From Maura:
Something happens in the middle of April – all of a sudden it seems as if we are rushing toward the end of the semester. I felt that way today and decided we needed to move on to chapter 9, Interest. As I read through the chapter to prepare I realized that the take-home message for today would be that linear growth is characterized by constant absolute change, while exponential growth is characterized by constant relative change. That set the tone for what I did in class.
We worked through the main example in the book, comparing a $1000 investment in Bank One, which offered $100 annual interest, to a $1000 investment with Bank Two, offering 8% interest compounded annually. I wrote a table on the board and the students worked out the balance in each case after a year. There was a good question: what does compounded mean? We talked about that and about how banks earn money. The next step was to look at year 2. I let them work it out and most of the students went on ahead to future years. When we talked about it as a group they seemed to see the constant change for Bank One and we even wrote out the linear function, B = 1000 + 100T. One student was puzzling over the function for Bank Two (good!) and I told them I could give them the answer, but that if I were them I’d rather see it worked out. I also said that by working it out we would see a nice pattern, which might help us remember the idea. So we broke down the Bank Two balances so that we saw the pattern: after two years, B = 1000(1.08)^2; after three years, B = 1000(1.08)^3; and we guessed that after four years it would be 1000(1.08)^4. We had the calculator do the arithmetic and then I asked them for the general function. What do they see that’s constant, what do they see that changes? We finally got to it: B = 1000(1.08)^T. Along the way we talked about the “1 + ” rule and why that’s a faster way to do this calculation. This was a good review of this concept.
I could have gone to Excel at this point, but decided to stick with the program of highlighting constant absolute vs. constant relative change idea. I had made a worksheet of examples (all fairly artificial) but they were examples that made the point. I asked students to quickly give a quick answer for each one – linear or exponential – then to find the constant absolute or relative change. They got most of them. The tricky ones were the doubling time and half-life which we hadn’t even talked about. We worked through all of the examples and they pushed me (and themselves) to figure out the functions, so that was good. Here are the examples we looked at:
- It costs $35 to rent a car for the day, plus 10 cents per mile.
- The population of my town is growing by 3% each year.
- The value of my car depreciates by 8% each year.
- I signed a contract at my new job that guaranteed a 2.5% raise each year.
- The salesman gets a bonus of $200 for every used car he sells.
- I contribute $50 to my retirement account every two weeks.
- The rabbit population doubles every 60 days.
- Plutonium-238, a radioactive element, has a half-life of 88 years.
The last few minutes were spent looking at the linearExponential file in Excel. This allows them to do some “what-if” examples with different simple and compound interest rates. I talked them through the main points of the spreadsheet and we did a few examples with the understanding that they will use it for homework.
From Ethan:
Maura covered my class last Thursday and the exam was last Tuesday so I haven’t taught for more than a week. I feel a little stale. I’d planned to spend the class having them start the regression homework due Thursday. When I got to the lab I saw the start Maura made on exponential growth, and decided to use her material. I explained that although there was no need in this class to cover particular material that might be required in a later course (no course follows this one) there was material to cover that might matter in life. In particular, we don’t want probability to fall off the end of the semester. We need to do exponential growth – and credit card debt! – before we can get there.
I think the mildly algebraic, mildly artificial problems on Maura’s worksheet were useful. The parallel linear:exponential::absolute change: relative change, repeated N times, may have sunk in. We also got to review percentage change and the 1+ trick.
Note: the linear models on Maura’s sheet all had positive slope, while the exponential ones both increased and decreased.
Spent the last 15 minutes getting them started on their homework.
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