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

Exponential Regressions: M&Ms

M+Ms by HazeyNut

I think this idea originally came from Virginia Tech, but I may be wrong.  In any case, here's how we did exponential regressions in College Algebra this semester.

Each student gets a "Fun Size" bag of M&Ms.  Students divide into teams of 3-4.  Each team gets a napkin that they're asked to unfold completely.  The teams spill out their M&Ms on their napkins, making sure all candy pieces are lying flat on the napkin.  Now for the math...

1. Count the total number of M&Ms on the napkin (this will correspond to x=0, where x is the number of "shakes").
2. Fold the napkin over the M&Ms and shake, shake, shake so that the candies get mixed up well.  When done, make sure all pieces are lying flat.  Take away any M&Ms that don't have the M facing up.  Eat them.  Now count how many M&Ms are left (this will correspond to x=1).
3. Fold up the napkin, shake, remove M&Ms that don't have the M facing up, eat them, and count the leftovers (x=2).
4. Repeat Step 3 until M&Ms are gone.

After each turn, I asked all teams to tell me how many M&Ms were left, and I inputted their results into an Excel spreadsheet, which was being projected on the board.  I created a template so that with each number I inputted, a scatterplot began to form.  [Download the template.]  After five turns, one group's data (whose M&Ms cooperated quite nicely) looked like this:

After the M&Ms were gone, I asked each team to find an exponential regression using their graphing calculators that fit their particular data (they could look up at the Excel spreadsheet, where the data had been recorded for them).  In a perfect world, their regressions would look something like y=a(0.5)^x, where a is the number of M&Ms they started with.  Of course, the number of M&Ms doesn't diminish perfectly to half its previous size every time, so we got results that looked more like this (again, this was a rather good trial):

But the imperfection is good.  For one, that's life.  For two, it makes it a little less obvious as to what's going on and creates a nice starting point for some discussion.

After the students gave me the regression equations, I plotted the regressions on Excel (which you can do easily in just a couple clicks).  I asked which team looked like they had the best regression and then we compared r^2 values to see if they matched the students' intuition.

I really wanted to use Skittles for this project so I could call it "Skittles:  Taste the Exponential Regression."  Alas, M&Ms were half the price of Skittles and my frugality won over.  Maybe next semester.

Quadratic Regressions with Angry Birds

Before we ended our discussion on quadratic functions, I wanted to introduce my students to quadratic regressions.  What better way to do this than modeling the trajectory of an angry bird?

This is the picture I projected on the screen.  There is more angry bird/quadratic function material out there on the world wide web that is better than this.  However, I couldn't find exactly what I needed with the materials I had (I wanted a screen shot of an angry bird, whose path wasn't yet finished, with a grid on top).  So, I made this one on Word using a picture I found on the internet.  Be nice.  It took much longer than one would think.

We chose some points that the top parabola passes through and then fit a quadratic regression using those points.  I then asked if the students thought the bird would hit the ice block (around (9.5, 4)).  The results for this varied depending on the class and what points they chose to input into their lists (which is fascinating).

Below gives a picture of the points we plotted in one of the classes, the quadratic regression, and the point (9.5,4), which was plotted post finding the regression.

There's more to be done here, I'm sure, but it's a fine start.