Instantaneous Rate of Change

Ten days from the end of the school year, it is not only the students who have lost some focus.  But a good lesson introducing derivatives to my precalculus students today got me pumped up like it was the first quarter again!

Although I didn't start the day with this idea, it's going to be a permanent addition to my instruction!  I try introducing the idea of instantaneous rate of change by talking about police radar detectors and how they must be measuring split-second changes in distance and time to calculate speed.  Inevitably, this leads not to thoughts of derivatives, but to police, speed traps, and how to argue one's way out of a ticket.


Apparently my explanation got through to some students.  Student comments of speed being zero/zero sparked some interest since we just talked about limits and the indeterminate form.  It was one student's comment about instantaneous speed being like a photograph that really led me to this new approach.  Lately I've been following the #anyqs project on Twitter, and this was a natural extension.  Here's how it works:



source

Present the above picture, and ask if anyone has any questions.  The photo is supposed to be enticing enough that most students will have the same natural question.  I plan to take a picture of a car driving past a speed limit sign to further encourage "How fast is the car going?"  (Today, a bad drawing of a photo of a car on the board even did the trick.)

Some questions I'll ask to lead the question:  How do we normally calculate speed?  So what's the distance and the time in this picture?  So the speed is 0 divided by 0?  What is that... undefined?  Zero?  (A small discussion today of shutter-speed ensued, but nothing compared to the side conversations about getting pulled over.)

Then they get another picture of the same car taken a small period of time later.  Again, any questions?  What do you want to know?  They answered today that they wanted some landmarks.  One kid cut to the chase saying he just wanted the distance.  Or a distance reference (like Mythbusters) in the background.  I suggested that it'd be especially convenient if the cars were at mile-markers on the highway (not to scale!).  So we agreed the distance would be useful.  One boy wanted to know the time interval.  I took that opportunity to introduce the "delta t" notation, which they seemed very comfortable with from science class.

The whole thing was such a nice, intriguing discussion that a senior even commented that it was really a great example.  Of course, I gave credit to my student in the previous period who gave me the idea, and they were equally impressed with him.  For a 7th period class in late May with senioritis spreading rampantly to the juniors, it was an amazing discussion with students showing clear, innate understanding.  It brought me joy that they were excited to see elements of limits in an actual real-life concept.  Great period to end the day with!  On top of that, they even liked my Geometer's Sketchpad secant-turned-tangent line example I made.

So, a thank-you to Dan Meyer and the Twitter community for keeping my imagination fresh.  I realize this probably isn't a monumentally new approach to this lesson, but it was the first time it occurred to me, and it was pure gold!!