M&Ms and the Population of Afghanistan

In Business Calc, we're currently studying exponential growth and decay.  I'm rather excited about this since it's something we study in College Algebra, too, and I feel--because it's material I've taught before--that I can expand a bit.  I'm learning that it's really, really hard to expand (i.e., go beyond an absolutely dazzling lecture *cough*) when I'm teaching a class for the first time.  I sort of feel like I did my first semester as a TA:  I just hope I don't screw something up too terribly. But--you gotta start somewhere, right?!

Anyway.  Back to exponential growth/decay.  In College Algebra, when we study exponential functions, I have my students model the decay of an M&M population.  I had planned to do this with my Business Calc class as well.  Then Bowman Dickson posted places to find awesome data, which made me want to use data the UN has on the world's populations instead.

The question:  How to relate M&M's to population growth or decay?

The answer:  I'm not entirely sure.  Here's what we did though...

We started out with the M&M project as in College Algebra.  Each team found the exponential regression (in the form y=ab^x) and the r^2 value for their data.  We talked about the meaning of a and b.  Then I asked them to convert their regressions to the form P(t)=P_0e^(kt), which turned out to be very close to the trendlines Excel found (yay!).  We talked about what k would mean if it this were a real population and how it's related to the derivative.

Now the challenge:  I asked them to do the same types of calculations for an actual population, using data from the UN.  They were on their own for this project, which may or may not have been a great idea.  Below is what they had to go off of.  I focused mainly on finding the exponential regression on a TI as well as understanding growth/decay rates.  But there's much, much more to do here (Bowman does a week-long project!).

And here's the project!

Population Growth or Decay Instructions

Average Rate of Change: Burgers

Summer classes started this week, which means I've started my two calculus classes (wohoo!).  One is essentially a survey course and the other is Calc I.  I'm enjoying it a lot, but I've also been continually up to my eyeballs in lesson plans, homework, and tests.

And it's only the first week.

In any case, as I was preparing for these classes last month, I came across this wonderful introduction to rates of change using (wait for it) a burger-eating competition.

So the first day of Calc I, I asked the class, "What do you think of when I say, 'How fast'?"

I got some good answers, like speed and rate.  I also got a suspicious answer:  average rate of change...way to steal my thunder.

We talked a bit about the importance of units when it comes to speed and then I asked them to consider speed in another setting--the speed of a person consuming an exorbitant amount of burgers:

And now for the questions...

• How many burgers were consumed in total?
• What does that mean that this man/woman's average speed was (in burgers per hour and burgers per minute)?
• Is this speed constant?  How can you tell?
• During what 10-minute interval was his/her speed the greatest?  How can you tell?
• Find the average rate of change (or speed) in burgers/minute between 50 and 60 minutes.
• Between 30 and 60 minutes.

So it's not revolutionary or anything, but it was a nice way to introduce calculus and ease everyone's nerves a little (including my own).