In this POW, called Just Count the Pegs, we were looking at the areas of polygons on a grid (we used geoboards, and geoboard paper). We were given three ideas for rules, in an in-out table format. One gave you the area of any polygon with no pegs on the interior, you just had to input the pegs on the boundary. The next needed exactly four pegs on the boundary, and you inputted the amount of pegs on the inside of the shape. The final one could give you the area of any shape on a geoboard, and you inputted both the amount of pegs on the inside, and on the boundary. Using various smaller problems, specialized to the two first formulas. After that, we had to find the “superformula”, or the third formula.
First, we had to find the first formula, as well as some variations. I did this by creating a table, with x being the number of pegs on the boundary, and A being the area of the shape. I found all of these things without using any formula, then after I was satisfied with the amount of shapes I had, I looked for patterns in my data. Here is my data:
x (number of pegs on the boundary) A (area)
8 3
6 2
3 1/2
4 1
A pattern that I found with this data is that the area was half of my x value, minus 1. I made this into my formula. A=1/2x-1. After this, we worked on variations of the original formula, like working with shapes that had 1 peg inside. With that one, I noticed that the A was half of the x. With that information in mind, I made my formula: A=1/2x. After that, we had to come up with our own numbers of pegs. I chose two, going along with the 0, 1, 2, pattern. After finding shapes and making an in-out table for that, I noticed that the A was half of x, plus one. Once I had made that into a formula, A=1/2x+1, I saw a final pattern. All of the formulas included 1/2 of x, and one minus however many interior pegs there were. I made this into a formula, A=1/2x+(y-1), with y being the interior pegs. After a bit, I realized that this might be the “superformula”. I tried plugging in a few shapes that I already knew the area of, and they all worked. I figured that that was it, but still tried to figure out the next formula.
For the next rule, we had to find a formula for the area of a polygon with exactly 4 pegs on the interior. When finding shapes and building the in-out table, I did the same thing as I did for the first formula. Then, I looked for patterns, and noticed that the number of pegs inside the shapes plus one was the area. Here is that data:
y (number of pegs on the inside) A (area)
1 2
0 1
The formula that I drew from that was A=y+1. I also used the “superformula” to make the formula 1/2(4)+(y-1). Next, we had to pick our own number of pegs on the border. I chose 8. I found the pattern relatively easily, the area was the y plus three, and got the formula A=y+3, as well as 1/2(8)+(y-1). After that, having already found a superformula (1/2x+(y-1)), I was finished with the main three problems.
Some Habits of a Mathematician that I used during this problem are looking for patterns, and solving a simpler problem. I looked for patterns when I was trying to find almost all of the formulas. I solved a simpler problem by looking for smaller formulas first, to help me try to find the “superformula”. Both of these things were very useful in the problem, as was staying organized.
Here is a scanned in copy of my work on the problem:
First, we had to find the first formula, as well as some variations. I did this by creating a table, with x being the number of pegs on the boundary, and A being the area of the shape. I found all of these things without using any formula, then after I was satisfied with the amount of shapes I had, I looked for patterns in my data. Here is my data:
x (number of pegs on the boundary) A (area)
8 3
6 2
3 1/2
4 1
A pattern that I found with this data is that the area was half of my x value, minus 1. I made this into my formula. A=1/2x-1. After this, we worked on variations of the original formula, like working with shapes that had 1 peg inside. With that one, I noticed that the A was half of the x. With that information in mind, I made my formula: A=1/2x. After that, we had to come up with our own numbers of pegs. I chose two, going along with the 0, 1, 2, pattern. After finding shapes and making an in-out table for that, I noticed that the A was half of x, plus one. Once I had made that into a formula, A=1/2x+1, I saw a final pattern. All of the formulas included 1/2 of x, and one minus however many interior pegs there were. I made this into a formula, A=1/2x+(y-1), with y being the interior pegs. After a bit, I realized that this might be the “superformula”. I tried plugging in a few shapes that I already knew the area of, and they all worked. I figured that that was it, but still tried to figure out the next formula.
For the next rule, we had to find a formula for the area of a polygon with exactly 4 pegs on the interior. When finding shapes and building the in-out table, I did the same thing as I did for the first formula. Then, I looked for patterns, and noticed that the number of pegs inside the shapes plus one was the area. Here is that data:
y (number of pegs on the inside) A (area)
1 2
0 1
The formula that I drew from that was A=y+1. I also used the “superformula” to make the formula 1/2(4)+(y-1). Next, we had to pick our own number of pegs on the border. I chose 8. I found the pattern relatively easily, the area was the y plus three, and got the formula A=y+3, as well as 1/2(8)+(y-1). After that, having already found a superformula (1/2x+(y-1)), I was finished with the main three problems.
Some Habits of a Mathematician that I used during this problem are looking for patterns, and solving a simpler problem. I looked for patterns when I was trying to find almost all of the formulas. I solved a simpler problem by looking for smaller formulas first, to help me try to find the “superformula”. Both of these things were very useful in the problem, as was staying organized.
Here is a scanned in copy of my work on the problem: