Crossroads in Mathematics

Example 6

A Puzzling Problem (Geometry and Deductive Proof)

While this document calls for the use of problems that are real-world applications, all problems do not have to be of that type. Problems that are easy to state, easy for students to understand, and that have an unusual twist can pique student interest. The following problem has a nonintuitive answer. Note that the proof that is requested is needed to validate conjectures that are made. In this sense it is an integral part of the solution process, and thus, enhances the mathematics involved.

Problem. The figure below shows two congruent overlapping 10 cm by 10 cm squares. The titled square is movable, but one vertex always remains at the center of the other square that does not move.

a. What is the largest possible area of the overlapping shaded region?

b. Validate your answer to part a by proving that your answer is correct.

Extension. In parts a and b the students should conjecture and then prove that the area of the shaded region remains a constant 25 cm. Does the perimeter of the shaded region also remain constant? If so, find the constant perimeter. If not, determine the minimum and maximum perimeters.

Using Manipulatives. Colored, transparent 10 cm by 10 cm plastic squares are commercially available. If students can experiment with, say, a red and a blue square, they are likely to gain insight into the problem.

Using Technology. The squares in this problem can be "constructed" using a computer and commercially available interactive dynamic geometry software. Once constructed, the overlapping polygonal region can be specified and its area and perimeter can be measured. The software will give a quasi-continuous readout of these two measurements as the top square is rotated about the center of the bottom square. This will enable the students to support previously made conjectures which would then be verified using traditional methods. The software can also be used to draw auxiliary line segments to aid in the construction of a proof.