I am a secondary math teacher and I do exactly this with my students. When working on retrieving the expression of an affine function, I tell them about Shamir'..., they choose a secret pin as the slope, generate two points, give them to two other students who have to pair together to find the pin again. The students are always very engaged.

I'm so glad it works. If you handwave at the generalisation to quadratics and higher polynomials, do the students follow it well? I assume you don't get into the finite field stuff, but it seems like it'd be cool to handwave at "there's actually other structures where most of this polynomial stuff works, even in somewhat weird ways".

It's the same idea as how GPS works. Fun to see someone's eyes light up when they get why that third satellite is needed to fix a 2D position.

(Actual GPS is a little more complicated than that because the real system needs accurate time information.)