Exponential Funtions

Date March 2006
Class Advanced Math class
Teacher Becky Bruening
Notes

Double Your Pleasure

I used “Double Your Pleasure” activity in my Advanced Math class to show a model of exponential growth.  We had just finished talking formally about exponential growth and how, while you don’t see much change at first, later there is an explosion!  We had discussed compounded interest formulas, half-life and populations, but this was a simple activity that THEY generated an exponential curve instead of the book or me doing it.

I did change the name of the activity to “M&M Reproduction” – generating a little more student interest and curiosity.  I had them start with 4 M&M’s.  We defined an M as a girl, and of course girls have babies.  We rolled, put a “baby” in for each girl (m), counted and recorded.  The original activity had them adding 2 for each m, but if I remember from last summer this took a LOT of M&M’s and got “big” even faster.  Our experiment lasted for about 6 years up to 10 years with only about 50 candies per person.  I liked this change, although I realize that the regression equation would be different.  It might be interesting when we do regression to have different groups use different rules and look at the equations. (Although I fear I am quickly running out of days this year!)
Date February 2006
Class Algebra 2 (Grades 10-12)
Teacher Mary Beth Weier
Notes

DOUBLE YOUR PLEASURE ACTIVITY

Objectives:  To collect data and analyze it to find the relationship between the “year” and the “population” of candies.
To explore exponential functions.

Materials:  “Double Your Pleasure” activity sheet, M&M candies, Cup and paper towel, and Graphing calculator

Procedure:
1.  Review with students the steps for entering data into the list menu on the calculator.
Example: Enter (0, 10.1) (5, 2.8) (10, 8.1) (15, 16.0) (20, 17.8)
Find a linear model, a quadratic model, and a cubic model and determine which is the best-fitting model for the data.
2.  Students collect their own data using the M&Ms as described on the activity sheet.
3. Students enter the data in their calculator, graph the data, and find both a linear and exponential equation.
4.  Students complete the questions on the activity sheet.

Notes:  We just finished studying quadratic and cubic equations and will be starting exponential equations next.  This was a very good activity to review the equations we have studied and connect those equations with the ones we will be working with in the next chapter.  The students were very involved and attentive and enjoyed the hands-on activity.  Of course having the candy was a big plus.  I reread the comments other teachers had written concerning this activity, and it helped me to plan the activity.  It is an excellent activity for studying exponential functions and I hope to use it again.
Date February 2006
Class Algebra
Teacher Patti Bailey
Notes For exponential growth I used M&Ms and had students add more for each one that was letter up.  I then had them do it again with doubling the population.  They made great exponential curves!  I also had them do exponential decay with for their final graph.  After the first class, I made modifications to the chart.  I had them roll the M&Ms 8 times rather that just 5.  It definitely gave them a better curve.  After another class did the activity, I again made modifications to the chart.  I will do this activity with an additional class next week.  We’ll see if any more modifications are needed.  It was a great to get students excited about class.  They knew they were going to get to have the M&M activity!  I will use this activity again!  It is a great way to show growth!  --->>> worksheet
Date January 2006
Class Calculus & Pre-Calculus classes
Teacher Stephanie Reynolds
Notes

M&M lab – Double Your Pleasure
With my calculus class, I used the form from class.  It went alright, but it needed more substance.  When Ray Weier posted his version of this lab, I knew that I wanted to use it with my Pre-Calc class when we got to exponential functions.  It went really well except that some students ran out of M&Ms before year 5.  I had them include year 5 with the table of years 6,7,8…
I will definitely do this lab in the future because it is a great way to visualize how exponential functions work.

Date November 30, 2005
Class precalculus classes
Teacher Ray Weier
Notes

In my precalculus classes, we were beginning to work on exponential functions.  We had just finished looking at some application problems on growth and decay, so I decided to try to do the Double Your Pleasure activity with them.  This activity involves using M&M’s to simulate a population model.  It requires that the students do some data collection and then try to fit regression equations to this data.
Each student was given a TI-83 Plus calculator, a paper cup and approximately 75 M&M’s.  They were then instructed to put just 4 M&M’s into the paper cup, pour them out, and record how many M&M’s had the M side facing up.  They then put the 4 M&M’s back into the cup plus double the number of M’s that they had on the previous “roll”.  This new total was then considered ‘year one’ of their experiment.  They then poured out this new total of M&M’s again, looked at the total number of M’s, doubled that number and put those M&M’s into the cup for ‘year two’, etc.
They repeated this exercise until they either reached ‘year five’ or ran out of M&M’s.  These results were recorded into a table similar to the one below.  This table shows the results from one  student’s experiment.

 Year Population 0 4 1 8 2 16 3 26 4 48 5 92

Using their tables, the students were told to use their calculators to StatPlot their data points and then were asked to try to estimate an exponential (growth) function for their data.  They then used their calculators to fit both a linear regression equation and an exponential regression equation to their data to see which gave them a better fit.
We also combined all the students’ results to make a table for the entire class, and then recalculated the exponential regression equation for this table.
Finally, each student was asked to write a few sentences comparing their experimental results to what the theoretical results should be.
Overall, I was pleased with this activity and will probably try it again.  I also plan to try to modify the activity next time so that we might also be able to model a decay problem (remove M&M’s).

Note:  The worksheet for this activity

For exponential growth I used M&Ms and had my algebra students add more for each one that was letter up.  I then had them do this again with doubling the population.  They made great exponential curves!  I also had them do exponential decay with for their final graph.  After the first class, I made modifications to the chart.  I had them roll the M&Ms 8 times rather that just 5.  It definitely gave them a better curve.  After the second class did the activity, I again made modifications to the chart.  I will do this activity with the third class next week.  We’ll see if any more modifications are needed.  It was a great way to get the kids excited to come to class.  They knew they were going to get to have the M&M activity!  I will use this activity again!  It is a great way to show growth!