Learning through Play: Angles of Polygons

One of my brothers and sister-in-law sent my oldest son this absolutely fantastic geometrigraph and polygraph set. We're just learning how to use it, but its already providing plenty of fun.

My oldest didn't want to do his normal math lessons the other day, so I suggested he use the tools and make a pattern of some sort and I would ask him some questions about it. He made a pentagon and connected the diagonals to create a five-pointed star.

Then he invented a cops and robbers type game where we moved along the edges to the vertices, the cop chasing the robber. If we both could move the same distance on a map where every vertice has four degrees then the robber was never caught, so we varied the rules. What if one triangle was off limits to the robber? Now the cop could be chased into an area from which he could not move as much. What if they moved at different speeds?

When we finished playing a number of games I assigned him the task of finding the angles of a number of triangles within the picture. Measuring a few and then calculating the rest by he quickly realized there was a "magic number" involved. Different angles could be one, two or three times the magic number. Did all stars made from completed graphs in polygons have a magic number?

We drew up a couple more polygons. He devised a strategy. For each shape we measure the angle of the ouside shape and divide it by the number of angles the diagonals divide it into. (So a square we measure a corner of the square, and divide it by two, since that corner has only one diagonal going through it. For a hexagon we measure the corner of the hexagon and then divide it by four since there are three diagonals connecting to that corner dividing it into four parts.)

We noted that the "magic number" keeps getting smaller the more sides the polygon had. We also noted that the "corner number" kept getting larger. He suggested that the magic number would keep getting smaller and smaller but never disappear completely and the corner number would keep getting larger and larger but never reach 180.

Since we had noted that all the angles in the different triangles created in our shapes were multiples of the magic-number, we realized that the number of multiples in any triangle always equaled to the number of sides in our outer shape. So the pentagon had a magic number of 36, and one triangle might have two angles with 36 and the other would have to have three times 36. My son was quick to realize this was another way of figuring out what the magic number is. It was 180 divided by the number of sides of the outer shape.

I'm sure to the more mathematically inclined what we did was simplistic foolishness but by now we were both happily exploring.