Problem Statement My task was to find 3 equations, that would give me an answer, if I had certain information. The first was to find one that if you knew that there were four pegs on the boundary, and none on the interior, you could get the area. The second was if you knew that there were 4 pegs on the boundary, and you knew how many were on the interior, you could get the area. And last, if you had the number on the interior, and the number on the boundary, you could get the area.
Process The first two equations, were a preparation for the final, building up towards the complete idea. This helped, because I could complete the first two pretty quickly. For Freddie I drew a 3 column T-Table, with a drawing of the figure, the number of Pegs (in), and the Area (out). I looked for a pattern between the in and the out, and quickly found one that made sense, and I worked it into a formula. I got X/2-1 = Y. Where X is IN (number of pegs) and Y is OUT (Area).
This works in all shapes with no interior pegs, like Freddie described. I attached this T-Table. For Sally I followed my luck of the 3 column T-Table, and drew another with the same guidelines. The figure, the interior pegs (in), and the area (out). After I filled in a few figures, and their properties, I noticed a pattern, and not long after, a formula, which worked for them. It was X+1 = Y.
This T-Table is also attached. Now... the next was not so easy. Frashy's required a long thought process, and several hours thinking it over, logically.
I thought that this next equation would be a combination of the two, it would have to incorporate what I had found out from both of the above. Especially the first. So I thought to myself what this equation, or formula, would have to include. And realized there wasn't 1 variable, but 2.
Because it has the variable from the first, and the second problem. 1: The number of pegs on the border, and 2: The number of pegs on the interior. 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. My answer is: Boundary Pegs / 2 - 1 + Interior Pegs = The Area of the Shape. I also attached my data from my T-Tables and drawings.
These were the building blocks of my answer. Evaluation This problem was a flashback to Algebra. Finding patterns, and then transferring what you see in the pattern into a formula is not always easy. Especially when you don't see a pattern. I learned that looking at a problem from different perspectives will sometimes help you, because I was looking at this problem in terms of the shapes for a very long time, but once I started looking at it in terms of numbers, my work was much more productive. It was educationally worthwhile for that purpose.
So I did enjoy working on this problem, I think it was a good mathematical exercise and I think see patterns and making equations out of them is very handy, and I seem to need to do it often, in every math class I take. I did think this problem was a little too hard. Even though I got a formula for each of them. The last one was very hard, and I think it should have been worked on in class so that we could get help when we were stuck. Because when I was stuck, it seemed like I had no where to go, and all I was doing was looking at a T-table for hours. I think this POW could be improved by maybe changing the first two problems.
Because I didn't really use the second formula to give me the last. I used ideas from it, but it wasn't a great help. If the second formula could be changed to maybe provide a bigger hint towards the final answer, I think that would be a good improvement for the problem. Self Assessment I think because the answer was so hard, that this POW should be graded on effort. And I put out a pretty hefty effort on this problem. It was my main concern for hours at a time, and a lot of hard work and frustration was put into it.
I believe I deserve an A, because I got an answer for each of them, that works, I included how I got them, and I made a good effort.