3
8
0.8 =
or 4
10 5
7
0.875 =
8
9
0.9 =
10
3
0.78 ≈
4
1
18. 3.14 ≈ 3
7
19. 12.96 ≈ 13
2
1
20. 0.42 ≈ or
5 2
c. Lou’s tree grew the most during the
first month.
Karen: 0.932 – 0.794 = x; sample
estimate: 0.9 – 0.8 = 0.1
21. 473.2 = 473
Lou: 0.678 – 0.510 = y; sample
estimate: 0.7 – 0.5 = 0.2
22. 0.33 ≈
Jeff: 0.903 – 0.788 = z; sample
estimate: 0.9–0.8 = 0.1
23. 4.25 = 4
Decimal Ops
1
5
1
3
1
4
2
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Investigation 1
A C E
Answers |
Investigation 1
24. a. 50 miles per hour
b. $20 for 8 pounds.
b. 400 miles in 8 hours; 600 miles in
c. 3 pounds
12 hours
26. a. The conventional unit rate in this
c. 9 hours to drive 450 miles
circumstance would be $14.80 per
student (rather than about 0.068
students per dollar of revenue).
25. a. $2.50 per pound (or 0.4 pounds per
dollar, the less common form of unit
price)
b. 60($14.80) = $888.
Connections
27. a. The indicated calculation does an
B represents 3.05
C represents 3.50
35.
addition (3(2.25) + 3(2.75)) before
multiplication (3(2.25) and 3(2.75)).
b. The result is correct because he used
the Distributive Property to replace
part of the indicated calculation with an.
equivalent expression.
3(2.25) + 3(2.75) = 3(2.25 + 2.75).
28. The figure is an isosceles triangle with
Set A
21 × 100 = 2,100
21 × 10 = 210
21 × 1 = 21
21 × 0.1 = 2.1
perimeter 1.832. You can estimate the
perimeter by adding 0.7 + 0.7 + 0.5 = 1.9.
This would result in an overestimation.
21 × 0.01 = 0.21
29. This figure is a quadrilateral with perimeter
26.63. You can estimate the perimeter by
adding 9 + 5.5 + 3 + 9.5 = 27.
30. This figure is a special kind of quadrilateral,
often called a kite. Its perimeter is 229.2.
You can estimate the perimeter by adding
80 + 35, which is 115. Then, double that
number to get 230.
21 × 0.001 = 0.021
Set B
2.1 × 100 = 210
2.1 × 10 = 21
2.1 × 1 = 2.1
2.1 × 0.1 = 0.21
31. This figure is a pentagon with perimeter
3.697. You can estimate the perimeter by
adding 0.7 + 0.8 + 0.4 + 0.9 + 1 = 3.8.
2.1 × 0.01 = 0.021
32. D; 81.91 is 0.01 larger than the other three
numbers. The other three numbers are all
equal to each other. The appended zeros
in 81.90 and 81.900 don’t increase the
value of 81.9.
2.1 × 0.001 = 0.0021
Set C
0.21 × 100 = 21
33. J
0.21 × 10 = 2.1
34. A represents 0.350
0.21 × 1 = 0.21
D represents 0.5
F represents 1.095
E represents 1.95
0.21 × 0.1 = 0.021
0.21 × 0.01 = 0.0021
0.21 × 0.001 = 0.00021
Decimal Ops
3
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Investigation 1
A C E
Answers |
Multiplying a decimal number by 10 moves
the product one place to the left with
respect to the decimal point (which often
seems like the decimal point moves one
place to the right). Multiplying a decimal
by 100 moves the product two places to
the left with respect to the decimal point
(which often seems like the decimal point
moves two places to the right).
Multiplying a decimal number by 0.1
moves the product one place to the right
with respect to the decimal point (which
often seems like the decimal point moves
one place to the left), making the number
smaller. Multiplying a decimal number by
0.01 moves the product two places to the
right with respect to the decimal point
(which often seems like the decimal point
moves two places to the left). Multiplying
a decimal number by 0.001 moves the
product three places to the right (which
often seems like the decimal point moves
three places to the left.
2
36. + 1 = 4 + 5
5 2 10 10
= 9
10
9 = 0.9
10
3
37.
+ 3 = 15 + 6
4 10 20 20
= 21
20
=1 1
20
1 1 =1 5
20 100
= 1.05
Investigation 1
41.
42. 0.07 is closest to 0.
Sample explanation: I know that 1 = 0.5
2
and 1 (half of 1 ) = 0.25. For a number
4
2
between 0 and 0.5 to be closer to 0.5 than
to 0, the number should be greater than or
equal to 0.25. 0.07 is much less than 0.25,
so it must be closest to 0.
43. 1.150 is closest to 1.
Sample explanation: 1.150 is greater than 1,
which is greater than both 0 and 1, so 1.150
2
must be closest to 1.
44. 0.391 is closest to 1.
2
2
45. 0.0999 is closest to 0.
Sample explanation: I know that 1 = 0.5
2
and 1 (half of 1 ) = 0.25. For a number
2
4
between 0 and 0.5 to be closer to 0.5 than
to 0, the number should be greater than
or equal to 0.25. 0.0999 is much less than
0.25, so it must be closest to 0.
46. 0.99 is closest to 1.
Sample explanation: I know that 0.99 is
between 1 and 1, but it is only 0.01 away
2
from 1, whereas it is 0.49 away from 1
2
(or 0.5). So, 0.99 is closest to 1.
2 10
3–1= 6 – 5
= 1
10
1 = 0.1
10
40. 5 2 10 10
Decimal Ops
1
Sample explanation: I know that 2 = 0.5
and 1 (half of 1 ) = 0.25. 0.391 is between
4
2
0.25 and 0.5, but closer to 0.5, so it must
be closest to 1 .
38. 11 + 7 = 1 5 + 7
10 10
= 112
10
=21
5
1
2 = 2.2
5
4
39. 2 – 3 = 2 16 – 15
5 4
20 20
=2 1
20
2 1 = 2.05
20
3
3
11 6
4 –1 = 3 –1
8
4
8
8
5
=2
8
5
2 = 2.625
8
47. 0.599 is closest to 1.
2
Sample explanation: I know that 0.599 is
between 1 and 1, but it is only 0.099 away
2
1
from (or 0.5), whereas it is 0.401 away
2
from 1. So, 0.599 is closest to 1.
48.
49.
273
10
273
100
2
= 27.3
= 2.73
273
= 0.273
1,000
273
51.
= 0.0273
10,000
50.
4
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Investigation 1
A C E
Answers |
Investigation 1
52. Answers will vary for the estimates.
57. These two number sentences do not give
Sample answer:
Estimate: 700 + 1,100 = 1,800
Exact result: 715 + 1,129 = 1,844
the same result. For the second number
sentence, 3.5 is not distributed to the
second addend within the parentheses,
so it is not the correct application of the
Distributive Property Over Addition.
53. Answers will vary for the estimates.
Sample answer:
Estimate: 1,500 - 500 = 1,000
Exact result: 1,543 - 512 = 1,031
58. These two number sentences give the
same result because of the Distributive
Property Over Addition. For the first
number sentence, you can take a factor of
3.2 from each addend, then add the two
addends together to arrive at 7.66 (3.2).
54. Answers will vary for the estimates.
Sample answer:
Estimate: 50 × 50 = 2,500
Exact result: 53 × 48 = 2,544
59. These two number sentences give the
same result because of the Distributive
Property Over Subtraction. The factor 3.48
is distributed to both the minuend and the
subtrahend in the first number sentence to
get the second number sentence.
55. Answers will vary for the estimates.
Sample answer:
Estimate: 8,100 ÷ 90 = 90
Exact result: 8,114 ÷ 92 = 88.1956522
56. The calculator doesn’t follow standard
order of operations rules. Instead, it
simply operates from left to right:
13.95 + 3 = 16.95, and
16.95 × 2.25 = 38.1375, which rounds
to $38.14. If the calculator followed the
standard order of operations, it would
calculate 3 × 2.25 first, which is 6.75, and
then add 13.95 to that, which would be
20.7, or $20.70.
Extensions
60. You cannot add time in the same way as
62. The number 3.002 is equivalent to 3.0020.
money because time units are not related
by powers of 10. 45 minutes is not equal to
0.45 hour since there are not 100 minutes
in an hour. Instead, 45 minutes is equal to
45
, or 34 of an hour, which is equivalent to
60
0.75 hour. In the same way, 57 minutes is
57
of an hour or 0.95 hour.
60
61. Again, you cannot add feet and inches in
the same way as you can add meters and
centimeters. 1 inch is not equivalent to 0.1
1
foot. Instead, 1 inch is equivalent to
,
12
or 0.83 foot. 7 inches is equivalent to
7
, or 0.583 foot, not 0.7 foot.
12
Decimal Ops
The 2 actually represents 2 thousandths, or
20 ten-thousandths. 20 ten-thousandths is
greater than 19 ten-thousandths.
63. a. 256 ÷ 16 = 16 pounds
b. 16 × 0.125 = 2 ounces;
16 × 3.375 = 54 ounces
64. a. 3.5 × 3 = 10.5 pounds
b. He needs slightly more than 10 pounds
of flour, so he has to buy three 5-pound
bags.
c. 2,000 ÷ 3.5 = 571.428571. He can
make 571 batches and have a bit of
flour left over.
5
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Investigation 1