Starting with Differentiability

My AP Calculus BC students will be starting off the academic year with me already having a few months of calculus under their belt.  They should already have a good understanding of limits and how that applies to the definition of the derivative.  They also will know most of the derivative shortcuts.  And it looks like one of the last topics they will have studied before starting this class is the idea of differentiability.  This topic basically comes at the end of Chapter 4, and BC calculus is supposed to start with chapter 5, which dives into the definite integral.

So before we start with the definite integral though, I want to ease in with some introductory assignments to gently jog their memories of calculus, without the boring beginning of the year review.  Bah!  I also want to establish a classroom culture of discussion and animated participation.  A great activity for the first or second day that I have used in the past involves one student describing a piece-wise graph to her partner that can’t see the graph.  The partner tries to draw the graph from the description, and then later as a class we discuss what sort of vocabulary is helpful and what is not.  I show them them several graphs and they take turns being the describer or the scribe.  I try to pick graphs that have several types of discontinuities so that class discussion leans towards limits and continuity.

I’m always surprised how much students enjoy this activity.  They get really animated in describing their graphs.  Sometimes I have to employ a “sit on your hands” rule so they focus on their vocabulary.  I also have students switch partners  once or twice.  I will definitely be using this again as an introductory activity.

I also found this great video by Numberphile during which Tony DeRose describes some of the math behind Pixar animation.  He talks about parabolas in an animation “meeting in smooth ways” and there are several points where continuity, limits, and differentiability are discussed without him actually using that calculus vocabulary.  I think I will have the students watch the video for homework and then write a paragraph or two about the calculus they might find in the video.  I think it may lead into some good discussions about differentiability.  I’m not sure if I will leave the assignment that general, or if I will give them more specific questions.  I’ll have to think about that over summer break.  Check out the video though.  It’s very interesting.