We always dabble a bit in quadratics at the start of Pre Cal. It's a nice pre-cursor the scary new world of trig to go over some things they've learned before. Plus there's elements like factoring and function behavior that help when we get to trig equations and stuff. I've done it by the book for a while, take some equations, determing the min or max, finding the vertex, all that fun stuff. But it lacked some relevance. The theme of the year is adding some meat to my lessons. Building in points of inquiry at the start, finding ways to use all my shiny iPads, and getting more relevant information across to the students. I have the content down at this point, I've finally hit the point where I'm comfortable enough with all the topics to "explore the solution space" as my Heat Transfer professor liked to say. Here's how I introduced quadratics this year in Pre Cal:
1. Borrow from Dan Meyer's Excellent "Will the ball hit the hoop?" 3ACTS:
It had all the desired effects. Kids going nuts waiting for me to show them the answer. We did some modeling in abstraction here. Making assumptions about release height, hoop height, ball height, etc. I used this to introduce the terms "minimum," "maximum," and "vertex." They were familiar with these terms already.
2. After a few more of the videos. I set them to work. Materials: iPad, tennis ball, $1 glowstick necklace. Their goal was to make a similar video on a smaller scale. One person plays hoop, one does the throwing, another mans the camera. We spent about 20 minutes playing around, trying to get 2-3 good takes from each group.
3. We then spent another 15 minutes or so having them scan through their results and finding the best one. Their instructions were to make some approximations about the scenario and produce a result. My exact words were "make a sketch in your notebook" but some of them remembered a little playing we did in SketchBook Express and I got super amazing things like this:
4. Recall this process later, because I will be giving them pictures of quadratic scenarios and they'll have to develop a full blown vertex form equation. More on that later.
They were enthralled with the video making process. I had several people trying trick shots. Then one clever kid was bouncing it off the wall to himself, since parabolas are symmetrical and all. The best part of the process was the CSI work they were doing on the videos. I sent them out there with no measurement tools, so it was fun to see the kids counting floor tiles. Then the fun moments of marking a tennis ball at 5ft in the air above the head of a person who was clearly taller than 5ft.
5. Complete the abstraction and talk about the specific calculation of a min/max through -b/2a and all that.
A++++++ lesson, would teach again