I'm noodling a lot about insight right now (another post coming out of that, hopefully), so when we've been playing with transformations of lines and parabolas the past week I had a thought that maybe you could think of solving equations or solving systems as a series of transformations.
So I made the two desmos graphs embedded below. They are not student ready. I'm not even sure they would help a student in any way. The second one has too many steps for it to really make much insight possible -- they'd just get lost in all the steps (as my office-mates have helpfully pointed out).
But they're really cool. Is there something there? Something that could lead to a genuine insight about what it means to solve an equation? Or a system?
What is that insight? What is the set of dense connections that a student could make, from a picture like this? Is it to prior knowledge and comfort with physical movement or transformation? Or is it to a hypothetical future experience with matrix transformations?
I'm just wondering what I might be able to do with this idea, what the objective would be, and then how to build it out in Desmos or some other tool.
The original Desmos graphs are here:
- Solve a one-variable equation as translations and rotations
- Solve a system of equations as translations and rotations