Just Count the Pegs POW
Problem Statement:
For this problem of the week, the goal was to find 'super-formulas' for three people: Freddie Short, Sally Shorter, and Frashy Shortest. We found formulas to find the area according to different polygons we could make with the geoboard. For Freddie, we found a single formula that could work for any polygon that has no pegs on the interior. For Sally, we found a formula that would fit any polygon that has only four pegs on the boundary, but you just have to tell her the number of pegs it has in the interior. And finally for Frashy, we had to find one single formula that could work with any polygon we make on the geoboard, and the only thing she needs to know is the number of pegs in the interior and the number of pegs on the boundary.
For this problem of the week, the goal was to find 'super-formulas' for three people: Freddie Short, Sally Shorter, and Frashy Shortest. We found formulas to find the area according to different polygons we could make with the geoboard. For Freddie, we found a single formula that could work for any polygon that has no pegs on the interior. For Sally, we found a formula that would fit any polygon that has only four pegs on the boundary, but you just have to tell her the number of pegs it has in the interior. And finally for Frashy, we had to find one single formula that could work with any polygon we make on the geoboard, and the only thing she needs to know is the number of pegs in the interior and the number of pegs on the boundary.
Process Description:
To find the answer to this problem, I started out by making 'in and out tables,' which is basically an easy way to organize a set of data. I began with Freddie's formula. We had to find a formula that could fit with no pegs in the interior, so I made some polygons. It became a pattern because the shape just grew one unit each time (as you can see from my work to the right). I stopped at about five polygons because it came clear to me that there was a pattern. I found out that the pegs on the boundary would count up by two's, and the area for them would count up by one. The formula that I found that would find the area for any polygon with zero pegs in the interior is located below in the 'solutions' column. After we found the formula for Freddie, our next task was to find a formula with only one peg in the interior. My work for that is also to the right. We had two more assignments for Freddie, and they were to pick a number bigger than one (as we had already found a formula for the number one), and find a formula for that number in the interior. I picked the number four for no apparent reason. As I was coming up with the different shapes with four pegs in the interior, I stopped at four polygons as the pattern/formula because clear to me. Our last task for Freddie was to complete a few more cases like the last one (which was finding a number bigger than one for the interior pegs, and finding a formula for them). I did three more cases, with the numbers: 3, 5, and 6 pegs in the interior. I had made random shapes for each of the cases, and they worked for three and six, but I could not, for the life of me, find a specific formula for five pegs in the interior. I saw no connection with any of the numbers in my in and out table. All of my evidence/work for completing Freddie's problem is located to the right, and all of the solutions is below. As I went onto Sally's problem, I took all of the work that I did for Freddie's problem into consideration. I began by specifically finding polygons with only four pegs on the boundary, and then making that into and in and out table. I made about seven polygons, when I realized that I had come to a conclusion, I found the pattern/formula. I went onto the next task with Sally. We had to find another formula to find the area of a polygon, but this time, we could pick a number other than four, and have that as our boundary number. I chose five pegs. After I found a formula for that set of pegs, I went onto the third and final task for Sally's formula. Just like Freddie's, we did more cases like the previous one. I completed three cases. I did one with 6 pegs, 7 pegs, and 8 pegs. There was an obvious pattern to me. When I added another peg onto the boundary, I found that the formula just adds up by 0.5. You can see the completed formulas down below, under the solutions column. All of the work that I did on Freddie and Sally's problems, was all leading up to Frashy's problem. We didn't really have a certain set of tasks to get to her 'super-formula,' but we basically just had to make an in and out table (like we've been doing), and find some sort of pattern to fit any shape of polygon. I just made about ten random shapes, that were all different sizes, and I collected the interior and the boundary pegs (the data), and calculated the area. After analyzing my data over and over again, I found the super-formula! All of my solutions for each of the tables, are located below. |
All of the in and out tables are shown first, and then the polygon shapes for each of the tables, are shown all in one group after the work.
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Note: The shapes on the geoboard are kind of in one huge group, so I color-coordinated all of them, according to their table, and after each person's work (Freddie, Sally, Frashy), I made a black line to show where the new shapes start and stop.
Solutions:
I will separate each of the formulas that I found by putting them into the given numbers/letters (for example: 1a, 1b, 1c...). NOTE: I substituted each of the ins and outs with variables: x, y, (z).
Formulas to find the area for Freddie's problems:
1a: (zero pegs in the interior) x / 2 - 1 = y
1b: (1 peg in the interior) x / 2 = y
1c: (4 pegs in the interior) x / 2 + 3 = y
1d: (3 pegs in the interior) x / 2 + 2 = y
(5 pegs in the interior) ? (couldn't for the life of me find an equation/formula for five pegs in the interior).
(6 pegs in the interior) x / 2 + 5 = y
Freddie's overall formula to find the area of any polygon with zero pegs in the interior is: x / 2 - 1 = y, or 1a.
Formulas to find the area for Sally's problems:
2a: (4 pegs on the boundary) x + 1 = y
2b: (5 pegs on the boundary) x + 1.5 = y
2c: (6 pegs on the boundary) x + 2 = y
(7 pegs on the boundary) x + 2.5 = y
(8 pegs on the boundary) x + 3 = y
Sally's overall formula to find the area of any polygon with exactly four pegs on the boundary is: x + 1 = y, or 2a.
Formula to find the area for Frashy's problems:
The one and only formula that I found that worked for the majority of my examples to find the area of any polygon, just by knowing the number of interior pegs and the number of boundary pegs is: x + ( y / 2 ) - 1 = z
I was unable to find a formula that would work for all of the polygons, but the one that I did find, worked for 8 out of the 10 polygons. If you look at Frashy's paper above, you can see that the numbers with the asterisk next to it, is the numbers that the formula doesn't work with. I could manipulate the formula to work for those two shapes, by just changing the last part of the equation: the minus 1. I could make it into plus 1, for one of them, and for the other, minus 2.
I will separate each of the formulas that I found by putting them into the given numbers/letters (for example: 1a, 1b, 1c...). NOTE: I substituted each of the ins and outs with variables: x, y, (z).
Formulas to find the area for Freddie's problems:
1a: (zero pegs in the interior) x / 2 - 1 = y
1b: (1 peg in the interior) x / 2 = y
1c: (4 pegs in the interior) x / 2 + 3 = y
1d: (3 pegs in the interior) x / 2 + 2 = y
(5 pegs in the interior) ? (couldn't for the life of me find an equation/formula for five pegs in the interior).
(6 pegs in the interior) x / 2 + 5 = y
Freddie's overall formula to find the area of any polygon with zero pegs in the interior is: x / 2 - 1 = y, or 1a.
Formulas to find the area for Sally's problems:
2a: (4 pegs on the boundary) x + 1 = y
2b: (5 pegs on the boundary) x + 1.5 = y
2c: (6 pegs on the boundary) x + 2 = y
(7 pegs on the boundary) x + 2.5 = y
(8 pegs on the boundary) x + 3 = y
Sally's overall formula to find the area of any polygon with exactly four pegs on the boundary is: x + 1 = y, or 2a.
Formula to find the area for Frashy's problems:
The one and only formula that I found that worked for the majority of my examples to find the area of any polygon, just by knowing the number of interior pegs and the number of boundary pegs is: x + ( y / 2 ) - 1 = z
I was unable to find a formula that would work for all of the polygons, but the one that I did find, worked for 8 out of the 10 polygons. If you look at Frashy's paper above, you can see that the numbers with the asterisk next to it, is the numbers that the formula doesn't work with. I could manipulate the formula to work for those two shapes, by just changing the last part of the equation: the minus 1. I could make it into plus 1, for one of them, and for the other, minus 2.
Self-Assessment and Reflection:
When I was working through this problem, I found that some parts were completely easy for me, and some other parts were impossible to complete, it was like there was no in-between for me. One part that I really struggled on, was trying to find a formula to find the area for having 5 pegs in the interior, back in problem 1d. I tried really hard to find a solution, and when I couldn't find one, I asked my peers for help, but they couldn't find one either. I still put my work up there, to show people what I was working on at that point. Other than struggling to find an equation to fit that set of data, I learned a lot throughout this problem. The thing I learned most about was patterns. I found that there are patterns everywhere. They're not just in a math problem, they are in our everyday lives too. Like when we wake up and do our daily routine, that's a pattern! I really liked this problem, because it wasn't too hard where I ended up stressing out too much over it, but it wasn't too easy that I didn't learn anything new. I think I believe that I deserve a 10 out of 10 for this problem. I worked really hard to find all of the answers, and I even went a little above and beyond compared to some of my other classmates, when it came to creating more cases in problems 1d and 2c; most of my classmates did only 1 extra table, but I did 3 for both. One habit of a mathematician that I think I conquered throughout this problem, would be 'stay organized.' I think I was very organized when completing this problem, because not only did I keep all of my in and out tables neat, but I also was organized when I color-coordinated my geoboard graphs to make it easier on the people that are looking at them. Another habit of a mathematician that I used was 'look for patterns.' This one is a given, but I wanted to put it down to show people that this problem is basically just about looking for patterns, while using an in and out table to organize the data. I mean, the whole point of this problem was to come up with an equation that would find the area for any polygon, so every polygon must have something in common, which would be the pattern. I had a lot of fun working on this problem, even though I struggled with it at times, as well.
When I was working through this problem, I found that some parts were completely easy for me, and some other parts were impossible to complete, it was like there was no in-between for me. One part that I really struggled on, was trying to find a formula to find the area for having 5 pegs in the interior, back in problem 1d. I tried really hard to find a solution, and when I couldn't find one, I asked my peers for help, but they couldn't find one either. I still put my work up there, to show people what I was working on at that point. Other than struggling to find an equation to fit that set of data, I learned a lot throughout this problem. The thing I learned most about was patterns. I found that there are patterns everywhere. They're not just in a math problem, they are in our everyday lives too. Like when we wake up and do our daily routine, that's a pattern! I really liked this problem, because it wasn't too hard where I ended up stressing out too much over it, but it wasn't too easy that I didn't learn anything new. I think I believe that I deserve a 10 out of 10 for this problem. I worked really hard to find all of the answers, and I even went a little above and beyond compared to some of my other classmates, when it came to creating more cases in problems 1d and 2c; most of my classmates did only 1 extra table, but I did 3 for both. One habit of a mathematician that I think I conquered throughout this problem, would be 'stay organized.' I think I was very organized when completing this problem, because not only did I keep all of my in and out tables neat, but I also was organized when I color-coordinated my geoboard graphs to make it easier on the people that are looking at them. Another habit of a mathematician that I used was 'look for patterns.' This one is a given, but I wanted to put it down to show people that this problem is basically just about looking for patterns, while using an in and out table to organize the data. I mean, the whole point of this problem was to come up with an equation that would find the area for any polygon, so every polygon must have something in common, which would be the pattern. I had a lot of fun working on this problem, even though I struggled with it at times, as well.