2. Exponential Astonishment
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3. Growth: Linear versus Exponential
Unit 8A
Growth: Linear versus Exponential
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4. Activity: Towers of Hanoi
The game is called Towers of Hanoi
The game begins with al the disks stacked on one peg in order of decreasing size.
The object of the game is to move the entire stack of disks to a different peg; following 2 rules
1. Only 1 disk can be moved at a time
2. A larger disk can never be placed on top of a smaller disk.
Play the game with seven disks, looking for the most efficient strategy for moving disks.
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5. Strategy CN (Game 1-6)
1. Look at the series of goals. How many total moves will it take to reach the goal of having 7 disks on another peg?
2. Find a general formula for the second and third columns after n steps.
3.Use the formula for the total moves to predict the number of moves required to complete the game with 10 disks.
4. The game is related to a legend…. How many total moves are required to move the entire set of 64 disks?
5. The legend holds that upon completion….. Based on your answer to #4, how many years will it take to move the entire stack of 64 disks. (moving 1 disk/sec and the universe is about 14 billion yrs. Old.
6. Briefly comment on what this game illustrates about the nature of exponential growth.
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6. Growth: Linear versus Exponential
Linear Growth occurs when a quantity grows by some fixed absolute amount in each unit of time.
Exponential Growth occurs when a quantity grows by the same fixed relative amount—that is, by the same percentage—in each unit of time.
Make sure students understand that the principles above also apply to linear and exponential decay as well.
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7. Growth: Linear versus Exponential
Straightown grows by the same absolute amount each year and Powertown grows by the same relative amount each year.
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8. Linear or Exponential? 1-5 CN (1)
In each of the following situations state whether the growth (or decay) is linear or exponential, and answer the associated questions.
The number of students at Wilson High School has increased by 50 in each of the past four years.
1. If the student population was 750 four years ago, what is it today?
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9. Linear or Exponential? CN (2)
The price of milk has been rising 3% per year.
2. If the price of a gallon of milk was $4 a year ago, what is it now?
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10. Linear or Exponential? CN (3)
Tax law allows you to depreciate the value of your equipment by $200 per year.
3. If you purchased the equipment three years ago for $1000, what is its depreciated value today?
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11. Linear or Exponential? CN (4)
The memory capacity of state-of-the-art computer hard drives is doubling approximately every two years.
4. If a company’s top-of-the-line drive holds 2.5 terabytes today, what will it hold in six years?
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12. Linear or Exponential? CN (5)
The price of high-definition TV sets has been falling by about 25% per year.
5. If the price is $1000 today, what can you expect it to be in two years?
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13. The Impact of Doublings Parable 1:
From Hero to Headless in 64 easy steps.
Legend has it that when chess was invented in ancient times, a king was so enchanted that he said to the inventor, “Name your reward.”
Grains of wheat for 64 spaces on a chessboard.
The king never finished paying the inventor and according to legend, had him beheaded.
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14. Parable 2: The Magic Penny
One lucky day you meet a leprechaun who promises to give you fantastic wealth, but hands you a penny before disappearing.
You place the penny under your pillow and the next morning you find two pennies..
By the end of 52 days you have enough to pay off the national debt of the US.
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15. Parable 3: Example
Bacteria in a Bottle: Suppose you put a single bacterium in a bottle at 11:00 a.m. It grows and at 11:01, it divides into two bacteria. These two bacteria grow and at 11:02 divide into four bacteria, which grow and at 11:03 divide into eight bacteria, and so on. Thus, the bacteria doubles every minute.
If the bottle is half-full at 11:59, when will the bottle be completely full?
Since the bacteria doubles every minute, the bottle will be full after one more minute. That is, the bottle will be completely full at 12:00 p.m.
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16. Doubling Lessons
The three parables reveal at least two key lessons about the repeated doublings that arise with exponential growth.
1. You’ll notice that the number of grains on each square is nearly equal to the total number of grains on all previous squares combined.
2. All three parables show quantities growing to impossible proportions.
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17. Key Facts about Exponential Growth
Exponential growth leads to repeated doublings. With each doubling, the amount of increase is approximately equal to the sum of all preceding doublings.
Exponential growth cannot continue indefinitely. After only a relatively small number of doublings, exponentially growing quantities reach impossible proportions.
Discussing each of the parables from the text is vital. Students should not miss this content and all the ramifications to the quantitative world around us.
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18. Homework CN (12)
12. Please answer the quick quiz multiple choice questions 1-10 on p. 478.
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19. Homework 8A CN 1-12 (Game 1-6, Notes 1-5, Quick Quiz) P. 478: 1-8
1 web
31. Computing Power
32. Web Growth
1 world
33. Linear Growth
34. Exponential Growth
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