just count the pegs
Problem Statement
In this problem we were given three different people to deal with Freddie, Sally, and Frashy. We are making polygons using a geoboard and all three people have shortcuts to finding the area. Fred's shortcut is only for the area of polygons with no pegs on interior, Sally's shortcut applies to exactly 4 pegs on the outside with x as the interior and Frashy well he's a smart one and he has a superformula that can just find you the area for any polygon. Our goal is to try and find Frashy's superformula by looking at the other two formulas.
Process
In order to try and find the superformula I followed the problem's advice, start small. I looked at Freddie and Sally's smaller more specific formulas and went from there. Starting with Freddie, to find out a way to find the formula I began to create my own squares following his requirements, first no pegs on the inside, then one peg, and then we could experiment with any number of pegs on the inside. I tried to create as many polygons as I could until I kinda got bored then I moved on. I used the same method to find Sally's formula. I followed the questions and then started creating my own polygons that fit her requirements. For both Sally and Fred another suggestion we were given was to set our data up like an In and Out table. This really actually helped to make everything more organized and neat so we could find similarities between data points to find a formula. After I finished with the smaller more specific formulas I moved on to finding Frashy's superformula. To be honest I kind of found the superformula on accident, my first thought was to look through my formulas and see any relations between most of them and as a matter of fact I did. I noticed that most of my formulas, actually just Freddie's all had a number that needed to be divided by two. I started there and then one of the formulas that I had written struck me. I looked at and it fit the requirements for a superformula I just had to test it with the other problems. I went to the data sets that didn't have similar formulas and tried there, it worked! I went to the remaining data sets and thankfully it worked.
Solution
In this solution I will just list all the formulas and key, for each variable means.
A= Area i= Interior b= Boundary
Freddie
A= b/2 -1
A= b/2
A= b/2 + (i-1)
Sally
A= i+1
A= i+1.5
A= i+2
Frashy's Superformula
A= b/2 + (i-1)
Reflection
The two Habits of a Mathematician I used were start small and stay organised. The suggestions really helped me start small and stay organised. To help me stay organised I created an In and Out table. I did this cause 1, it was a requirement and 2, because it actually helps you compare and find similarities between two data sets to find formulas. Starting small was also kind of started through the suggestions of the problem. I started small by looking at each the simpler formulas first also when finding the formulas I usually looked for the most obvious similarities and went from there. These obvious similarities helped me narrow down my options for formulas to help me stick with one.
In this problem we were given three different people to deal with Freddie, Sally, and Frashy. We are making polygons using a geoboard and all three people have shortcuts to finding the area. Fred's shortcut is only for the area of polygons with no pegs on interior, Sally's shortcut applies to exactly 4 pegs on the outside with x as the interior and Frashy well he's a smart one and he has a superformula that can just find you the area for any polygon. Our goal is to try and find Frashy's superformula by looking at the other two formulas.
Process
In order to try and find the superformula I followed the problem's advice, start small. I looked at Freddie and Sally's smaller more specific formulas and went from there. Starting with Freddie, to find out a way to find the formula I began to create my own squares following his requirements, first no pegs on the inside, then one peg, and then we could experiment with any number of pegs on the inside. I tried to create as many polygons as I could until I kinda got bored then I moved on. I used the same method to find Sally's formula. I followed the questions and then started creating my own polygons that fit her requirements. For both Sally and Fred another suggestion we were given was to set our data up like an In and Out table. This really actually helped to make everything more organized and neat so we could find similarities between data points to find a formula. After I finished with the smaller more specific formulas I moved on to finding Frashy's superformula. To be honest I kind of found the superformula on accident, my first thought was to look through my formulas and see any relations between most of them and as a matter of fact I did. I noticed that most of my formulas, actually just Freddie's all had a number that needed to be divided by two. I started there and then one of the formulas that I had written struck me. I looked at and it fit the requirements for a superformula I just had to test it with the other problems. I went to the data sets that didn't have similar formulas and tried there, it worked! I went to the remaining data sets and thankfully it worked.
Solution
In this solution I will just list all the formulas and key, for each variable means.
A= Area i= Interior b= Boundary
Freddie
A= b/2 -1
A= b/2
A= b/2 + (i-1)
Sally
A= i+1
A= i+1.5
A= i+2
Frashy's Superformula
A= b/2 + (i-1)
Reflection
The two Habits of a Mathematician I used were start small and stay organised. The suggestions really helped me start small and stay organised. To help me stay organised I created an In and Out table. I did this cause 1, it was a requirement and 2, because it actually helps you compare and find similarities between two data sets to find formulas. Starting small was also kind of started through the suggestions of the problem. I started small by looking at each the simpler formulas first also when finding the formulas I usually looked for the most obvious similarities and went from there. These obvious similarities helped me narrow down my options for formulas to help me stick with one.