So, I’ve been trying to teach Algebra this year by modeling. I needed a good way to build the quadratic function model, so I started (about two weeks ago) by having the students analyze a basketball shot video. They used Logger Pro and found that vertical position vs. horizontal position made a nice curve. Like this:

But then I was lost as to how to get them to come up with the equation for the curve. They could get Logger Pro to tell them the equation, but that wouldn’t really help them. So, how can they come up with the equation when they’ve never seen one like this before?

I decided to take a different approach. We started by examining the function y=x^2 and then examining different ways to shift the function around (using Desmos). The handout I used had them graph different shifts (like shift y=x^2 up 3 units). Then I asked them to find the equation for that shift using Desmos. They had to figure out which piece of the equation to change.

Then after doing the activity (and a bunch of practice with it, making whiteboards, etc) they knew how to build a quadratic function from scratch. I think now it’s time to go back and find the equation of the original basketball shot we analyzed (BTW, I also teach these kids physics, so we’ve been studying *why* the ball behaves this way as well).

When I started writing this blog post, it was going to be about a failure to stick to teaching with modeling. As I typed it though, I realized that this was still a modeling unit. The kids recognize that the shape of the basketball graph is a symmetrical curve. We just didn’t have the appropriate skills yet to write the equation for the curve. Now we do, so it’s time to complete the model. I’ll let you know how it goes!

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I’m very interested in this series of lessons. I’m starting a similar series this week — actually, we started on Friday. I’m using a catapult made with a plastic spoon and we are going to graph the motion of a small ball. Can you remember the series of functions you gave them to graph so that they started to make sense of transforming the basic y = x^2 graph? Do you have some functions that you recommend or an order in which to present the ideas? Thanks!

It helped that we did absolute value functions right before this unit (I should’ve mentioned that in the post). So the kids already had a grasp on vertex form because we did a very similar activity with those.

I had them shift 3 units up (y=x^2 + 3) then shift 2 to the right (y=(x-2)^2) then make the parabola face down (y=-a(x^2)). After that, I asked them to plot y=-2(x-3)^2+4 by analyzing where the vertex would be located. It worked out pretty well. We’ll see how they do with applying this skill to the basketball shot. I think they may have trouble finding the “a” values still.