"There is no one way"

Sunday, June 29, 2014

World Cup Fever

The World Cup happens mostly during summer vacation, which is unfortunate, because it provides a great context for a geometry exploration.

I start my book Geometry Labs with lessons about angles, starting from the very basics, and moving on to the inscribed angle theorem. (The book is available in its entirety as a free download here. Some highlights of Section 1 of the book can be previewed here.) The reason I like teaching the inscribed angle theorem early in the course is that it is interesting to my stronger students, but at the same time it provides an environment with few prerequisites for students who don't have a strong sense of what an angle is. In other words, it is helpful to a wide range of kids. Along the way, there are some great lessons using the CircleTrig Geoboard, a device I designed for this very purpose, and also to introduce basic trig ideas.

I mention all this now because it turns out that one context in which one can preview (or apply) the inscribed angle theorem is soccer (or as they call it elsewhere: football, or futbol). In the book, I propose a hands-on approach, which requires scissors to cut out angles, and pins stuck in cardboard to represent the goalposts. ("Soccer Angles" is Lab 1.10 in Geometry Labs.)

A recent post by Scott Steketee on the Sine of the Times blog offers some Geometer Sketchpad applets to facilitate an initial exploration in a browser (on a computer or, presumably, a tablet.) I recommend you check out the applets yourself, and then use them with students. They are well thought out, and well-crafted. Making a version of them in GeoGebra is now on my to-do list. Here is a screen shot, showing in blue an area where the shooting angle is 15°. The basic mathematical question is finding the spot that gives the greatest shooting angle as one runs down the pitch.


After playing with Scott's sketches, you can wander to my soccer angles page, in which I present a full analysis of the problem, and some extensions, including an unexpected connection to one of the conic sections. That page includes a number of Flash animations of Cabri figures. I hope to replace them with interactive GeoGebra applets when time allows, but until then, I provide two GeoGebra files you can use to follow the argument interactively on your computer. (The links for those files are near the top of the page.)

One interesting feature of this problem is that it can be studied profitably in a geometry class (as an illustration of the inscribed angle theorem,) then again in algebra 2 or precalculus (focusing on that conics connection,) and finally as an optimization problem in calculus.

--Henri

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