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!

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:

 

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 Photos, Muybridge Cat Worksheet, Muybridge Cat Key.

 

Volumes, Calculus, and Cucumbers

My calculus students have always had a hard time visualizing three-dimensional objects. So, a couple of years ago, I started using cucumbers to help them understand the theory behind using integration to calculate the volume of rotated solids. Here’s a more detailed explanation of the activity.

Actually cutting the cucumbers into disks, measuring each disk’s volume, and adding them up to approximate the total volume is a great hands-on exercise that reinforces the concept of using an integral to add up lots and lots (an infinite number!) of disks to calculate the volume of an irregular solid. Plus, they always enjoy having a snack after all that brain work!

Building a Better Box – An Optimization Math Lab

In our calculus class today, we did an activity that takes advantage of optimization techniques. We spent the three previous classes learning how to use the derivative to find maximum and minimum values of functions.

I have 12 students in this section, so I split them into 6 pairs. I handed out one piece of paper to each group. There were 3 different sizes of paper. The task I assigned them was to construct a box of maximum volume by cutting off squares from the corners and folding up the tabs to form the sides of the box:

I also provided scissors, tape, and a ruler to each team.

Since each pair of teams had  the same size of paper, they were able to compare results, and check each others’ work.

The pieces of paper we worked with were 8.5″ x 11″, 8.5″ x 14″, and 11″ x 14″. Each pair of teams came up with very close matches for dimensions of boxes that contained maximum volume.

The 8.5″ x 11″ results were 1.6″ x 7.8″ x 5.4″ for a volume of 67 cubic inches. The teams with the 8.5″ x 14″ sheets of paper came up with 10.7″ x 5″ x 1.7″ for a volume of 91 cubic inches. Those working with the 11″ x 14″ sheets built 12.6″ x 6.6″ x 2.1″ boxes for a volume of 183 cubic inches.

Here are their masterpieces:

Here are the steps I provided on the student handout:

Calculus Activity: Optimization          

Name____________________________

Name____________________________

 

In this activity, you will figure out the dimensions of an open box that will maximize its volume, given a specified amount of paper to work with.

1. Measure the initial dimensions of your sheet of paper.

Length = ____________________________ inches

Width = ____________________________ inches

Area = ______________________________ in²

To construct your box, you will cut a square x inches wide off of each corner, fold up the remaining tabs, and tape them together to make the box’s sides.

2. Figure out a volume function in terms of x based on your given sheet of paper.

Volume function V(x) = ____________________________________

3. Use calculus to determine the value of x that maximizes the volume.

x = ___________________________ inches

Max volume = ________________________ in³

Cut off the corners, and build your box!

4. Measure the dimensions of your box, and see if the volume matches the one you calculated in Step 3.

Length = _________________________ inches

Width = __________________________ inches

Height = _________________________ inches

Volume = ________________________ in³

5. One other group was given a piece of paper the same size as yours. Compare their results to yours.

Their volume = ________________________in³

% difference = ______________________

An interesting followup to this activity would be to investigate the relationship between the area of the given sheet of paper and the volume of the constructed box, and then try to generalize that result!

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 Photos, Muybridge Cat Worksheet, Muybridge Cat Key.