After pouring out your cube into three tetrahedra, and before losing focus on your cube, quick switch to another cube model and demonstrate our 24-MITE dissection (perhaps with CubeIt! or maybe with something homemade) i.e. nail it right there, before proceeding to your octahedron or other basic shape.Sounds like a (lesson) plan.
And don't forget the Coupler while you're at it? Lots of ways to go, per hypertoons.
Anyway, let's keep those suggestions flowing. I'll be incorporating some of them, but hey, don't wait for me to hog all the fun.