So this means that there are 2 IN's. And operations on the two variables will give me my out. So then I went to the T-Table, the perfect tool for seeing patterns. This time I had 4 columns and extra for my extra variable. I had the figure in one column, then the pegs on the interior, pegs on the boundary, and finally, the area. I started drawing shapes, and then filling in their properties, as I did earlier. Until I had about 5 or 6. I then started to look at the numbers, hoping to find a pattern of some sort. I didn't. I then put the table entries I had, into order, by the area. This helped a lot, so I could see how the different shapes related. I noticed patterns in groups, I saw a pattern if there were 0 pegs on the interior, or 1 peg on the interior, but not one that flowed through-out them all. It was very frustrating.
Then I tried to work with the equations I already had, from the other two. I started with the first, X/2-1=Y and I thought, if I were to add the other variable (which I called I, standing for interior) where would it go in? So I tried placing it in the equation, because I saw that it would fit in, and make sense. So I had X/2-1+I=Y . This worked beautifully. I tried plugging-in different values to checked it from my table, and they worked. So I have the answer.
Solution.
So I have found each persons formula. Freddie, Sally, and Frashy. I found each of them using mental thought process by looking at the values I got on an in/out table. For Freddie's, I had to find a formula that gave me the area if I knew the boundaries, if the interior was empty. And I got: The Number of Boundary Pegs / 2 -1 = The Area of the Shape. For Sally's I had to find a formula that gave me the area if I knew how many pegs were on the interior, if the boundary was four. I got: Interior Pegs + 1 = The Area of the Shape. Then, for Frashy's I had to find a formula that gave me the area if I knew how many pegs were on the interior, and how many were on the boundary.
