Cats and Calculus

I’ve done this activity once before, and I wrote about it here, but I’m continuing to tweak it. In the late 1800s, Eadward Muybridge published several time-lapse photo essays of animals and people in motion. Because he used a background with a grid on it, it is easy to see how much distance the subject covers in between each frame (which are snapped at 0.031 sec intervals). This sets up a great lesson on calculating average velocity, and approximating instantaneous velocity!

Muybridge Cat

While I still asked my students to plot the cat’s position vs. time by hand, we also used the desmos online grapher to plot the data as verification of their work. It was great to see the light bulbs go off in my students’ heads as they worked out difference quotients for smaller and smaller time intervals.

Here’s the plot generated using desmos:

Muybridge Data

 

We had an excellent discussion of how the cat’s motion breaks down into two distinct parts, and how the slope of each corresponds to the velocities of the cat walking and running.

Once again, I must thank  Dr. Nell Rayburn, Professor and Chair of Mathematics at Austin Peay State University for sharing this activity with other calculus teachers in Tennessee. Here are her original documents: Cat PhotosMuybridge Cat WorksheetMuybridge Cat Key.

 

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Feline Calculus for Everyone

Eadward Muybridge (1830 – 1904) was a British photographer who did several studies of animals and humans in motion. He set up a bank of cameras with fast shutters that would take pictures of the subject while it moved in front of them. The result is a series of stop-motion frames that allowed him to analyze the gaits of horses, cats, dogs, and other animals.

To introduce the concept of instantaneous velocity, my students look at Muybridge’s frames of a cat breaking into a run:

The background is divided into 5 cm blocks, so they can count the distance the cat covers over the frames. Each frame represents 0.031 elapsed seconds, so they know the total time the cat is moving. By looking at two or three consecutive frames, they can even estimate the instantaneous velocity of the cat at that time by calculating a difference quotient!

What I like about this is the fact that even Algebra I students could work through it and still get a nice understanding of the distinction between average and instantaneous velocity.

I learned about this activity from a presentation by Dr. Nell Rayburn, Professor and Chair of Mathematics at Austin Peay State University. Here are her original documents: Cat PhotosMuybridge Cat WorksheetMuybridge Cat Key.