Two identical squares can be made to overlap in many ways. Here are two such methods.
Varying the overlap slider varies the degree of overlap. In the right hand panel the
area in common for each method is plotted as a function of the degree of overlap.
Which function corresponds to which method? How do you know?
Why is one function linear and the other not?
What is the non-linear function? How do you know?
What conjectures do you have about the nature of the common area function for
convex polygons other than squares?
What conjectures do you have about the common area function if the two squares
are not related by a simple translation along a line of symmetry?
What questions could/would you ask your students based on this applet?