## The asymptote as more than an invisible line

Suppose you ate Chinese food that was left on the kitchen counter for four days. You didn’t know, and now you’re sick. You go to the doc and get some medicine. You’re diligent and take the prescribed dosage each day, but what if you don’t? What if you take the meds one day and stop?

With help from Lois and Max that was the problem I posed to my students recently during my exponential functions unit.

What initiated it was asymptotic behavior. I no longer want this powerful constraint to be “an invisible line that a graph approaches infinitely.” In a world of NYS Regents exams, I want that line to have meaning. Note: my students have not graphed exponential functions in my class (it’s part of the algebra 1 curriculum), but they have explored function transformations.

When the kids walk in, I have three containers in the front of the class. Thank you science department for the borrow.

The bell ringer opens things up with an exponential functions exploration. I want them to relate the value of b with either growth or decay. This will help later with constructing a model for our medicine.

We discuss that over time the medicine gets diluted and you must consume it everyday to maintain its potency. Otherwise, you’re never going to feel better. I mention that we can model this situation with mathematics and represent it with the containers of water.

Then I put 5 drops of food coloring into the “body” container and mix it up. I originally wanted to use 5ml, but realized in preparation that this was way too much to see a difference in color for our trial. To represent the medicine leaving the body each day, I tell them that I will remove 1 cup (10%) of the body water and replace it with 1 cup of clean water. Before we start, I have them estimate, based on our fictional simulation, how much medicine would remain in their body after 2 months if they only consume one dose of meds.

We start diluting. I do it 7 times (to represent a week) and they calculate the amount of drops remaining in the container after each day. They graph. The decreasing nature of the graph is noted and I ask them to find an exponential model for this data.

They use the model extrapolate 2 months, with one week intervals, and graph the data.

At this point a few students start to get it. Bam. The asymptote is born.

Reflections:

• Pretty cool activity. Somewhat teacher centered, but highly engaging. The design was such that the intervals of estimation, graphing, and modeling helped to put their focus on the mathematics and not me.
• My focus was asymptotes, but it also served as an introduction to exponential functions in algebra 2. I’m not sure the exponential relationship between the variables was really understood. To that end, I’m not convinced that this was the best activity for the job – at least how it’s designed right now. I should tweak things.
• I was pleased with the bell ringer. There was some good debate around b being fraction and whether that’s why y2 was decreasing.
• My goal after the demo was to go into graphing exponentials with transformations, so I included a series of questions that are typically found on NYS Regents exams. It tanked. I didn’t make a strong connection between the Desmos activity and this one. The 2nd class did slightly better because I bridged the two concepts a little better on day 2 (with an improved bell ringer), but for the goal I set out here, I still need to adjust my planning to better relate things like f(x) = 2^x and f(x+2) = 2^(x+2). They knew what transformation f(x+2) meant, just not how to interpret it as an exponential function.
• Neither class answered the last writing prompt (stop & jot), mainly because I wasn’t feeling it during the lesson. Still thinking about how to wrap things up.
• I started this on one of our shortened days, so I’m not sure if we could have done this in one day. Probably not.
• I didn’t realize it until now, but this whole thing has a 3-act feel to it. I like!

bp