Let me go back a step and talk about why I do this project on projectile motion. In my class, we did Graphing Stories and other tasks that helped students go through the modeling process, as outlined by the Common Core:

1) identify important variables in the situation
2) formulating a model that represents the relationship between the variables
3) analyze and perform operations to draw conclusions
4) interpret the results in terms of the original situation
5) validating the conclusions, then either improving the model or, if it is acceptable…
6) reporting on the conclusions and the reasoning behind them

“Graphing stories” was terrific early in the year because it didn’t overwhelm kids with each of these steps. I chose and set the scale for one of the two variables; they chose the other. Little if any calculation took place. That lightened the cognitive load for the better.

By April, students have been working with modeling two variables for a while now. One thing I love about the Projectile Motion Project is that Step 1 of the modeling process leads to THREE  variables that kids are curious about: distance, height and time. My requirement was that they represented all three variables in their project. “THREE VARIABLES!!?!?” Before kids started flipping tables upside-down, I shared one way this could be done:

After a little explanation, this was enough direction for most.

I got some pretty cool final products. I’ll share a few that I think are a representative sample of the class:
Sample 1
Sample 2
Sample 3

There’s so much to talk about in all 3 of those, but I want to highlight one aspect in particular. Look at the height-time graphs in Sample 2 and Sample 3 (about a soccer kick and a potato toss, respectively):

 

Both students use my favorite equation, h(t) = -16t² + vt + y₀  , and both decide: “Thanks, but no thanks! Your equation doesn’t give me something that matches reality, so I’ll make my own instead.” On the one hand, I’m thrilled that they are interpreting these graphs, attempting to validate them and after analyzing realizing that the model I gave them is inaccurate. On the other hand… remember the good old days when you could just tell kids that h(t) = -16t² + vt + y₀ would work for projectiles and kids would have to have to really, really care and do a ton of work if they wanted to disprove that? Thanks a lot, Desmos.

This is what I’m talking about when I say kids just weren’t satisfied with the tools I had initially provided them to represent the relationships between these three variables. Most students took the route of the samples above and just gave my equation the proverbial finger. As I said, there’s a lot I like about that, and I’m fully content – thrilled, actually – with the work-around that those students chose.

But some students pushed this project much further than I could ever imagine. Those results coming up next.