Here’s a really interesting topic for middle school math teachers: origami constructions. Whereas straight-edge and compass (SE&C) constructions are equivalent to solving quadratic equations, origami can solve any SE&C problem and solve cubic equations. As “K’s Origami” puts it:

Source: http://origami.ousaan.com/library/conste.html
Now, let's solve the cubic equation x3+ax2+bx+c=0 with origami. Let two points P1 and P2 have the coordinates (a,1) and (c,b), respectively. Also let two lines L1 and L2 have the equations y+1=0 and x+c=0, respectively. Fold a line placing P1 onto L1 and placing P2 onto L2, and the slope of the crease is the solution of x3+ax2+bx+c=0.

Very nifty! The site linked above has the best explanation of the theory behind origami constructions that I’ve found. I’m still looking up resources and trying to absorb this. The best examples for the classroom I’ve found so far are at Tom Hull’s website. It has examples of trisecting an angle, and doubling a cube (ie finding the cube root of 2).