(POW #3)
The purpose of this problem is to create formulas that work for different scenarios. We were given three different scenarios in which we had to create three different formulas that work every time. Freddie Short has a formula that can find the area of any polygon that has no pegs on the inside on a geoboard. Sally Shortest has a formula that can find the area of any polygon quickly with exactly four pegs on the exterior. Frashy Shortest has a superformula that can find the area of a polygon no matter how many pegs are on the boundary or in the interior.
To start off, I had to find the formula for Freddie. I used the geoboard paper to come up with different shapes that had no pegs on the inside. I gathered the information I found and put it into an In and Out table.
To start off, I had to find the formula for Freddie. I used the geoboard paper to come up with different shapes that had no pegs on the inside. I gathered the information I found and put it into an In and Out table.
Pegs
2 4 6 8 |
Area
0 1 2 3 |
With this information, I saw that the pegs always went up by two while the area went up by one. Therefore, I was able to create the equation: (p - 2) / 2 = a p = pegs & a = area. In question 1b, we were told to find an equation that works for any polygon that has one peg in the interior.
Pegs
4 5 6 8 9 |
Area
2 2.5 3 4 4.5 |
The information in this In and Out table tells me that the area is exactly half of the number of pegs. So my formula is: p/2 = a. I also conducted this test on questions 1c and 1d and I found that when the number of interior pegs increases you must add one to the equation. For example, 2 pegs: p/2 + 1 = a, 3 pegs: p/2 + 2 = a and so on.
The next scenario was to find Sally’s formula. The formula would have to find the area of any polygon with exactly four pegs on the boundary. I drew out various shapes on the geoboard paper and then created an In and Out table with the information that I have collected.
The next scenario was to find Sally’s formula. The formula would have to find the area of any polygon with exactly four pegs on the boundary. I drew out various shapes on the geoboard paper and then created an In and Out table with the information that I have collected.
Interior Pegs
0 1 2 3 |
Area
1 2 3 4 |
In the table, the interior pegs go up by one while the area goes up by one. So my formula came out to be: i + 1 = a, i = interior pegs. I did the same for questions 2b and 2c with different numbers of exterior pegs. For example, 6 pegs: i + 2 = a, 8 pegs: i + 3 = a.
The last scenario was to create a superformula that works for all polygons. The equation must include the number of pegs on interior and the number of pegs on the boundary. With some help, I was able to come up with a formula: i + (p/2) - 1 = a. I also created an In and Out table that tests the formula.
The last scenario was to create a superformula that works for all polygons. The equation must include the number of pegs on interior and the number of pegs on the boundary. With some help, I was able to come up with a formula: i + (p/2) - 1 = a. I also created an In and Out table that tests the formula.
Interior
0 0 1 2 2 3 |
Pegs
4 6 4 4 6 6 |
Area
1 2 2 3 4 5 |
I used the information from the previous tables to test if the superformula works or not. It proves to be an effective formula and can find the area for any polygon.
Throughout this problem, I used two Habits of a Mathematician: Stay Organized and Look for Patterns. I wasn’t very organized with the first problem, but once I started seeing patterns, I became more organized with my work to clearly show what I was thinking. I also used the habit of looking for patterns by using the In and Out tables to my advantage. The tables helped me see the patterns easier and then in turn find the formulas faster. I do think I could’ve used some more Habits of a Mathematician for this problem, but I certainly will use more in future problems.
Throughout this problem, I used two Habits of a Mathematician: Stay Organized and Look for Patterns. I wasn’t very organized with the first problem, but once I started seeing patterns, I became more organized with my work to clearly show what I was thinking. I also used the habit of looking for patterns by using the In and Out tables to my advantage. The tables helped me see the patterns easier and then in turn find the formulas faster. I do think I could’ve used some more Habits of a Mathematician for this problem, but I certainly will use more in future problems.