First Day, Part II


I should be taking down our tree since it’s Epiphany but instead I am writing…

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”.

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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.

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Partial screen shot of the circuit.

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).

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Partial screen shot of the second circuit.

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!