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.