Problem Statement
What I have to do
What I have to do for this write up is to learn to discover a equation for different types of polygons by creating tables for them. The question that we are challenged to find the solution for is to find Frashy’s amazing equation that can be used to find the area of any polygon when given the number of pegs on the inside and the number of pegs on the boundary.
Question
Freddie Short has a shortcut. He has a formula to find the area of any polygon on the geoboard that has no pegs in the interior. His formula is like a rule for an in-and-out table in which the IN is the number of pegs on the boundary and the OUT is the area of the figure.
Sally Shorter says she has a shortcut for any geoboard polygon with exactly four pegs on the boundary. All you have to tell her is how many pegs it has in the interior, and she can use her formula to find the area immediately.
Frashy Shortest says she has the best formula yet. If you make any polygon on the geoboard and tell her both the number of pegs on the interior and the number of pegs on the boundary, her formula will give you area in a flash!
Your goal in this math write up is to find Frashy’s “superformula” but you might begin with her friends’ more specialized formulas. Here are some suggestions about how to proceed.
Process and Solution
The first step I had taken was to create random polygons onto the geoboard. I then found the areas of those polygons and inserted them onto a in-and-out table then searched for a equation for the table. I had done this just so I can become more comfortable with the process.
This is the geoboard that I used for mostly all of my polygons that I had created for this problem.
What I have to do
What I have to do for this write up is to learn to discover a equation for different types of polygons by creating tables for them. The question that we are challenged to find the solution for is to find Frashy’s amazing equation that can be used to find the area of any polygon when given the number of pegs on the inside and the number of pegs on the boundary.
Question
Freddie Short has a shortcut. He has a formula to find the area of any polygon on the geoboard that has no pegs in the interior. His formula is like a rule for an in-and-out table in which the IN is the number of pegs on the boundary and the OUT is the area of the figure.
Sally Shorter says she has a shortcut for any geoboard polygon with exactly four pegs on the boundary. All you have to tell her is how many pegs it has in the interior, and she can use her formula to find the area immediately.
Frashy Shortest says she has the best formula yet. If you make any polygon on the geoboard and tell her both the number of pegs on the interior and the number of pegs on the boundary, her formula will give you area in a flash!
Your goal in this math write up is to find Frashy’s “superformula” but you might begin with her friends’ more specialized formulas. Here are some suggestions about how to proceed.
Process and Solution
The first step I had taken was to create random polygons onto the geoboard. I then found the areas of those polygons and inserted them onto a in-and-out table then searched for a equation for the table. I had done this just so I can become more comfortable with the process.
This is the geoboard that I used for mostly all of my polygons that I had created for this problem.
My next step then was to find the equation that Freddie had said he had created to find the area for any polygon that has no pegs on the interior. The first thing I had done was to create polygons that had no pegs on the inside. I created about 4 or 5 of them then I moved on to find the areas of the polygons so I can create a in-and-out table. After I had found the areas of the polygons I then constructed a table to help me discover the equation that helps find area for any polygon that has no pegs on the inside.
My table for this step:
My table for this step:
I then looked at the table and attempted to figure out what the equation was. I used the guess and check method to find it. I chose a possible equation and plugged in the (x) variable into them to see if they worked. Some equations didn't work for any plug-ins and some worked for a couple plugins but not all so because of that those equations are not true. I then discovered the equation (x-2)½=A (in this instance A stands for area). This equation then proved to be true for all the plugins and therefore is a true equation that works for finding area of a polygon that has no pegs on the interior.
The next procedure I had taken was to find the shortcut equation that Sally stated she has to find the area of any polygon that possesses exactly 4 pegs on the boundary. She stated that the only thing that she needed to be given was the number of pegs on the interior of the polygon. I used the same process that I had used to find Freddie’s equation with the one with Sally’s. I used a different geoboard to create my polygons for this portion of the problem.
The next procedure I had taken was to find the shortcut equation that Sally stated she has to find the area of any polygon that possesses exactly 4 pegs on the boundary. She stated that the only thing that she needed to be given was the number of pegs on the interior of the polygon. I used the same process that I had used to find Freddie’s equation with the one with Sally’s. I used a different geoboard to create my polygons for this portion of the problem.
I then created my graph for Sally’s equation based off the polygons in the image above.
I then continued the same process that I had used for Freddie using the same method of guess and check by plugging in for the x variable. I eventually came up with the equation
A= x+1. This equation worked for all the plugins that I had, therefore this equation is true too the problem.
Even though finding the equations that Freddie and Sally made technically was not part of the problem requirements. It still came as a big help. This is because those two problems came as a practice or warm-up to the actually problem. Also both of those equations came as a section of the new equation for the bigger, more complex equation that Frashy states that she has.
Frashy states that she can find the area of any polygon when given the numbers of pegs that it contains on the interior and around the boundary. I used the same process that I had used to find the equations for the Freddie and Sally sections. I first created a couple polygons that fits the description of what the equation is capable of handling and in this case, it would be any polygon. I then measure the number of pegs on the interior and on the boundary along with finding the area of the polygon. Then I had plugged them into a graph to make finding the equation simpler, cleaner, and neater. This table is slightly different from the table I had created for Freddie and Sally because it possesses three columns instead of two. One column is for the number of pegs on the interior, one for the number of pegs on the boundary, and the third is for the area of the polygon.
A= x+1. This equation worked for all the plugins that I had, therefore this equation is true too the problem.
Even though finding the equations that Freddie and Sally made technically was not part of the problem requirements. It still came as a big help. This is because those two problems came as a practice or warm-up to the actually problem. Also both of those equations came as a section of the new equation for the bigger, more complex equation that Frashy states that she has.
Frashy states that she can find the area of any polygon when given the numbers of pegs that it contains on the interior and around the boundary. I used the same process that I had used to find the equations for the Freddie and Sally sections. I first created a couple polygons that fits the description of what the equation is capable of handling and in this case, it would be any polygon. I then measure the number of pegs on the interior and on the boundary along with finding the area of the polygon. Then I had plugged them into a graph to make finding the equation simpler, cleaner, and neater. This table is slightly different from the table I had created for Freddie and Sally because it possesses three columns instead of two. One column is for the number of pegs on the interior, one for the number of pegs on the boundary, and the third is for the area of the polygon.
It was a long and headaching process to find the equation for the probably but with the help of a classmate of mine I discovered the formula that Frashy claims she has to find the area of any polygon when given the number of pegs on the interior and on the boundary. The formula is A= I(# of pegs on the interior)+(B[# of pegs on the boundary]/ 2) -1. What I had also discover was that this equation is actually a quite famous equation. Its known name is Pick’s Theorem.
Evaluation
This problem I believe was a fairly complex problem that taught a very important theorem to us in a clear and self-involved manner. The problem really had us working hard and a long time on solving it but it wasn’t as hard to the point of not being solvable. I would say that the habit of a mathematician that I used the most would clearly be looking for patterns. When I had created my tables for Freddie, Sally, and Frashy I spent a long time just staring at the numbers and searching for a equation that works, then I would use the method of guess and check to prove whether my chosen method was true or false. I also used the habit of starting with solving a simpler problem. I portrayed this habit by solving Freddie’s and Sally’s equations first as because they are simpler then Frashy’s. I also had visualized, I created different polygons on the geoboard and then created tables for the statistics of them.
Evaluation
This problem I believe was a fairly complex problem that taught a very important theorem to us in a clear and self-involved manner. The problem really had us working hard and a long time on solving it but it wasn’t as hard to the point of not being solvable. I would say that the habit of a mathematician that I used the most would clearly be looking for patterns. When I had created my tables for Freddie, Sally, and Frashy I spent a long time just staring at the numbers and searching for a equation that works, then I would use the method of guess and check to prove whether my chosen method was true or false. I also used the habit of starting with solving a simpler problem. I portrayed this habit by solving Freddie’s and Sally’s equations first as because they are simpler then Frashy’s. I also had visualized, I created different polygons on the geoboard and then created tables for the statistics of them.