Get Rolling with Exponential Decay
One lesson I really enjoy in my Exponential Functions unit is the “10-Sided Dice” lesson. Most Algebra teachers know about the M&M exponential decay lesson, where students get a cup full of M&M’s and roll them. After every roll, students get to eat the M&M that faces UP and then record how many M&M’s are left. That lesson is fun and students love getting the snack, BUT I love this lesson because it still allows them to get their own data, but it’s different than the typical half-life decay model where b = 0.5. In this activity, the decay factor is b = 0.9, which they wouldn’t run into as often. Also, they love rolling the dice and being the last person in their group to run out of dice!
I got the idea for this lesson from NCTM: National Council of Teachers of Mathematics, back when they had a separate magazine subscription just for high school math content and the final page of each magazine featured a teacher’s description of their favorite lesson:
Well, it has also become one of my favorites! As soon as I read this article almost a decade ago, I knew I had to do this with the classes. The problem was: where to get almost 300 10-sided dice?! I forget which website I used, but I went ahead and bought them since I knew I’d get years of use out of them.
This year, I updated my Word Doc of my version of this activity, which leads groups through the entire process, from data gathering to calculator modeling to calculating class averages. I clarified some of the directions and spaced things out more with the formatting.
The button link above is this year’s print-out for students, while the button link below is a Word compilation of all of my past versions so far (pages 10 - 13 are the most recent version).
How I run this in my class:
Students work in groups of 3 or 4 and split the dice up as evenly as they can to begin
Students roll their dice to collect data until they run out of dice
Using a graphing calculator, plot just the data (no regression curve yet) and analyze their scatter plot (domain, range, intercepts, and more)
Use the Exponential Regression feature of the graphing calculator to find an equation to fit the data, and then analyze just their curve (domain, range, intercepts, asymptotes, and more)
Using probability, create a theoretical model to represent this experiment.
Calculate the class averages of “number of dice remaining” for the first 22 rolls. Compare this to their individual group's data and to the theoretical model’s data points. State the Law of Large Numbers to explain why the class average is closer to the theoretical model
Overall, the experiment went well, and groups were able to finish in one and a half classes (47 minutes long). Ideally, it would be done in a long block to avoid losing the momentum.
Here are some things to look out for if you do this with your students:
Make sure students count the dice BEFORE they begin collecting data! In one of my classes, one bag had 42 dice and one had 38, so both groups had issues with their data
Make sure all students begin with a handful of dice to roll (ideally 10 dice). You don’t want one person rolling all the dice while everyone else just watches
Emphasize recording every roll, even if the amount of dice remaining doesn’t change, and even at the end, when you might have to roll a single die 15 times until you get a ‘1.’ This is what will make your data look exponential!
Walk around and read students explanation of why their data looks exponential. Do NOT allow answers such as: “Because it’s decaying” or “Because it doesn’t have a constant rate of change” etc. Students need to be able to justify clearly and demonstrate that they know what exponential functions look like
Make sure students realize that although their data has an x-intercept, their exponential model will NOT have an x-intercept because of the asymptote
If you run this with your classes, let me know if you find any other ways to improve it! Enjoy!