This past weekend, on Saturday afternoon, I got to be part of a working group at Desmos headquarters in downtown San Francisco. The drive up there wasn’t too long – it stayed pretty dry, although I was sure the downpours would start any minute the entire trip. I wasn’t sure exactly what to expect, just that the plan was to work with other math education professionals, and have a Desmos activity for my Algebra I class by the end, ready to pilot.
There were 8 of us there, and after a while discussing some possible ideas, with a focus on quadratic equations and graphs, we split into two groups. The other group put together a really nice activity, exploring the symmetry of quadratics, and they included a paper and pencil worksheet to accompany the online portion of the activity. I’m pretty excited to use this activity when we get into the graphing of parabolas, as I’m sure it will make it easier for students to anticipate and simplify their graphs.
My group took on something that I was really excited about, mostly because I couldn’t quite see how it was going to work. We were going to use Desmos to explore the concept of completing the square graphically. I’ve covered the concept with algebra tiles, which students seem to get, and I’ve covered it algebraically, building a perfect square trinomial, but had no idea how to approach it using parabolas. Working with three other bright, creative, and experienced math teachers, we went to town, coming up with some interesting ideas and approaches. There were connections that we were making between vertically shifting a parabola to “sit” on the x-axis and adding a number to the equation to complete the square, and hopefully noticing the pattern between the b and c values.
It seemed so clever, so creative, until we started wondering – what are students going to get out of this? Will this help them make connections and understand the process of completing the square? Will it make things more complicated for them? In short, were we delving into this because it’s a good educational idea or because it’s interesting to us as people who enjoy mathematics?
For my upper level math courses, I’ve been more willing to experiment with them, to show them things that I wasn’t sure about, to see where things would go, just because the math itself was fun, and it was fun for the sake of itself. I’ve been less inclined to do that with my Algebra 1 students, by far the most differentiated of my classes in terms of previous knowledge, comfort with math and with their own abilities. Is this the same idea? Maybe showing something that is interesting, not because it reinforces a standard or because it introduces a new standard, but simply because it’s what I, as a mathematician, found interesting and curious. Isn’t there value in that, in and of itself?
Mind you, our working group WILL be getting a complete activity/lesson out of this, and it will be taught in a few weeks in my Algebra 1 class. Will it, in the end, be all about completing the square? Where will it fit? We will see.