Pennies, Circles, and Quadratics

We have been learning about mathematical models, and I came across this very nice activity put together by Dan Meyer, and tweaked by Andrew Shauver.

After showing this brief video Dan made:

we talked about what questions it raises, which eventually led to the main one, “How many pennies will it take to fill the big circle?” Then we discussed what information we needed to answer that, namely see how many pennies it takes to fill some smaller circles. I handed out sheets with 1 inch, 2 inch, 3 inch, 4 inch, 5 inch, and 6 inch diameter circles, lots of pennies, and let them go at it:

We plotted the data using the online grapher Desmos and decided that a quadratic function would model it best:

Using the QuadReg feature on our graphing calculators, we found the quadratic model for the data, and used it to predict how many pennies would fill the big circle. Here’s Dan’s answer:

We had varying results, which led to a good discussion of how small changes in data can lead to big differences in results, but several students were within 10 pennies of the answer.

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Conics Hide and Seek

We just wrapped up our study of conic sections, which can be a pretty dry topic. So to liven things up, I had my precalculus students go on a scavenger hunt. These days, everyone has either a digital camera or phone with a camera, so everybody could participate. Here is the handout I gave them outlining the rules:

Conics Hide and Seek

Names____________________________________

Math is all around us, and in this activity, you are going to find some places where it is hiding. We have finished our study of conic sections (parabolas, circles, ellipses, and hyperbolas), so it’s time to have a scavenger hunt! Using your phone or digital camera, you and a partner will explore the campus and take a picture of at least one example of each type of conic. The team with the most points will get a prize. You are on your honor not to share your finds with other teams.

Here’s how many points each type of conic is worth:

Circle: 1 pt (maximum of 5 examples)

Ellipse: 2 pts

Parabola: 3 pts (remember, a parabola is not the same as a “U”!)

Hyperbola: 5 pts

You can submit your photos via email. Happy hunting!

They spent more than half of an 80-minute block combing the campus for examples of conics. Did this activity involve rigorous mathematics? No, but it was a lot of fun for the girls, and it opened their eyes to some of the ways math can describe the world around them. When we reviewed the teams’ submissions, there was a lot of discussion about whether certain shapes actually were parabolic, or ellipsoid, etc. All in all, a very useful activity.

Here is a sampler of the best submissions. The winning team took over 100 photos!

Light Hyperbola

Light Hyperbolas

Eyes & Nose Hyperbola

Arch Parabolas

Window Ellipse

Basketball Hyperbola

Clock Circle

Doorbell Ellipse

Light Fixture Parabola

Printer Parabola

Pull Tab Ellipse

Outlet Ellipse

Planter Circle

Ford Ellipse

Escher Hyperbola

Water Fountain Parabola

Courtyard Circle

HH Circle

Smiley Parabola

Sign Ellipse

Railing Hyperbolas

Vectors and Dancing – A Dangerous Mix

I came across this activity from Jim Noble at www.teachmaths-inthinking.co.uk the other day and tried it out with my precalculus class. We defined four different dance steps with vectors, and combined them into a routine.

We then tried out the routine while listening to Donna Summers’ “Hot Stuff”. As you can see, we could have spent a little more time practicing!

I liked this activity, because it got girls up and moving. We had a good discussion of the various ways to combine the steps to reach the same destination, and they mapped them out using the vector definitions. Jim Noble has made available some excellent worksheets that go along with this project.

If you’re trying to teach simple vector addition and scalar multiplication give this activity a try!

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.