The Twittersphere recently made me aware of Desmos, a really fancy online graphing calculator. Fortunately, I was not introduced to it as such: I found it through a blog by a math teacher that now and then features a “maths gems” post — little tidbits of interest to mathy people and math teachers. The gems post that turned me on to Desmos also told me how minutes and seconds got their names!
Back to Desmos: this HTML5 web app is much more than a calculator. Their activities created by and for teachers include some winners. I checked out the “Polygraph: Parabolas” activity and it addresses vocabulary and understanding of quadratic functions in what looks to be an effective way. How do students learn to talk about parabolas using words like root, vertex, intercept, etc? By practice! And yet even with a group worksheet (and I love groupwork with worksheets!) structuring that conversation can be difficult. Here with a computer game we can get students to use their math vocabularies, without having to print out a lot of matching graphs, make copiers on the copy machine on another floor, cut them up using that paper cutter that gets stuck halfway, and then trying to collect them all after class so you can do the activity with the next group. A nice use of technology to lower the activation energy for a cool activity.
Besides activities for students, Desmos allows some nifty interactive demonstrations. This demo shows that if the complex roots of a quadratic are a+bi and a-bi, then the real roots of the reflection of its parabola are at x=a-b and x=a+b. This is easy to prove, of course. Look at the quadratic formula and see that the change of signs brought about by the reflection simply flips the sign of the discriminant. I knew this. But I had never seen it. Visualization can make a fact viscerally true.
Last advertisement for Desmos: a return to good old graphing calculator art. Did you, oh reader around my age, ever sit around drawing complicated pictures on your TI? Did you ever either make or give a beautiful polar coordinates Valentine’s Day card? If your student can use equations to draw Keroppi or a Disney-looking princess, I must say I think they’ve learned to play with math.