I had my honors kids do a series of constructions in class with compass and a straight edge. To help them overcome the temptation to "cheat", I gave them popsicle sticks as straight edges. Most kids figured out right away how to do an isosceles triangle (all on their own), and about half of the class figured out on their own how to make a kite. (I figured it was a small hop from being able to do isosceles triangles.) Some kids fumbled their way to an equilateral triangle while trying to get the isosceles one. Then, everyone struggled with constructing a pair of parallel lines, so after they struggled for a while, I let them open up to a part of the textbook that describes the "rhombus method" for constructing parallel lines. Except, the way that the book describes does not allow them to construct 5 equally spaced parallel lines as I had requested. So, either the kids had to fiddle and figure out on their own a modified "rhombus method" (which a handful of kids did manage to do), or they had to get a hint from me.
All in all, the kids liked the activity so much that I decided to turn it into a project. So, the next day I gave them a list of specs, and they brought me clean final drafts with explanations for justifying why the sides are indeed congruent.
I tried to scan in the best piece of student work, but the scanner doesn't pick up on the arc marks all too well. For a kite, he started with two intersecting circles of different radii, and connected the circles' centers to the points of intersection in their arcs. He constructed equilateral triangle using two intersecting circles of the same radius, and his parallel lines are formed from a network of congruent circles.
Neat, eh? Lots of math with no numbers.
We'll be seeing construction again, very soon...