In this problem, Freddie Short, Sally Shorter, and Frashy Shortest are coming up with formulas of polygon area. All of the polygons given to these 3 people are created on geoboards. Geoboards are basically pegs where you can create polygons and shapes. Freddie can find you the area of any polygon with no pegs in the interior. He can do this by creating an In/Out table, in being the number of pegs on the outside, and out being the area he calculates for the figure. Sally can find you the area of any polygon with exactly 4 pegs on the outside. No more, no less. All she needs to know is how many pegs are in the interior of the polygon, and she can tell you the area. Frashy can tell you the area of any polygon with any number of pegs on the interior and exterior. You just need to tell her the interior peg count and exterior peg count, and she can give you the area immediately. Our goal in doing this problem was to find Frashy's formula which claims to be the best, but starting by finding the two other formulas by Freddie and Sally.
process
To come up with different formulas for Freddie, Sally, and Frashy, I started by drawing polygons depending on their requests. Freddie wanted no pegs in the interior, Sally wanted 4 pegs on the outside, and Frashy wanted any number of pegs on the interior and exterior.
An example of a polygon for Freddie
An example of a polygon for Sally
An example of a polygon for Frashy
I then put the values the different people requested into In/Out tables and analyzed the results. I then came up with a formula for Freddie and Sally.
solution
My two beginning formulas were y = x/2 -1 for Freddie, and y = x + 1 for Sally. I got these formulas by comparing the In/Out tables and coming up with different equations. Frashy's formula actually came easier to me, since it could be formed by combining Sally's and Freddie's formulas. I started by noticing that there was a negative 1 and a positive 1, which could be used to cancel out each other. This didn't work because I ended up with y=x/2 and y=x; which I couldn't combine. I then changed Y to A to represent the area. After experimenting more I ended up with A = (Y/2) - 1 + X; which solved the equation.
reflection
My experience doing this problem has swayed a lot. At first it seemed very complicated and I had no way of finishing it by the deadline, but throughout the problem it became easier, and then hard again. I think one of the most difficult parts of this problem was finding how to combine Sally's and Freddie's formulas to get Frashy's superformula. A lot of the content on this problem was discussed in class, and that made it easier for me to understand by hearing many people's perspectives instead of just one or two people. Some habits of a mathematician I used during this problem were solve a simpler problem, seek why and prove, and stay organized. I used all of these habits together to lessen stress on myself, and make the problem and write up a simpler process, buy chunking up the problem and completing each part one by one. In class we used a post it note with a checklist, and I think using those a lot helps me stay more organized and make my mind less cluttered with information and due dates.