Natalie Smith

EMAT4680/Sheehy

Functions of Proof

Verification:

Given the problem of finding a point equidistant from the three vertices of a right triangle, Mr. White’s class observes that the point seems to be the midpoint of the hypotenuse. A proof is needed to verify the class’s observation that the distances of the three new segments extending from the point are equal.

Explanation:

From the above problem, a proof might be needed to give further explanation of why they got their results. The explanation proof in this situation might provide insight to the Pythagorean theorem or other commonly used theorems of geometry.

Discovery:

From the triangle proof that Mr. White’s class verified before, it might be seen that there is a further relationship between the three newly formed segments and the equidistant point. Proof by discovery can lead the class to the conclusion that the point is actually the center of circle with given radius the same as the new segments. Proof can also show that the circle circumscribes the triangle and that this concept can be applied to other types of triangles as well.

Systemization:

The proofs from Mr. White’s class create the footwork for a broader understanding of some key mathematical concepts. The proofs show the connections between triangles, circles, and other key theorems. The proofs particularly reveal connections about how trigonometric functions might be derived from the right triangles and the circles.

Communication:

When Mr. White’s class heard that Mr. Green’s class was trying to solve a similar problem to their own about the triangle but with a rectangle they decided to share their proofs. Because proofs are generally constructed as universally understood, Mr. Green’s class was able to use this information to spring board their own conclusions. Having the foundation helped Mr. Green’s class to look even further into the relationship between circles and polygons.

Intellectual Challenge:

Now that both classes feel confident in their knowledge of the relationship, they are ready to apply their proof to new shapes---not just regular polygons. They can try L-shapes and combinations of arcs and triangles like a baseball diamond. The foundation they created before with proofs should stimulate new thoughts about the shapes and how they relate to one another.