# Modeling Exponential Equations Continued

Yesterday I talked about our introduction to exponential equations, which had the students create patterns and find the connections between the different representations of the pattern. The next lesson was the M&M lab. I have a problem which seems exactly opposite of most teachers, but I have too few students in this class. There are about 15 enrolled, but day to day I have often have about 3 show up. (I teach at a public, alternative school serving at risk youth. We have lots of challenges, but attendance is up there with the best of them.) I had them each do the task alone so we’d have a few different models to talk about. The low numbers are great for self guided learning and lots of hands on time, but they really kill some cool group projects that benefit from lots of voices. I’ve had to rethink a lot of my favorite lessons to try and make up for the low numbers. Again, this is the first year I find myself wishing for rich student mistakes.

Even though the students knew we were studying exponential equations, they all seemed delightfully surprised when the scatter plot had the same shape they’d found the day before. We did one round of growth (Start with 2, add one for every M) and one round of decay (Start with total, take out every M) per student.

S1: It looks like the graph from yesterday, but the numbers don’t have a pattern.

S2: I bet they do.

S1: Then why are our numbers not the same?

Enter, find a pattern time! As well as experimental versus theoretical models. Students started to talk about how they could check for a pattern.  They all wanted to jump to slope, but at least one remembered the division pattern in the table from yesterday and they were off to find the growth rates.

Today we are going to pick up where we left off. They each have a growth rate and a decay rate. Warm up will be answering some questions about how there graphs behave as well as writing an equation for their model. Then we are going to use Desmos to input all our points to find a group pattern and model equation o see how it compares with the individual models and the theoretical model.

Resources: I used the first page of a M&M lab I found here. I only used the first page because I didn’t want to give the students the percent change formula to calculate the growth factor. We had been doing patterns (Multiply by 3 for example) the day before so I wanted them to use that knowledge to come up with a way to find the growth factor. That way there is no equation to memorize and growth and decay are found the same way as opposed to remembering 1+r and 1-r. The students used the first sheet to create there own recording table and graph for the decay part of the experiment. And our class discussion from yesterday was my guide for their warm up reflections today.

Side Note: Spring colored M&Ms are pretty, but they are really hard to read the Ms on. Especially the light yellow ones. Either that, or my eyes are old. But… they also added to our discussion on experimental trials. Did all the M&Ms really have an M on one side or might some have rubbed off? Would that skew our models at all? Could we have skipped some of the lighter ones? Where else does error show up?