Only four classes left in the semester after today. All the more reason to focus on what I’d like students to take away with them rather than on what the syllabus says I should “cover”.
So today was for developing/sharpening intuition about exponential growth (and decay). Started by asking the class to figure out (in groups, by any method they chose) how long it would take to double a $1000 investment that earned 7% compounded annually. They found the answer – about 10 years – several independent ways: building a spreadsheet (on the fly), searching the web for an investment calculator, solving 2000=1000(1.07)^x for x (by cut-and-try). None of them thought to use the exponential growth spreadsheet that comes with the course materials, which we’ve been working with for several weeks.
They found the doubling times for different initial balances at the same 7% interest rate to be the same 10 years. I pointed out that that was an easy consequence of the algebra since in all cases it meant solving 1.07^x = 2, but acknowledged that an explanation using algebra pleased me but probably didn’t please them. The evidence does that better.
Working out doubling times for interest rates of 1%, 2% and 10% led directly to the Rule of 70. I pointed out that it failed for high interest rates – it predicts a doubling time of 0.7 years for 100% interest, when the correct answer is clearly one year.
I told them that to find out why the Rule of 70 worked they’d have to take a course in calculus. The reason is, of course, that the natural log of 2 is 0.6931 (see page 20 of Robert Heinlein’s Time for the Stars)– nearly 70%. I believe that if I prepared carefully I could provide a one hour explanation they could follow, if not remember. But that wouldn’t be a good use of an hour of class time.
Moved from growth to decay. A few students saw right away that a $20,000 car depreciating at a rate of 7% per year would lose half its value in 10 years. I think I convincingly made the point that exponential decay was just exponential growth with time running backwards. The graphs help.
I asked who knew the term “half life” – most had heard it but few were willing to venture a definition. But we came up with one pretty easily. When I asked where the term might come up in the world someone said “drugs”, others said “nuclear something”. We did drugs first – amount left in the bloodstream over time following an initial dose. The first guess about what would happen after two half lives – “all gone” – was a wonderful hook on which to hang a review of the difference between linear and exponential. In exponential decay it’s never all gone, though it will be negligible after a while.
I showed why 10 half lives left 1/1000 of the original, so 20 half lives one one-millionth.
What followed was a long discussion of nuclear chemistry (simplified). Radioactive decay: X -> Y + particle + energy. (Did say that the mass decreased, which was where the energy came from, but didn’t write e=mc^2. Should have. Should put a problem in the book about that.) Someone asked about the sun’s energy, so I wrote H + H = He + energy, and talked about atomic (fission) and hydrogen (fusion) bombs. Finally got to radioactive waste disposal of things with really long half lives.
Just started probability – ten heads in a row will happen with probability 1/1000, so it’d be rare for you but would happen to someone if many thousands of people tried the experiment.
The bottom line from the day – lots of common knowledge and common sense. So worthwhile, I think, even though it won’t “prepare them for the exam in the course.”
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