Full disclosure: What I’m about to share is not typical of 9th grade students in Algebra 1. My school offers one 9th grade math class (mine), so the class is heterogenous, especially with regard to prior math experience. Some aced Algebra 1 in 7th grade, yet here they are in a class called Algebra 1. Needless to say, this subset of students was eager for me to spice things up for them and I was happy to oblige.
So, Projectile Motion. As mentioned in an earlier post, students pushed me to give them the REAL math – not a simplification – of how to model the flight of a projectile with respect to variables like distance, height, time, initial velocity, gravity, starting height and angle of launch. I created this document to help explain this and gave my blessing to any kids crazy enough to go down this road.
The students that took this on made enough sense of my worksheet to create this Desmos graph that allowed them to control initial velocity, angle of launch, starting height and gravity:
When looking for the EXACT POSITION at any given time, they simply added a slider for time (t) and pressed play:
Parametric equations in Algebra 1. You’d think they’d be satisfied, right? Wrong. These kids wanted the floating dot AND the graph at the same time. GREEDY! They also pointed out that the curve looked like a parabola and so they wanted to know its regular (i.e. not parametric) equation.
Fine. Figure it out on your own (keeping in mind that I had a handful of kids in each class still struggling with adding negative numbers).
Well they did, with relative ease. Which they could confirm without me, because the graph of the parametric matched the graph of their new equation:
So you think it would end here, right? A height function in terms of distance that incorporates velocity and launch angle – where could it possibly go from here? Well, if you’re as nit-picky as my students, you’re probably not happy with the domain and range of these graphs, both of which should be greater than or equal to 0. You’d think there would be an easy way to fix this on Desmos. And you’d be right. But four months ago (April 2014), you’d be wrong. Today, in August 2014, you can easily restrict domain and range:
Four months ago, Desmos could not do both. It could restrict the domain OR the range. I thought this was a flaw. It turned out to be a better math problem than I ever could have thought up.
Before I tell you how they did it, put yourself to the test: how would you make the graph above, where the domain and range both exclude negative numbers, if you could only restrict the domain OR range. Yes, it must work as any of the parameters (gravity, launch angle, velocity and starting height) are changed. Good luck! I’ll check back soon.