Yesterday, January 5th, was the first day of our second semester. The last time I had seen my AP Calculus students was on December 14th. That meant they hadn’t done any calculus (let alone any rigorous thinking, last-minute college application essays aside) in 22 days. [I don’t believe in assignments over break, but that is the subject of a different blog post.] So, what to do for that first day back, especially when it’s a FRIDAY? In the first semester we actively reviewed precalculus and then learned limits, more limits, rules of differentiation, and applications of the derivative. I always like a clean break between the semesters: first semester the derivative; second semester the integral. But I didn’t want to start with slicing and dicing area like I usually do. I wanted my AP Calculus students to practice what they had learned in the first semester but with a twist.

First, I gave them this for a “Warm Up”. All students were able to get it mostly correct without ever hearing the word “antiderivative” or “integral”.

The students were also able to articulate which ones they didn’t remember or couldn’t do without looking back in their notes (mainly the 5th and 6th terms). Then, I said, “Well, that was almost what my original function was, but not quite. It’s missing a term.” The students then came up with the idea that a constant was missing. I mentioned that mathematicians write “+C” at the end of their antiderivative (first time I use that vocabulary word) to indicate that there is a constant too. They queried, “How do we find it?” “You’ll need more given information,” I replied. Then I did go off on a little bit of a tangent about how it should be +K instead of +C because we really need to get rid of the letter Cc altogether in English (except for konsonant blends). I mean really, wouldn’t it be easier to teach reading if we spelled “a locus of points equidistant from a fixed point” a sirkle? But, I digress.

I distributed a quick (10-20 minute) circuit What Could the Original Function Have Been? They immediately dug in. After a minute I heard, “Ms. Cornelius, I ain’t gonna lie to you. My brain is dusty.” I ignored the comment.

As students finished the circuit amid ample discussion / arguing on their parts and a modicum of help on my part, I began to distribute a more involved circuit on Antidifferentiation. In this circuit students had to actually calculate the constant term given a point on the original function. Again, they dug in and commented, “I actually like this one better!” And, “My brain is smokin’!” There was a lot of discussion and I had to help some students get unstuck (as usual calculus was not the miscreant but trig and algebra).

Most of my students got all the way through the first circuit and about 1/3 to 3/4 of the way through second one by the time the bell signaled the end of our 50 minutes. I felt like it was a really good use of the first day of the second semester and I am giving them a quiz on Monday to ensure they finish their circuits and go back over their must-memorize derivatives.

Read more about teaching with mathematical circuits by clicking here . I am honored to be presenting on this idea in Washington, D.C. at the national NCTM conference in April. Click here for registration information.

Find 40+ calculus circuits (lots of them are free!) by clicking here .

And finally, because you teach not only calculus, here is the link to my store:

Virge Cornelius’ Mathematical Circuit Training .

Happy Training!