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

Everything You Wanted To Know About Circles

This latest screencast is for my Geometry students. We are beginning a unit on circles, so this one defines all the terms we will be using. I use Geometer’s Sketchpad to demonstrate the various parts of circles.

I’ve found that in using the reverse classroom method with my younger geometry classes, it is a little harder to hold them responsible for watching and learning from the screencasts. So, first thing next class, they are going to have to complete a really short worksheet that covers the lesson material.

Conic Sections Are Everywhere

My Honors Precalculus students just wrapped up a unit on conics sections: circles, parabolas, ellipses, and hyperbolas. As a final project, I had them find examples of each type, and take pictures of them. They came up with some very creative responses:

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