From Ethan

I taught both classes today. There was some good news and bad news.

Good news: I started both classes with the penny-payment-doubling-monthly auto deal. Writing out the sequence 1, 2, 4, 8, … , 1024 showed them that ten doublings was a factor of 1000 – that is, 2^10 ~ 10^3 = kilo. We saw that a long time ago (Chapter 1) so no one remembered. I asked about 20 doublings and heard “2000” and “10,000”. They did understand after a while that 20 doublings would add 6 zeroes instead of 3, so a factor of 1,000,000 (mega). So in 20 months the penny payment becomes a million pennies, or $10,000. With four more doublings to go.

In each class I solved an exponential prediction problem by guess-and-try. I hope I convinced them that was a really good strategy, not a poor substitute for not knowing the formulas (in this case that would have called for logarithms, which I didn’t say). It’s a strategy worth knowing since you can use it for lots of problems.

Then in each class I mangled an attempt to build the exponential curve through two points using Excel. I planned to have that exercise force us to consider e, so that I could show how e^rt =(e^r)^t  allowed us to rewrite the equation Excel produces for its exponential trendline in a form that allowed us to recover the growth rate in terms we understood. I still don’t quite understand what went wrong, and can’t seem to reproduce the problem. I will work on that. There is one curricular bottom line: e will be subtle no matter how/when it’s introduced. The only reason for mentioning it in this course is that it appears in Excel when you ask for exponential trendlines.

 

More blog later today.

 

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