Name ____________________________________________ Date ____________ Class ____________
Investigation 1
1ACE Exercise 22
Growing, Growing, Growing
22. Many single-celled organisms reproduce by dividing into two identical cells.
Suppose an amoeba splits into two amoebas every half hour.
a. An experiment starts with one amoeba. Make a table showing the number
of amoebas at the end of each hour over an 8-hour period. A table is
started below.
HINT Drawing a diagram may help to
show what happens when cells divide. Use
it to help you think about what goes on in
the table.
Start = 0 hours
1
2 hour
1 hour
152
Hours
0
1
2
1
Number of
Amoebas
1
2
4
1 12
2
2 12
3
4
64
256
5
6
7
8
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Amoeba Reproduction
Name ____________________________________________ Date ____________ Class ____________
Investigation 1
1ACE Exercise 22 (continued)
Growing, Growing, Growing
b. Write an equation for the number of amoebas a after t hours.
Note that when t = 1, amoebas (a) = 4
when t = 2, amoebas (a) = 16
when t = 3, amoebas (a) = 64
a=
c. After how many hours will the number of amoebas reach 1 million
(a = 1,000,000)?
HINT Extend your table or use your
equation.
d. Make a graph of the (time, amoeba) data from part (a).
Use the table to help you think
about scales for the axes.
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HINT
153
Name ____________________________________________ Date ____________ Class ____________
1ACE Exercise 22 (continued)
Investigation 1
Growing, Growing, Growing
e. What similarities do you notice in the pattern of change for the
number of amoebas and the patterns of change for other
problems in this investigation?
154
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What differences do you notice in the pattern of change for the
number of amoebas and the patterns of change for other
problems in this investigation?
Name ____________________________________________ Date ____________ Class ____________
Investigation 2
2ACE Exercise 3 (continued)
Growing, Growing, Growing
3. Leaping Leonora just signed a contract with a women’s basketball team. The
contract guarantees her $20,000 the first year, $40,000 the second year, $80,000
the third year, $160,000 the fourth year, and so on, for ten years.
a. Make a table showing Leonora’s salary each year of this contract.
Leonora’s Salary
Year
Salary
Year
1
$20,000
6
2
3
4
$40,000
$80,000
$160,000
7
8
9
5
10
Salary
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b. What total amount of money will Leonora earn over the 10 years?
c. Describe the growth pattern in Leonora’s salary.
d. Write an equation for Leonora’s salary s for any year n of her contract.
s=
155
Name ____________________________________________ Date ____________ Class ____________
3ACE Exercise 1
Investigation 3
Growing, Growing, Growing
1. In parts of the United States, wolves are being reintroduced to wilderness
areas where they had become extinct. Suppose 20 wolves are released in
northern Michigan, and the yearly growth factor for this population is
expected to be 1.2.
HINT What is meant by growth factor?
a. Make a table showing the projected number of wolves at the end of each of
the first 6 years.
Year
1
2
Number of
Wolves
20
24
(20 x 1.2)
3
4
5
W=
c. How long will it take for the new wolf population to exceed 100 (W > 100)?
156
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b. Write an equation that models the growth pattern of the wolf population
W for any year n.
6
Name ____________________________________________ Date ____________ Class ____________
Investigation 4
4ACE Exercise 1
Growing, Growing, Growing
1. Latisha has a 24-inch string of licorice (LIK uh rish) to share with her friends.
As each friend asks her for a piece, Latisha gives him or her half of what she
has left. She does not eat any of the licorice herself.
a. Make a table showing the length of licorice Latisha has left each time she
gives a piece away.
Latisha’s Licorice
Number of
Friends
Size of
Licorice
Left (in.)
1
2
3
4
5
12
6
3
112
3
4
6
7
8
9
10
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b. Make a graph of the data from part (a).
157
Name ____________________________________________ Date ____________ Class ____________
Investigation 4
4ACE Exercise 1 (continued)
Growing, Growing, Growing
c. Suppose that, instead of half the licorice that is left each time, Latisha gives
each friend 4 inches of licorice. Make a table and a graph for this situation.
Latisha’s Licorice
Number of
Friends
Size of
Licorice
Left (in.)
1
2
20
16
3
4
5
6
Explain the similarities.
Explain the differences.
158
8
9
10
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d. Compare the tables and the graphs for the two situations.
7
Name ____________________________________________ Date ____________ Class ____________
Investigation 5
5ACE Exercise 31
Growing, Growing, Growing
31. Manuela said it must be true that 210 = 24 ⫻ 26 because she can group
2⫻2⫻2⫻2⫻2⫻2⫻2⫻2⫻2⫻2
as
(2 ⫻ 2 ⫻ 2 ⫻ 2) ⫻ (2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2).
a. Verify that Manuela is correct by evaluating both sides of the equation.
210 = 24 ⫻ 26
=
2⫻2⫻2⫻2⫻2⫻
2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2 = 1,024
(2 ⫻ 2 ⫻ 2 ⫻ 2) ⫻
(2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2 ⫻ 2) =
16 ⫻ 64 = 1,024
210 = 1,024
HINT
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b. Use Manuela’s idea of grouping factors to write three other
expressions that are equivalent to 210.
Evaluate each expression you find to verify that it is equivalent to 210.
The first one has been done for you.
1.
210 = 21 ⫻ 29
1,024 = 2 ⫻ 512
1,024 = 1,024
2. 210 =
3. 210 =
159
Name ____________________________________________ Date ____________ Class ____________
Investigation 5
5ACE Exercise 31 (continued)
Growing, Growing, Growing
c. The standard form for 27 is 128 (meaning 27 = 128), and the standard form
for 25 is 32 (meaning 25 = 32). Use these facts to evaluate 212.
HINT What do you notice about the
relationship between 27, 25, and 212?
HINT
What is the standard form of 212?
Explain your work.
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160
Name ____________________________________________ Date ____________ Class ____________
Investigation 5
5ACE Exercise 31 (continued)
Growing, Growing, Growing
d. Test Manuela’s idea to see if it works for exponential expressions with other
bases, such as 38 or (1.5)11. Test several cases.
HINT
Test 38.
38 =
38 =
6,561 =
6,561 =
6,561
34 ⫻ 34
81 ⫻ 81
6,561
Test 3 more cases with different bases.
1.
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2.
3.
Give an argument supporting your conclusion.
161
Name ____________________________________________ Date ____________ Class ____________
Unit Test
Growing, Growing, Growing
1. Several species of whale have been declared endangered. When the
populations of a particular whale species fall dangerously low,
biologists encourage governments to agree to a ban on hunting the
species.
Suppose that, in the year 2000, there were only 5,000 whales
of a particular species and that the population was predicted to
continue to decline as shown in the table.
a. Which equation below models this population pattern?
A. W = 5,000(0.1y)
C. W = 5,000 - 500y
B. W = 5,000(0.9y)
D. W = 5,000y
b. What is the decay factor for the relationship? Explain how you
determined your answer.
Year (y)
Whales (w)
0
(2000)
5,000
1
(2001)
4,500
2
(2002)
4,050
3
(2003)
3,645
4
(2004)
3,281
5
(2005)
2,952
6
(2006)
2,657
d. Suppose the danger point for these whales comes when the population falls
below 2,000 whales.
When will this happen?
Explain.
162
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c. According to the prediction, what will the whale population be in 2007?
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
Growing, Growing, Growing
2. a. On Grid I, sketch and label graphs of y = 2x and y = 2.5x. On Grid II,
sketch and label graphs of y = 0.5x and y = 0.8x.
Grid I
y
40
30
Grid II
y
20
1.0
10
0.5
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0
x
0 1 2 3 4
0
x
0 1 2 3 4
b. In Grid I, which equation represents the faster rate of growth?
c. In Grid II, which equation represents the faster rate of decay?
d. How does the graph help you to answer parts (b) and (c)?
e. How do the equations help you to answer parts (b) and (c)?
163
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
Growing, Growing, Growing
3. Belinda has a plan for distributing prize money for a trivia contest. For the
first correct response, the contestant will receive $500. For the second correct
response, the contestant will receive an additional $100, for a total of $600. For
the third correct response, the contestant will receive $100 more, for a total of
$700. Belinda’s plan continues in this pattern.
a. Make a table showing the amount of money a contestant would receive for
answering questions 1 through 6 correctly.
Trivia Contest Prize Money
Number of Correct
Responses
Total Money
Received
1
2
3
$500
$600
$700
4
5
6
b. Make a graph of the data in your table.
164
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c. Write an equation for the relationship between the number of correct
responses c and the amount of money the contestant will receive m.
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
Growing, Growing, Growing
4. Monty has a different plan for distributing prize money for the trivia contest.
The contestant will receive $5 for the first correct response. For the second
correct response, the total winnings will increase to $25, for the third correct
response, the total winnings will increase to $125, and so on.
a. Make a table showing a contestant’s earnings for answering questions
1 through 6 correctly.
Trivia Contest Prize Money
Number of Correct
Responses
1
2
3
Total
Earnings
$5
$25
$125
4
5
6
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b. Make a graph of the data in your table.
c. Write an equation for the relationship between the number of correct
responses c and the amount of money the contestant will receive m.
165
Name ____________________________________________ Date ____________ Class ____________
Unit Test (continued)
Growing, Growing, Growing
5. How are the patterns of change in Belinda’s and Monty’s plans
(Exercises 3 and 4) alike?
How are they different?
6. Decide whether each of the following statements is true or false.
Explain your reasoning.
a. 25100 ⫻ 2510 = 251000
b. 49 ⫻ 59 – 99
d.
106
10 2
= 103
e. 70 = 1
7. Write 11 ⫻ 11 ⫻ 11 in exponential form.
166
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c. (36)8 = 348