Sunday, August 19, 2012

What's a k-gram?

Have you ever had a lesson just rolling around in your head?  Just lurking, waiting, for the one day that you finally get a chance to teach it?  For nearly two years now, ever since I started my teaching program, I've had a half-formed idea for a project/lesson/unit on stars.  Not the ones in the sky, but the ones you can draw on paper.  You're probably familiar with five and six-pointed stars, but stars can also be drawn with any (natural) number of points.  Now 2, 3, and 4 pointed stars aren't very exciting [try to draw them!], but when you get up to seven points a neat thing happens - you can actually form multiple types of stars!  (Try it if you don't believe me - draw two sets of seven points, then draw a star skipping one point each time.  Draw another star skipping two points each time.)  In general, shapes like this are called n-grams (or pentagrams, heptagrams, dodecagrams, etc.)  There's lots of neat math that you can do with these shapes: angle sums, prime numbers, modular inverses.  There's a lot of material that can make a good lesson here too - it's numerical, visual, and tactile.  You can draw a star, or make a star with sticks and string, or even make a human star. 

Of course for now these are all just ideas, that I've not had the chance to put into practice.  But they represent something that I can aspire to in my teaching - to make my lessons investigative, artistic, interactive, and mathematically rich.  Maybe one day I'll even have the joy of teaching real actual students about n-grams!  When naming the blog, I couldn't resist substituting k for n, since it's my initial and all.  I also like the association with a tele-gram - a form of communication - which is what I hope for this blog to be: a way to send ideas and feedback back and forth between others who think math and teaching are awesome and who want to deliver the best instruction possible.

3 comments:

  1. Interesting connection! I like that your desire is to emphasize the "artistic" part of mathematics. Definitely, technique we all would like to integrate more. When you do that lesson, post it here with pictures. I'd like to see it!

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  2. Reading this I couldn't stop thinking of Vi Hart's star video (http://www.youtube.com/watch?v=CfJzrmS9UfY). And now I'm excited to do some lesson on stars too!

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  3. Hi commenters! Welcome :)

    @ Mr. Carby - I'll definitely post some more info if I ever do a real lesson on this stuff. One day...

    @ Andrew - <3 for Vi Hart, and that video. And since we're linking, I'd be remiss if I didn't link the Mathematics Teacher article that started me on n-grams: http://www.nctm.org/publications/article.aspx?id=26026

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