It’s that time of year again, when Valentine’s Day candy is all over store shelves. Grab some “conversation hearts” – SweeTart™ Hearts are my favorite – and explore exponential models with your students!
First, let’s model exponential growth. Begin with 5 conversation hearts in a small cup for each student or group. The independent variable is “Toss Number” and the dependent variable is “Total Candy” as shown in the data collection table below. Before any tosses have happened, we have 5 candies, so fill in 5 in the top row.
Toss the candy (gently) onto a flat surface. Count the number of candies with writing FACE UP, and add that many more to the original 5. So if 2 hearts show writing, add 2 more candies to the cup, and record 7 total for toss number 1. Toss the new total candies onto the table, count the number with writing face up, add in that number of candies, and record the new total.
Continue tossing and adding until you have used up your candy supply.
Once the data is collected, create a scatterplot on your technology of choice (or graph paper). Students can use the general exponential model with sliders to model the data.
Let students experiment with different values of A and B to get a good fit. If desired, use an exponential regression to generate the equation that models the simulation data.
What is the “right” model equation? Theoretically, half of the candies will land face up, so there is 50% rate of increase expected, so the base should be 1 + 0.5 or 1.5. The initial amount is 5, giving the theoretical equation
Discuss with students how close their simulation model is to the theoretical equation. How could we improve our experimental results?
The second part of the activity is “Disappearing Desserts!” to explore exponential decay. Begin with 40 candies, combining groups of students if necessary. This time, remove the candies that have writing face up, continuing for several tosses until the candy is depleted or the data chart is filled up. The theoretical model would be
since there is a 50% expected rate of decrease, so the base should be 1 – 0.5 or simply 0.5.
If you’d like to try a different probability experiment, use dice. Add in or take away dice with a certain number, 6 perhaps. The expected rate of increase or decrease would be ±1/6, so the base of the exponent would be 7/6 in the growth scenario and 5/6 in the decay experiment.
Here is the student activity sheet for both the CANDY TOSS and DISAPPEARING DESSERTS explorations, along with a brief teaching guide. Be sure to discuss the results and activity questions with the class afterwards to solidify understanding and clarify misconceptions.
If you don’t have candy1, or don’t want to do the activity that way, you can also use a random number generator on your technology platform of choice. If you are working with a whole class, be sure to set the “random seed” first so that all the students don’t generate the same set of numbers!2
The random number command for each technology platform is:
- All TI Calculators: RandInt(lower bound, upper bound, number of trials).
- Desmos: enter the list of potential values in square brackets: [lower bound,…,upper bound].random.
- GeoGebra: RandomBetween(lower bound, upperbound).
And there are many random number generators available on the internet. Many simulations are possible using a random number generator that aren’t easy to do in a physical experiment: roll 1000 dice, perhaps?
For further details on each technology platform, check out the notes & resources below. Enjoy a sweet Valentine’s Day ❤️, and hope that your math students’ knowledge grows exponentially!
Notes and Resources
1 Variations include using coins, dice, polyhedral dice, spinners, other candies with writing on one side (like M&Ms), etc.
2 Most calculator and computer random number generators are “Pseudo Random Number Generators” because they have an algorithmically-determined list of numbers that is nearly indistinguishable from random. However, they default to the same beginning number, so it is important to “seed” the random command so it generates a number from somewhere else in the list. Imagine my surprise when I didn’t do this, and my entire Algebra 2 class generated the same “random number” after I had explained why randomness was important!
Here are some more details on lists, scatterplots, sliders, regression equations, and random numbers on each of these technology platforms: