1. Section 8A Growth: Linear vs. Exponential
Pages

2. Growth: Linear vs Exponential
Imagine two communities, Straightown and Powertown, each with an initial population of 10,000 people. Straightown grows at a constant rate of 500 people per year. Powertown grows at a constant rate of 5% per year.
Compare the population growth of Straightown and Powertown.
Make sure students understand that the principles above also apply to linear and exponential decay as well.

3. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2 3 10 15 20 40

4. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2
11,000 3 10 15 20 40

5. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2
11,000 3
11,500 10 15 20 40

6. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2
11,000 3
11,500 10
(10x500) =15000 15 20 40

7. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2
11,000 3
11,500 10
(10x500) =15000 15
(15x500) =17500 20 40

8. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2
11,000 3
11,500 10
(10x500) =15000 15
(15x500) =17500 20
(20x500) =20000 40

9. 8-A
Straightown: initially 10,000 people and growing at a rate of 500 people per year
Year
Straightown
10,000 1
10,500 2
11,000 3
11,500 10
(10x500) =15000 15
(15x500) =17500 20
(20x500) =20000 40
(40x500) =30000

10. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2 3 10 15 20 40

11. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2
10000 x (1.05)2 = 11,025 3 10 15 20 40

12. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2
10000 x (1.05)2 = 11,025 3
10000 x (1.05)3 = 11,576 10 15 20 40

13. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2
10000 x (1.05)2 = 11,025 3
10000 x (1.05)3 = 11,576 10
10000 x (1.05)10 = 16,289 15 20 40

14. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2
10000 x (1.05)2 = 11,025 3
10000 x (1.05)3 = 11,576 10
10000 x (1.05)10 = 16,289 15
10000 x (1.05)15 = 20,789 20 40

15. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2
10000 x (1.05)2 = 11,025 3
10000 x (1.05)3 = 11,576 10
10000 x (1.05)10 = 16,289 15
10000 x (1.05)15 = 20,789 20
10000 x (1.05)20 = 26,533 40

16. 8-A
Powertown: initially 10,000 people and growing at a rate of 5% per year
Year
Powertown
10,000 1
10000 x (1.05) = 10,500 2
10000 x (1.05)2 = 11,025 3
10000 x (1.05)3 = 11,576 10
10000 x (1.05)10 = 16,289 15
10000 x (1.05)15 = 20,789 20
10000 x (1.05)20 = 26,533 40
10000 x (1.05)40 = 70,400

17. Population Comparison
Year
Straightown 1
10,500 2
11,000 3
11,500 10
15,000 15
17,500 20
20,000 40
30,000
Powertown
10,500
11,025
11,576
16,289
20,789
26,533
70,400

18. Growth: Linear versus Exponential

19. Two Basic Growth Patterns
Linear Growth (Decay) occurs when a quantity increases (decreases) by the same absolute amount in each unit of time.
Example: Straightown each year
Exponential Growth (Decay) occurs when a quantity increases (decreases) by the same relative amount—that is, by the same percentage—in each unit of time.
Example: Powertown: -- 5% each year
Make sure students understand that the principles above also apply to linear and exponential decay as well.

20. Linear/Exponential Growth/Decay?
The number of students at Wilson High School has increased by 50 in each of the past four years.
Which kind of growth is this?
Linear Growth
If the student populations was 750 four years ago, what is it today?
4 years ago: 750
Now (4 years later): (4 x 50) = 950
Make sure students understand that the principles above also apply to linear and exponential decay as well.

21. Linear/Exponential Growth/Decay?
The price of milk has been rising with inflation at 3.5% per year.
Which kind of growth is this?
Exponential Growth
If the price was $1.80/gallon two years ago, what is it now?
2 years ago: $1.80/gallon
Now (2 years later): $1.80 × (1.035)2
= $1.93/gallon

22. Linear/Exponential Growth/Decay?
Tax law allows you to depreciate the value of your equipment by $200 per year.
Which kind of growth is this?
Linear Decay
If you purchased the equipment three years ago for $1000, what is its depreciated value now?
3 years ago: $1000
Now (3 years later): $1000 – (3 x 200)
= $400

23. Linear/Exponential Growth/Decay?
The memory capacity of state-of-the-art computer hard drives is doubling approximately every two years.
Which kind of growth is this?
[doubling means increasing by 100%]
Exponential Growth
If the company’s top of the line drive holds 300 gigabytes today, what will it hold in six years?
Now: 300 gigabytes
2 years: 600 gigabytes
4 years: 1200 gigabytes
6 years: 2400 gigabytes

24. Linear/Exponential Growth/Decay?
The price of DVD recorders has been falling by about 25% per year.
Which kind of growth is this?
Exponential Decay
If the price is $200 today, what can you expect it to be in 2 years?
Now: $200
2 years: 200 x (0.75)2
= $112.50

25. 8-A
More Practice
The population of Danbury is increasing by 505 people per year. If the population is 15,000 today, what will it be in three years?
16,515
During the worst periods of hyper inflation in Brazil, the price of food increased at a rate of 30% per month. If your food bill was $100 one month during this period, what was it two months later?
$169
The price of computer memory is decreasing at a rate of 12% per year. If a memory chip costs $80 today, what will it cost in 2 years?
$61.95

26. 8-A
The Impact of Doubling
Parable 1 From Hero to Headless in 64 Easy Steps
Parable 2 The Magic Penny
Parable 3 Bacteria in a Bottle

27. From Hero to Headless in 64 Easy Steps
Parable 1
From Hero to Headless in 64 Easy Steps
Parable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.

28. 8-A
Parable 1
Square
Grains on square 1
1 = 20 2
2 = 21 3
4 = 22 = 2×2 4
8 = 23 = 2×2×2 5
16 = 24 = 2×2×2×2
. . .

29. 8-A
Parable 1
Square
Grains on square 1
1 = 20 2
2 = 21 3
4 = 22 4
8 = 23 5
16 = 24
. . . 64
263

30. Formula for total on board
Parable 1
8-A
Square
Grains on square
Total Grains on chessboard
Formula for total on board 1
1 = 20
21 – 1 2
2 = 21
1+2 = 3
22 – 1 3
4 = 22
3+4 = 7
23 – 1 4
8 = 23
7+8 = 15
24 – 1 5
16 = 24
= 31
25 – 1
. . . 64
263

31. From Hero to Headless in 64 Easy Steps
Parable 1
From Hero to Headless in 64 Easy Steps
Parable 1 “If you please, king, put one grain of wheat on the first square of my chessboard,” said the inventor. “ Then place two grains on the second square, four grains on the third square, eight grains on the fourth square and so on.” The king gladly agreed, thinking the man a fool for asking for a few grains of wheat when he could have had gold or jewels.
264 – 1 = 1.8×1019 =
≈ 18 billion, billion grains of wheat
This is more than all the grains of wheat harvested in human history.
The king never finished paying the inventor and according to legend, instead had him beheaded.

32. Parable 2 The Magic Penny
Parable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies.

33. 8-A
Parable 2
Day
Amount under pillow
$0.01 1
$0.02 2
$0.04 3
$0.08 4
$0.16
. . .
Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

34. 8-A
Parable 2
Day
Amount under pillow
$0.01
$0.01 = $0.01×20 1
$0.02
$0.02 = $0.01×21 2
$0.04
$0.04 = $0.01×22 3
$0.08
$0.08 = $0.01×23 4
$0.16
$0.16 = $0.01×24
. . . t
$0.01×2t
Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

35. Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)
Time
Amount under pillow
1 week (7 days)
$0.01×27= $1.28
2 weeks (14 days)
1 month (30 days)
50 days
Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

36. Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)
Time
Amount under pillow
1 week (7 days)
$0.01×27= $1.28
2 weeks (14 days)
$0.01×214= $163.84
1 month (30 days)
50 days
Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

37. Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)
Time
Amount under pillow
1 week (7 days)
$0.01×27= $1.28
2 weeks (14 days)
$0.01×214= $163.84
1 month (30 days)
$0.01×230= $10,737,418.24
50 days
Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

38. Parable 2 1 week (7 days) $0.01×27= $1.28 2 weeks (14 days)
Time
Amount under pillow
1 week (7 days)
$0.01×27= $1.28
2 weeks (14 days)
$0.01×214= $163.84
1 month (30 days)
$0.01×230= $10,737,418.24
50 days
$0.01×250= $11.3 trillion
Although many financial institutions will provide the individual with an amortization schedule upon initiating the loan, there may be two important reasons that students should be able to follow the simple mathematics of a schedule. 1) Once they gain a little confidence with the flow of the columns, they are in a position to modify their own schedules using Excel or some other software and customizing it to facilitate their personal strategies of making additional payments of principal. 2) It would also be important to periodically check the accuracy of the lending institution’s monthly or annual statements to verify that everything is legitimate and up to date.

39. Parable 2 The Magic Penny
Parable 2 A leprechaun promises you fantastic wealth and hands you a penny. You place the penny under your pillow and the next morning, to your surprise, you find two pennies. The following morning 4 pennies and the next morning 8 pennies. Each magic penny turns into two magic pennies. WOW!
The US government needs to look for a leprechaun with a magic penny.

40. Parable 3 Bacteria in a Bottle
Parable 3 Suppose you place a single bacterium in a bottle at 11:00 am. It grows and at 11:01 divides into two bacteria. These two bacteria each grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on.
Question0: If the bottle is full at NOON, how many bacteria are in the bottle?
Question1: When was the bottle half full?
Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?
Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?

41. Single bacteria in a bottle at 11:00 am 2 bacteria at 11:01
Question0: If the bottle is full at NOON, how many bacteria are in the bottle?
Single bacteria in a bottle at 11:00 am
2 bacteria at 11:01
4 bacteria at 11:02
8 bacteria at 11:03
. . .
At 12:00 (60 minutes later) the bottle is full and contains ≈ 1.15 x1018
Make sure students understand that the principles above also apply to linear and exponential decay as well.

42. Question1: When was the bottle half full?
Single bacteria in a bottle at 11:00 am
2 bacteria at 11:01
4 bacteria at 11:02
8 bacteria at 11:03
. . .
Bottle is full at 12:00 (60 minutes later)
and so is 1/2 full at 11:59 am
Make sure students understand that the principles above also apply to linear and exponential decay as well.

43. 8-A
Question2: If you (a mathematically sophisticated bacterium) warn of impending disaster at 11:56, will anyone believe you?
½ full at 11:59
¼ full at 11:58
⅛ full at 11:57
full at 11:56
At 11:56 the amount of unused space is 15 times the amount of used space.
Make sure students understand that the principles above also apply to linear and exponential decay as well.
Your mathematically unsophisticated bacteria friends will not believe you when you warn of impending disaster at 11:56.

44. enough bacteria to fill 1 bottle at 12:00
Question3: At 11:59, your fellow bacteria find 3 more bottles to fill. How much time have they gained for the bacteria civilization?
There are . . .
enough bacteria to fill 1 bottle at 12:00
enough bacteria to fill 2 bottles at 12:01
enough bacteria to fill 4 bottles at 12:02
Make sure students understand that the principles above also apply to linear and exponential decay as well.
They have gained only 2 additional minutes for the bacteria civilization.

45. Question4: Is this scary?
By 1:00- there are 2120 bacteria.
This is enough bacteria to cover the entire surface of the Earth in a layer more than 2 meters deep!
After 5 ½ hours, at this rate . . .
the volume of bacteria would exceed the volume of the known universe.
Make sure students understand that the principles above also apply to linear and exponential decay as well.
Yes, this is scary!

46. Key Facts about Exponential Growth
• Exponential growth cannot continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions.
• Exponential growth leads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.
Discussing each of the parables from the text are vital. Students should not miss this content and all the ramifications to the quantitative world around us.

47. 8-A
Repeated Doublings

48. Homework :
Page 496
# 8, 10, 12, 14, 18, 26