Power Play · Lesson 1 of 7
Experiencing the Power Play ...
“Explore exponential growth through repeated paper folding and powers of 2.”
• Explain repeated doubling as exponential growth. • Represent paper thickness after repeated folds using powers of 2. • Compare multiplicative growth with simple additive growth. • Use tables and powers to predict values after many folds. • Develop intuition for how quickly repeated multiplication changes scale.
An Impossible Venture!
The chapter begins with a sheet of paper assumed to be 0.001 cm thick. Each fold doubles the number of layers, so the thickness doubles too. The early values look small, but the same multiplication keeps repeating.
| Folds | Thickness | Power form |
|---|---|---|
| 0 | 0.001 cm | 0.001 × 2⁰ |
| 1 | 0.002 cm | 0.001 × 2¹ |
| 2 | 0.004 cm | 0.001 × 2² |
| 10 | 1.024 cm | 0.001 × 2¹⁰ |
| 20 | ≈ 10.485 m | 0.001 × 2²⁰ |
| 30 | ≈ 10.737 km | 0.001 × 2³⁰ |
Three additional folds multiply the current thickness by 2³ = 8. Ten additional folds multiply it by 2¹⁰ = 1024. The multiplier depends on how many extra folds are made, not on which fold number we start from.
Problem
Find the thickness after 10 folds.
- 1.Use T₁₀ = 0.001 × 2¹⁰ cm.
- 2.2¹⁰ = 1024.
- 3.0.001 × 1024 = 1.024 cm.
- 4.So the thickness is just over 1 cm.
Problem
The paper is 0.016 cm thick after 4 folds. Find the thickness after 7 folds.
- 1.From fold 4 to fold 7 there are 3 more folds.
- 2.Three doublings give a multiplier of 2³ = 8.
- 3.0.016 × 8 = 0.128 cm.
- 4.So the thickness after 7 folds is 0.128 cm.
Repeated multiplicative growth is called exponential growth. The surprising size after many folds is not caused by one huge jump; it comes from applying the same multiplication again and again.
Adding 2 repeatedly gives an additive pattern such as 2, 4, 6, 8. Doubling repeatedly gives 2, 4, 8, 16. Only the second pattern is exponential.
Quiz
A quantity doubles after every step. By what factor has it changed after 5 steps?
Which expression gives the paper thickness after 8 folds?
Ten more folds multiply the current thickness by:
Which sequence is exponential?
Why does repeated doubling become large so quickly?
Practice Problems
- Calculate the thickness after 6 folds starting from 0.001 cm.
- A quantity is 12 now. What will it be after three more doubling steps?
- Write an expression for a sheet of starting thickness v after 15 folds.
- Compare the multiplier from 4 extra folds with the multiplier from 8 extra folds.
- Explain why repeated doubling eventually outgrows repeated addition by a fixed amount.
Key Takeaways
• Exponential growth is repeated multiplication by the same factor. • In the paper model, each fold doubles the thickness. • After n folds, the multiplier is 2ⁿ. • Ten extra folds multiply the current value by 1024. • A small starting value can become enormous under repeated multiplication. • Powers make long repeated-multiplication patterns compact.
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Next · Lesson 2
Exponential Notation and Operations