A basic premise of science is that much of the physical world can be described mathematically and that many physical phenomena are predictable. This was part of the scientific revolution that took place in Europe during the late 1500’s. So, we started asking questions in my math classes and started looking for models of data that they were interested in.
Many years ago my Calculus students were exploring models of population growth. They worked with some computer programmers from State Farm Ins. and ran some data through their computers on the hunt for a good model. What they came up with is that there was no one good model but that the exponential model showed some promise. So, now when I get students ready for creating models or introduce exponential functions – growth and decay, I make it a Sweet Treat!
Begin with an experiment. You’ll need a small cup or of small bag of skittles (there is printing on only one side vs. m&m’s with two), 2 paper plates, and a graphing calculator for each group of 3 or 4 students.
Part 1: Growth (hint: don’t eat the skittles!)
- Place one skittle on a paper plate. Make sure that this skittle has an “s” on one side. (Sometimes there are a few when the “s” is missing or too light to read.) Start to collect data – trial # vs. # skittles on the plate. Start with trial #0 and 1 skittle on the plate.
- Place the other plate over the top and gently shake it. Take the plate off and if you see an “s”. If you do, place another skittle on the plate and record trial 1 with two skittles. If not, don’t add any skittles and record trial 1 with only 1 skittle.
- Place a paper plate on top and again, gently shake the skittles between the plates. when you take the plate off, add one skittle for each “s” that you see, record the trial and number of skittles on the plate. Repeat step 3 until you don’t have enough skittles to add. However, make sure to include in your last trial the number of skittles you should have had if you could have added enough!
Part 2: Decay (hint: time to start eating the skittles!)
- Place all of your skittles on the paper plate. Count all of them. Start to collect data – trial # vs. # skittles on the plate. Start with trial #0 and the number of actual skittles on the plate.
- Place the other plate over the top and gently shake it. Take the plate off and if you see an “s”. If you do, take the skittles with the “s” showing, off the plate and record trial 1 with the number of skittles that remain on the plate. Eat the skittles you take off the plate!
- Place a paper plate on top and again, gently shake the skittles between the plates. when you take the plate off, again take off each skittle showing an “s” that you see, record the trial and number of skittles left on the plate. Repeat step 3 until you have 1 (or zero) skittles left on the plate. However, make sure to include in your last trial 1 skittle! Why might this be important? Think about the domain of the functions you choose!
Working with the data
For each part of this experiment, create a separate scatter plot. Look at the data for part 1 and choose a regression to try. Graph the regression and see how it fits. Try another regression and see if it fits better. When you are satisfied with your mathematical model (function, regression equation), write it down and repeat this for the data in part 2.
Look at your models and explain the numbers in the equation. Explain why or why not it was important to have one skittle recorded in your last trial in part 2.
It is a sweet treat for students and makes a delicious way of introducing modeling and/or exponential functions! Enjoy!
Download some supporting materials on my wiki
- PowerPoint, SMART Notebook, and/or .pdf – http://greenapples.wikispaces.com/Hands-on+Lessons
- ActivInspire flipchart – http://activinspireadventures.wikispaces.com/Download+Flipcharts
- Graphing Calculator instructions – http://greenapples.wikispaces.com/Graphing+Calculator+Help
- Scatter plots and linear regressions – an ActivInspire flipchart – 2 sample problems, directions for the ti-84 calculator to do scatter plots and regressions, an activity to gather data and predict Shaquille O’Neal’s arm span, and find the model that can predict temperature from cricket chirps. Preview the lesson in a .pdf file at: http://activinspireadventures.wikispaces.com/Download+Flipcharts Download the flipchart at: http://www.prometheanplanet.com/en-us/Download.aspx?ContentId=167738