Math write up: COUNTING THE PEGS
Problem Statement: Freddie, Sally, and Frashy are friends. They are playing a game with some geoboards. Freddie is the one that goes first. He says that he has a formula to find the area of a polygon without any pegs in the interior. The In of his table is the number of pegs in the boundary and the Out on the table is the area. Then Sally said that she had a formula for any geoboard with only four pegs on the boundary. Now Frashy's is the most challenging because she claims that she can give you the area of any polygon using the boundary number and interior number of pegs used. The key is to find a formula that will work to solve for polygons with certain numbers of pegs in the interior.
Process: First of all, I drew some different examples of polygons on the geoboard paper for each amount of pegs in the interior. I made sure to follow Freddie, Sally, and Frashy's directions. After I sketched out the shapes, I made an In & Out table for Freddie and Sally. X is the number of pegs. In and Out is the Y, which is the area. I counted the number of pegs in each example and organized them as much as I could. I did the same for the area. After that for Freddie and Sally I had to find a formula for them each. Frashy's table looked a bit different. I then added the interior as well as the boundary and looked for a formula that fits the area best.
Solution: This is the formula for Freddie is : x/2-1
This is the formula for Sally is: y=x+1
This is the formula for Frashy is: x/2-1+y
Example:
This In and Out chart represents a polygon with no interior pegs.
A=I+B/2 -1
=0+3/2-1
=1/2
A=0+4/2 -1
=2-1
=1
a=1+3/2-1
=1+1/2- 1=1.5
a=5+8/2-1=4+5-1=8
The formula works with all types of polygons with different boundaries and interior pegs in order to solve for their area.