This demonstration simulates the construction of a parabola by paper folding. Try the demo, then try it with actual paper. Try it with different focal lengths.

Proof that this method constructs a parabola
Folding point X to the focus guarantees that the two triangles shown are congruent because they are made to coincide. Therefore P is equidistant from the focus and the directrix. That is the criterion for P to be on the parabola. The set of all possible locations for P is therefore a parabola.
This construction can be extended to prove the reflection property of a parabola. A ray of light from the focus to P reflects off the fold at equal angles to go vertical. You can show that the fold is tangent to the parabola at P by showing that any other point on the fold is closer to the directrix than the focus. Showing this is left as a challenge for the student.