P.o.W. 3: Just Count The Pegs
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In this problem, Freddie Short has a formula to find the area of any polygon on a geoboard that has no pegs within the interior, using an In-Out table. The In is the number of pegs on the boundary and the Out is the area of the polygon. Sally Shorter also has a formula. She can find of any polygon on a geoboard with four pegs on the boundary. All she needs to know is the number of pegs in interior. Like Freddie, she also use an In-Out table, where the In is the amount of the pegs within the interior and the Out is the area. Frashy Shortest has a formula that can find the area of any polygon on a geoboard. All she needs to know is the number of the pegs on the boundary and the number of pegs within the interior. My goal was to find the find Frashy's formula by finding and manipulating Freddy and Sally's formulas.
I first started out this problem by finding Freddie's formula. I did this by constructing four polygons with no pegs on the inside. I then constructed an In-Out table, as seen to the left, to find Freddie's formula. I noticed a pattern and after a while, I managed to find a formula, y = x/2 - 1. In this formula, the y substitutes the In and x substitutes the Out. I then manipulated t he restrictions by making polygons with one peg within the interior. The formula I found for the areas of these polygons is y = x/2. I then did the same process with polygons with 2, 3, 4, and 5 pegs inside of it. The formulas I got for the polygons are y = x/2 +1, y = x/2 + 2, y = x/2 + 3, and y = x/2 +4, respectively. I then decided to work on figuring out Sally's formula. I first did this by constructing four polygons with four pegs on the boundary. I then constructed an In-Out table to find the formula for their area. This In-Out table led to the formula y = x + 1. I then explored the problem further by creating polygons with different amounts of pegs within the interior. I created polygons with five, six, and eight pegs within the interior. By using In-out tables, I found out that the formulas for these areas of these polygons were y = x +1.5, y = x + 2, and y = x + 3. I used these formulas to finally reach Frashy's master formula. |
Freddie: 0 Pegs
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Freddie: 1 peg
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Sally: 4 pegs
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Sally: 5 pegs
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Sally: 6 pegs
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Sally: 8 pegs
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I noticed while working on the Sally's formula that the number added to the x was always one less than the half of the number of pegs on the boundary. So, if x = the number of pegs on the boundary, y = the number of pegs within the interior, and z = the area of the polygon, then the master formula is (x/2 - 1) + y = z. I used these formulas to finally reach Frashy's master formula. I then decided to test this formula with polygon with ten pegs on the boundary and three pegs in the interior. The area of the polygon equaled 8 units and (10/2 - 1) + 3 = 8, so my formula seemed to work. I then tested it out with a complicated shape with 16 pegs on the boundary and 2 in the inside. The area of the figure was 9 and the formula went as following: (16/2 - 1) + 2 = 9, proving my formula once more. I then tried out a complex polygon my teacher, Mr. Corner, provided. My formula found that polygons area as well. After all this testing, I feel confident in my formula.
Working on this problem helped me become better at finding patterns. While I was working on the different formulas, it became easier and easier for me to draw recurring similarities between the formulas. Using these patterns, it became easier for me to see how the amount of pegs were affecting the constants within the formula. I feel that I deserve a 10/10 for my work on these problem. I put a lot of effort into these problems and managed to actually find a solution to the problem. I also put some time into finding all the formulas. A habit of a mathematician I used was being systematic. Once I had found one formula for a series of polygons, I decided to slightly change the polygon and explore a new formula. The changes and permanence within the formulas helped me find specific patterns, which led to me finding the master formula. So, if I hadn’t been systematic, I would have never found Frashy’s formula. This is just one example of the importance of this habit. |