I’m teaching a section of Calc A this semester (getting a head start on differential calculus for kids going into BC Calc next year). We’re ahead of the pacing calendar, which has given me a bunch of freedom to dive deep into the big ideas of differential calculus and give students a ton of practice looking at it from a variety of perspectives. Here are three activities that I’m decently proud of.

**Mean Value Theorem Desmos Activity**

The Mean Value Theorem is pretty dry on the surface. A function that is differentiable and continuous on a certain interval must have a point at which the derivatives is equal to the average rate of change over that interval? Boring, and easy to forget.

I put this together in Activity Builder in an attempt to help make more sense of the Mean Value Theorem, use it to get some practice visualizing tangent and secant lines, and generally be more interesting than what my textbook has to offer on that front.

Here are two screens from the activity:

Students work through a few other functions, as well as non-examples where the Mean Value Theorem doesn’t apply.

They need an introduction to the theorem before launching into this activity, but I found that a brief, five minute introduction, with one clear example left on the board as a reference, was enough to get students into the activity in partners. A few had questions that needed to be clarified, but at the same time, the amount of active thinking and experimenting I got out of this activity felt like a big win, and none of the misconceptions students ran into along the way brought the activity to a halt — and likely would have persisted without something like this to bring them out.

**Graphles to Graphles**

Kate Nowak wrote a while back about a great game for practicing domain and range, “Graphles to Graphles“. Students get a few qualities of a function — domain, range, x-intercept, etc — and have to draw a function that fits the description, or show that it is impossible. Students take turns as the “referee”, draw lots of silly pictures, and have fun. Individual whiteboards make this a good time, but it’s not hard to generate a little worksheet with some coordinate planes if necessary.

I decided to adapt Graphles to differential calculus. Here are my cards. There are three pages (the fourth were my notes when I created it). The first page describes characteristics of the function (restrictions for points or asymptotes, or symmetry). The second page gives constraints for the first derivative, and the third page gives constraints for the second derivative. We played a few different ways. Turns out that, when drawing a card from all three categories, it’s often impossible to create the function as described. I think this is fine — it leads to some great conversations, and proving rigorously that some of them are impossible are useful exercises. That said, it tends to slow down gameplay, and drawing two cards at a time yields a much higher success rate while maintaining some useful practice with the principles of derivatives. Either way, I found this to be a good balance of fun and challenge — some are relatively simple, but many are challenging to draw, and require an attention to detail that is challenging for many students. And they love correcting each other. They just love it. The amount of engagement I got out of this, compared with the few minutes to pick out my constraints and introduce them to the paper cutter, felt awesome. I ended up playing three times, in shorter chunks, to reinforce these ideas over time.

**Derivative Analysis**

I don’t have a great name for this, but it was spurred by a lesson where the kids were bored, I was bored, and I needed something to mix things up and get them thinking and discussing. I scrapped whatever I was doing, and threw up this Desmos graph, with the equations hidden.

It’s a function, and it’s first, second, and third derivatives. Students had to figure out, in small groups, which function was which, and be prepared to defend their opinion to the class.

I loved the discussions that came out of this. It created a fun intellectual need for talking about maximums and minimums, and connecting features of a derivative with features of the original function. It also created some great disagreement and discussion using calculus language. One important features is that this activity helps students focus on the parts of functions that we are particularly interested in calculus — the x-values of maximums, minimums, and points of inflection — all while putting aside a bunch of competing, less relevant information.

Here are a bunch more functions I used — I opened two windows of the Desmos graph, and I would project one while I put a new function on the other on my computer and changed the colors. Then, call on a few students to offer their reasoning for the different functions, put the next graph up, and repeat.

One of the big ideas of differential calculus I want students to come away from this class competent in is connecting the graphs of functions and their derivatives. Something I love about these activities is that there are lots of chances for students to practice thinking about derivatives, in lots of different ways, and to get some good feedback on that thinking. Plus these were marginally more fun for students than most of what I have in my toolbox for a calculus class.

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