[wbhart]

The proof starts with two of the "string-art" lines, a line L1 (line Q'R' in the video) with parameter s and a line L2 (line QR in the video) with parameter t, and computes the intersection point P of these two lines. Keeping line L2 fixed, one gets a series of intersection points P as one varies the parameter s. One then notices that the point of intersection of line L2 and the parabola (the "touching point") is the limit of these intersection points P as s tends toward the value t. The proof finishes by showing that this limit point is just what it was hypothesised to be, namely the point along line L2 that divides it in the same proportions as its endpoints divide the control lines.

I agree that some students will not recognise the crux of the proof, but will simply see something ends up being equal to something else in some identity without understanding how that relates to what was being proved.

Euclid's Elements (an ancient Greek textbook on Geometry) is a good model in this regard. He first states what he is to prove. He then starts from the assumptions and finishes with what he was to prove. Each step in between is justified.

[britcruise]

These are great points. You'll notice that video is actually a "bonus" step which isn't required to complete the final exercise in that lesson. We are still experimenting with how to go about "bonus" steps. In other lessons you'll see they are done in a multi-step article style. Such as in Animation: https://www.khanacademy.org/partner-content/pixar/animate/pa...

We also do this for the bonus step in Character Modeling (my favorite lesson!)

Our plan is to continue collecting more feedback on these different styles and find out what works best.