Wow! Thanks for all that information! It is a little (okay, a lot) more than I can digest, but I bet my son knows exactly what you are talking about!

A couple of years late, but I just wanted to mention for anyone reading this that this project does indeed create a genuine parabola. It takes a bit of calculus to verify, (or at least that's the only way I could think of to prove it) so unless your kids are towards the end of high school it's probably not worth it to verify. This with them. If they have started calculus though it's actually an interesting project involving optimization to find the equation of the curve.

Without calculus you can at least do a pretty cool experiment to see the properties of a parabola. The 90 degree angle where your two original guide lines meet lies on the directrix of the parabola. So if you fold your paper in half on that corner so that the two original guidelines torch you'll get the axis of symmetry of the parabola on that fold. Then draw a line perpendicular to the axis stating at your original corner (I just used another sheet of paper to get a 90 degree angle and put it at the corner where the two guide lines meet with one edge on the axis). This line that is at a 90 degree angle with the axis is the directrix. Lastly, you can find the focus by going along the axis from the intersection of the guidelines to the curve then doubling that distance. If you look up some things about parabols you'll find that any point on the parabola is the same distance from the focus and the directx.

Finding the formula for your curve is much more involved and requires calculus I think. Briefly outlined, each of the lines you draw goes through a point "t" on the x axis and "1-t" on the y axis. High school kids taking calc should be able to derive the equation y = [(t - 1)/t]x +1-t for any given line. The curve is generated by the top line for any given vertical value, so you can have them find the maximum value over all "t" for a fixed value of "x". It helps to simplify the linear equation to "y = (1 - 1/t)x + 1 - t before taking the derivative. Solving should give them the critical point t = the square root of x. They can plug that back into the linear equation to get the formula for the curve as "y = x - 2sqrt(x) +1.

They may or may not know how to do a linear change of variables at this point (it's not always taught in algebra 2 or pre calc) but if they do know how, have them change variables to u and v with x = 1/2 (u+v) and y = 1/2 (u - v). This will result in the traditional quadratic for a parabola.

Thanks for that! I'll have to figure out the difference between a parabolic and a geometric curve. Good luck with your lesson!

I don't homeschool so this was primarily for fun. I like to show my son how math can produce wonderful art. But of course you can certainly discuss the characteristics of parabola, their relationship to conical structures, and also symmetry.

String art is very cool.