Thursday, March 29, 2012

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.

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