I introduced quadratics today. We’re not going too crazy anytime soon — in particular, no factoring for awhile. But I want to introduce quadratics to give students another example of what a function is and let them figure out some of its properties — shape, vertex, symmetry, ways to graph it, etc. Goal for day one was just to have students graph with a table, make some inferences based on symmetry, and get an idea for the shape and structure of quadratics.
Clearly some significant confusion about squares, negatives, and some sloppy arithmetic in general. We’ve got lots of work to do. This should’ve told me in big neon letters that I need to take a step back from my big ambitious plan for day one of quadratics. But instead, I decided to add a bit of scaffolding and go ahead.
I was pretty proud of this. I got plenty of good noticings from question one. Awesome. Number two, I got lots of answers with the right shape and without a lot of precision. In one class, one student picked out the points and reflected them. I compared this with another graph that was a bit lazier, and had students discuss with each other which one was better. I think it lead to some great understanding of the symmetry of a quadratic.
In my other classes, without great student work to go off of, I added a dotted line of symmetry to the graphs above, and then we used the symmetry to refine students’ graphs for number two. One way I can make this better for next time is to have an anonymous-looking teacher version that uses the points, and compare it with a piece of student work to spur the same conversation we had in one class whether or not anyone figures out the symmetry.
Anyway, we talked about symmetry, and I made a joke about “It’s symmetric! Boogie woogie woogie woogie woogie.” Then we moved into graphing. I had students graphing with a table, just plotting points and sketching the graph. But, as I should’ve been prepared for, many students’ arithmetic was not up to the challenge. Lots of little mistakes, including one student who I think at the end of class still didn’t believe me that negative four squared is positive four. I slowed down, did some more as a class, had them try simpler examples. We still had trouble. I think the most frustrating part was that plenty of kids had the arithmetic down and were graphing right every time, no problem. And lots of kids who were making mistakes knew pretty well at that point what a quadratic was supposed to look like, so when theirs turned out asymmetric or just not U-shaped they knew it was wrong, and were frustrated picking through their arithmetic to find a mistake. Too many kids with too many different challenges, and not enough confidence to go around. Still, the fact that kids knew their answers didn’t make sense was powerful — that kids have an idea what their answer should look like, and they’re gut-checking it against their calculations.
Plenty of good stuff came out of this lesson, but I finished feeling pretty defeated. The tone in the room felt negative, because the class as a whole knew they weren’t successful, and I couldn’t help everyone who needed it. More than that, I think the difficulty increased too quickly — we went from making pretty simple observations about curvy graphs to complicated calcuations with negatives and squares and negative squares.
Tomorrow, I’m going to be a lot more deliberate in my scaffolding, and in particular mix looking at pictures with graphing to try and help stem the frustration and provide some more opportunities throughout the lesson for kids to feel successful, instead of feeling good at the start and hitting a brick wall for the last 20 minutes. Here’s hoping.