“What-If-Not” strategy

“What-If-Not” strategy from the book “The Art of Problem Posing” is a very good math method in many ways. First, it opens up the door of opportunities for people to investigate many different paths to find some more useful corollaries or attributes. Sometime, it can even lead to another important theorem. Second, by thoroughly investigating many possible variations, we can get a deeper insight as to what this theorem can do, where this theorem can be applied and how this theorem is used. These ideas will insure that the theorem can reach its full potential and be applied in as many areas as it can. However, the major problem for this method is that by creating so many different cases, most of them will be proved to be dead end and useless. The procedure can be very time consuming, although prudential choice of some related attributes might significantly reduce the number of different attributes created and thus reduce the investigation time. For our micro-teaching, we will be working on Pythagorean Theorem, one of the most well known and widely used theorems. After presenting a^2 + b^2=c^2, we will ask the students to link the formula to the graph. We will ask students to think what if the triangle is not a right triangle. The angle facing c side is smaller or bigger than the right angle. How does the formula change intuitively just by looking at the graph? By investigating these variations, students will have a better understanding that a^2 + b^2=c^2 is not only a necessary condition of the right triangle, but also a sufficient condition of the right triangle. By checking the 3 sides of a triangle fit the formula, we can be sure the angle c side faces is right angle. This seemly easy understanding has extensive use to check or to get a right angle from an existing line in all civil engineering areas.