Session7 Notes: Nonlinear Functions

A B C D

Notes for Session 7, Part B

Note 5

In this section, we'll explore situations that give rise to increasing exponential functions.

Groups: Begin by putting up an overhead of the description of two different salary options for Problems B1 and B2. Discuss your gut reactions: Which is the better choice? No need to justify your answers; go instead with your first instincts.

Work with a spreadsheet or calculator to figure out the final value of each salary scenario. Remember that the most important factor in this comparison is the total amount earned over the 25 weeks.

A calculator can be used to create the same table as in a spreadsheet. Even if you are filling in the table by hand, there are shortcuts to using the calculator. For example, type in ".01" then "Enter." Then type "x 2" and "Enter." Repeatedly hitting the Enter key will produce the list of outputs for the second salary option. Keep track of each output, and then sum the outputs at the end. For the first salary option, because the salary never changes, simply multiply the salary ($2,000) by the number of weeks (25) to get the total.

Groups: When finished the comparison, share your results and reactions. Many will probably be surprised that starting with such a small amount -- a penny compared with $2,000 -- you could end up with more than 10 times as much total money, and in only half a year.

<< back to Part B: Exponential Growth

Note 6

The salary activity segues nicely into the population model. If a population reproduced by doubling, it would quickly run out of resources, even starting with a small number. Though exponential models are used for some populations, the bases (the constant multiple between outputs) is usually much closer to 1.

Groups: Read through the whale problem. Spend a moment discussing why increasing by 14 percent is mathematically equivalent to multiplying by 1.14. Percents are not a focus of this session, but it's worth spending a little time on this idea in order to understand why it is true. You can also relate it to other things the students probably know:

•	If you have a 10 percent decrease, you can calculate 0.9 x (original number). This is the same as (1 - 0.1) x (original number), or (original number) - 0.1 x (original number).
•	To compute the final price of an item when you have to pay 10 percent sales tax, you can use 1.1 x price, which is the same as (1 + 0.1) x price, or (price) + 0.1 x (price).
•	3 x (number) + 5 x (number) = (3 + 5) x number
•	or in symbols: 3n + 5n = (3 + 5)n = 8n
•	or in words: If you have 3 of something and add 5 of that thing, you end up with 8 of the thing.

These are specific cases of an important algebraic idea: the distributive property of multiplication over addition.

Here, we have
1 x (population) + 0.14 x (population) = (1 + 0.14) x (population) = 1.14 x (population)

<< back to Part B: Exponential Growth

Note 7

If working on a computer, open a new worksheet to model the situation.

Groups: Work in pairs on Problems B4-B6.

Think about a strategy for calculating how long the population takes to double. The Fill Down command on the spreadsheet can be used until a population of 700 is reached. To answer the question of whether it depends on the initial population, change that starting number in the spreadsheet and see if it doubles in the same place. The doubling time does not depend on the starting value; thus an exponent n can be found so that 1.14ⁿ = 2.

Here's one way to see that the time to double doesn't depend on the starting value, and it also highlights some important algebraic thinking.

Years		Population growth

1		1.14
2		1.30
3		1.48
4		1.69
5		1.93
6		2.19
7		2.50
8		2.85
9		3.25
10		3.71

You're looking for a year where 1.14 x 1.14 x 1.14x ... x 1.14 x n = 2 x n. There is an n multiplied on each side, so the only thing that could possibly make the multiple of 2 is all those 1.14s multiplied together. You just have to find the right number of them, and the number of 1.14s only depends on the year.

This also tells you that, for example, if you get a 5 percent raise at your job every year, the number of years it takes you to double your salary is fixed, and it doesn't depend on how much you start out earning.

<< back to Part B: Exponential Growth

Note 8

Groups: Problem B7 relates back to earlier work with toothpick patterns, but requires a bit of creativity. You may want to have actual toothpicks available. If you have difficulty coming up with patterns that work, think of a particular case. For example, try to come up with a pattern that uses twice as many toothpicks at each successive stage. To maintain a pattern visually, it helps to think about making copies of the shape at any stage, and arranging them in some regular way.

Here are a couple of possible solutions:

toothpicks

blocks

Groups: To wrap up this part, talk about how "growing exponentially" is used as slang in the press to mean "growing very fast," but in fact "exponentially" has a specific mathematical definition. Think of possible definitions for "growing exponentially" in your own words, and add exponential functions to the list of nonlinear functions started at the beginning of the session.

<< back to Part B: Exponential Growth

Session 7: Index | Notes | Solutions | Video