# Exponentials, Differentiation, and Real-World Connections

Exponential growth and decay is one of the best units because there are so many real-life examples. There’s carbon dating – we listened to this NPR story about using radiocarbons to tell if ivory is illegal or not. This connected to one of my favorite field studies of the year, a trip to the National Geographic museum. They offer free movies on Tuesdays, and the one we saw was a film by Dereck and Beverly Joubert about lions and the impact of poaching. The filmmakers were actually there at the museum, and some of the students even got to ask them questions – not bad for a last-minute government shutdown scramble trip! It also connected to a trip I planned the the Smithsonian Museum of African Art where we viewed ivory artifacts and had another discussion about the ethics of ivory.

There’s also – tragically – the exponentially-growing incarceration rate in this nation. We spent some time talking about mass incarceration, though not as much as last year, when we watched all of The House I Live In. That was time well spent, absolutely, but possibly not as mathematically rigorous as it could have been.

I gave the students a list of some common chemicals, like ADHD medications and antidepressants, as well as caffeine. This outlined their chemical makeup just because that was fun to draw, as well as the equation for their decay inside the bloodstream. I figured this out from their half-lives, readily available online. I would have had them do it but we’re doing logarithms next, so they don’t yet have the knowledge to do so.

Sometimes students want to choose a medication not on the list that they are actually taking, which is so great! It doesn’t get much more real-world than a process taking place inside the student’s own body.

While I love this assignment, it has its challenges. It surprises me how much difficulty some students have with the exponential growth formulas. When I tell them “plug in t=1,” it’s not always immediately obvious to them how to do that. I spent a lot of the instructional time explaining exactly which keys to press on the calculators. As straightforward as this assignment seems, this class could have seriously benefited from more explicit, written-out directions, which I did nothing in the way of.

This particular class has a wide variety of mathematical skill levels, prior knowledge, ages and grade levels, learning needs, and personalities. At any point this week, students were working on any one of several different potential things. Some breezed through the chemical graph, the assessment, this other activity I wasn’t even planning on using, teaching themselves the next topic, and that assessment. Others required what some may consider outrageous levels of hand-holding through the chemical graph, and are just now finishing up the assessment for that. I’m still going to track the second assessment for the students who made it that far, but I won’t hold it against the ones who didn’t. I know those guys are more than capable of learning the second small chunk of material, but it doesn’t seem fair to keep everyone else in a holding pattern.

I can’t decide if the huge range of the class’ products is a problem to be solved with more explicit direction-writing and tighter behavior monitoring, or if it’s an inevitable result of the differences in the learners and should be taken as a sign of my flexibility and ability to differentiate. I’m sure it’s some combination of both – things like this often are.