I loved teaching AP Calculus BC last year! I was a little intimidated at first… I hadn’t taught or even used Calculus in 15 ish years. It was a great experience and I can’t wait to do it again! But alas… not enough sections this upcoming year. I do want to reflect on what I did and what I would like to do in the future so I don’t forget.

My favorite lesson was actually the first day of school. I showed three linear graphs and asked students to write down everything they knew about them.

After talking about key features – intercepts, end behavior, domain and range – and linear vs. constant functions and a line that’s not even a function, I surprised them: these aren’t lines!

Students worked in teams to match each graph with an equation and come up with a convincing reason why. Then teams presented their solutions via gallery walk.

We touched on domain and range, even and odd functions, and other key features of functions; we also began exploring the ideas of limits and local linearity. Students really latched onto the idea of “zooming in” on the graph, which ultimately paid huge dividends throughout the course. I found myself saying weekly, if not daily, “remember the first day of school when we zoomed in on those three graphs?”

Another win last year was practicing derivatives and (some) antiderivatives at the same time. This is one of my favorite teaching strategies in general; if you can do something one way, you should be able to think “backwards.” I only did basics, like the table below, but it really got students thinking both directions.

I like to give a little peek into what’s coming, and since we already knew that the derivative of a constant is zero, it wasn’t too challenging to sneak in conversation around this “+C” business early.

Here’s another one, this time after trig derivatives and chain rule:

In the future I would like to do this even more intentionally and often.

Around the same time, we were sketching derivatives given a graph, so I threw in some antiderivatives in the same way: if this is the derivative graph, what could the original graph look like? Here they could really see the “+C” as teams came up with different vertical translations of the same graph.

As much as I loved exploring and wondering about these different ideas, the biggest struggle I had was the order in which to introduce concepts. Everything is so connected, and I didn’t want to muddle their understanding by doing too much at once. I ended up sticking pretty close to our textbook; especially in my first year, especially since my PLC teammate would also be following the textbook, especially since the instructor of my APSI used the same textbook at his school and provided me with a bunch of already aligned supplemental resources (!!!), it felt important to stay consistent.

Now that I have a little more “big picture” vision, I would do things a little differently.

First, standards-based grading! I started this in second semester last year, when we began studying series. It was a natural break to begin something new, but the content turned out to be perfect for the kind of self-reflection and improvement that I needed from students. (I’ve been implementing various versions of SBG every year I’ve been teaching; this might be my favorite version so far, but still far from perfect.)

Learning Objective 4.1A – Determine whether a series converges or diverges.

This learning objective encompasses A LOT of different tests for convergence, and what I found is that our textbook gave very bare answers to homework problems, so students thought they were doing great when they were actually missing big chunks of explanation that would communicate sound mathematical reasoning (and let’s be honest, would cost them points on the AP exam). By changing my grading strategy, I was able to better communicate my expectations and give feedback for improving.

After our first quiz on the topic, I gave students their highlighted quizzes back (no points marked, my typical M.O.) along with a Google doc for reflection. This included a rubric for them to score their own progress on this learning objective.

I showed an example of a 4 to the whole class, emphasizing three parts to communicating whether a series converges: conditions, evidence, conclusion. Many students were providing the evidence, but not first checking that the conditions for using the convergence test have been met or providing an appropriate conclusion; we looked at two examples of 3s that illustrated these omissions. Then students made corrections to each problem on their quiz and gave an overall score for their understanding with evidence to support their score. Ultimately they were very accurate and I only had a couple of conversations with students whose evaluation of their learning was different from mine.

The best part – every student earned a 4 on their next assessment of the same learning target. After a whirlwind first semester, I finally felt like I was doing something right!

I always give cumulative quizzes/exams, so this quiz also included derivatives from first semester, along with a rubric for self-evaluation similar to the one above. I got some very honest, “I thought I was a 4, but this evidence says that I’m currently at a 2” evaluations. They realized for themselves that they still had some work to do on the “easy” stuff! I really hope to start this from the very beginning of the year when/if I get to teach this course again… I think the potential for continuous learning here is huge.

The other thing that I would really want to do is address the sequence of content; I still don’t have a good feel for this yet. In my other courses, I try to space out related content so that students have time to practice content before trying to build on it. For example, in EM3, we teach unit circle and trig functions (all no calculator) in unit 2, practice it throughout homework in units 3 and 4, then unit 5 brings in inverse trig functions after they have built up fluency. I don’t know yet what this looks like in Calculus, but I would like to attempt it. Thoughts so far:

- Do some work with series earlier in the course. Chapter 10 in our textbook is all about series and it is HEFTY. It could easily be two units with all of the convergence tests taught earlier and practiced before needing them to test endpoints for intervals of convergence later.
- Teach derivatives and integrals simultaneously??? Maybe start with graphical and tabular interpretations of both, then move to analytic approaches? Can this be done??? I have more questions than answers.
- I felt like I needed more time for teaching and practicing integration techniques, so find a way to do this earlier. My students sucked at u-substitution, honestly because I think I sucked at teaching it. Maybe by teaching chain rule and “backwards chain rule” together, students will become more fluent. Maybe by separating it from other integration techniques and letting them get good at one technique at a time, they can feel more equipped when I ask them to compare and contrast strategies. Maybe Coaching Questions would be a good routine to use practice using integration techniques.
- Polar area – I feel like I can sneak this in earlier with other area integration problems. Still not sure.
- The construct of teaching all AB content first, then moving to BC content, seems forced. Is there a reason for doing this, other than my textbook said so? Yes, the AB content is foundational for BC content, but can/should it be better integrated?

And there we have it folks – a lot of partially formed ideas about AP Calculus BC! I really do hope to teach this class again. Shoutout to my department chair – if you’re reading this, in case I haven’t said this enough, I would LOOOOOOOOVE to teach AP Calc again in the future. 😉 😉 😉