We reviewed polygon nomenclature today. We built with the geometric sticks on a white board for ease of labeling and drawing diagonals. I showed them the diagonals of a pentagon. Me Too asked to draw the diagonals of the heptagon and was eager to try it out on more shapes. Kal-El said he wasn't interested in doing any more shapes and suggested Me Too continue alone while he did something else. Me Too continued on and did the octagon, nonagon and decagon.
Meanwhile, Kal-El reappeared with some graph paper. Our work in Life of Fred recently covered placing points on a graph. Unprompted, Kal-El drew a graph and plotted the points we already knew (number of sides, number of diagonals): (5, 5) and (6, 8). Then he connected the points (with a geometric stick at first, but it got bulky so he eventually used it as a straight edge) and used his graph to try to predict what Me Too would discover on the white board.
He was disappointed that it didn't work. Me Too was getting different results. I explained that this kind of graph works well when a progression is linear but that the relationship between number of sides and diagonals isn't linear. I told him I think that the word for the relationship would be exponential but I'm not sure. He suggested that there might be a formula. So, I looked it up.
Here she is: n(n-3)/2
We worked the formula together for the heptagon we had already completed to make sure it worked. Then Kal-El used it to predict Me Too's results for his remaining polygons.
I suggested he try to think of a polygon relationship that would be linear and he decided on the number of sides and the number of diagonals leaving each vertex. We drew this graph together.