POW #6: Just Count the Pegs
The “Just Count the Pegs” problem covered concepts such as finding area, identifying mathematical patterns, and the application of various equations. Each topic correlated to the same general idea of compiling multiple pieces of information in order to find one final solution. The majority of my work was done in a precise and meticulous manner, for I needed to make sure that each value that was being “plugged into” my formula was accurate. I found that I had to be very patient throughout the process, but that virtue was valuable in the end. By taking the multistep problem piece by piece, I could more effectively organize my ideas, and was efficient about finding a conclusive answer.
Part 1: Problem Statement
In this problem, I had to analyze three components related to one main question. Freddie Short, Sally Shorter, and Frashy Shortest had different ways to go about calculating the areas of polygons on a geoboard, which is a grid of pegs on which rubber bands can be stretched in any direction. These three individuals also derived formulas from their reasoning, with each one yet to be discovered, and only variables for both sides of an In-Out table present. Freddie had a formula to find the area of any polygon that did not have any pegs on the interior. The input for his table would be the number of pegs on the boundary of the figure, while the output would be the area of the figure. Sally had a similar process to Freddie, except her shortcut only applied to polygons with exactly four pegs on its border. The input for her formula would be the number of pegs on the interior of the figure, and the output would be the area of that figure. Even more refined than the other two students, Frashy Shortest had a formula in which one could tell her both the number of pegs on the boundary of the figure along with the number of pegs in the interior, and she could provide the area of the polygon using that information. The goal of the problem was to try and solve for Frashy’s “superformula” after starting off with her friends’ methods. Freddie and Sally’s equations were designated to allude to the way in which Frashy’s formula would be presented.
Part 2: Process
To start off, I went about solving for Freddie’s formula. Using the information stated previously, I knew that the In values would correspond with the number of pegs on the boundary for a polygon with no pegs in the interior, and the Out values would correspond with their respective areas. Keeping in mind that a polygon had to have at least three sides, I started my In-Out table with 3 for my first “x” value. I would then move down the line and draw a quadrilateral with an empty interior, a five-pegged figure with no pegs for its interior, and so on.
Part 1: Problem Statement
In this problem, I had to analyze three components related to one main question. Freddie Short, Sally Shorter, and Frashy Shortest had different ways to go about calculating the areas of polygons on a geoboard, which is a grid of pegs on which rubber bands can be stretched in any direction. These three individuals also derived formulas from their reasoning, with each one yet to be discovered, and only variables for both sides of an In-Out table present. Freddie had a formula to find the area of any polygon that did not have any pegs on the interior. The input for his table would be the number of pegs on the boundary of the figure, while the output would be the area of the figure. Sally had a similar process to Freddie, except her shortcut only applied to polygons with exactly four pegs on its border. The input for her formula would be the number of pegs on the interior of the figure, and the output would be the area of that figure. Even more refined than the other two students, Frashy Shortest had a formula in which one could tell her both the number of pegs on the boundary of the figure along with the number of pegs in the interior, and she could provide the area of the polygon using that information. The goal of the problem was to try and solve for Frashy’s “superformula” after starting off with her friends’ methods. Freddie and Sally’s equations were designated to allude to the way in which Frashy’s formula would be presented.
Part 2: Process
To start off, I went about solving for Freddie’s formula. Using the information stated previously, I knew that the In values would correspond with the number of pegs on the boundary for a polygon with no pegs in the interior, and the Out values would correspond with their respective areas. Keeping in mind that a polygon had to have at least three sides, I started my In-Out table with 3 for my first “x” value. I would then move down the line and draw a quadrilateral with an empty interior, a five-pegged figure with no pegs for its interior, and so on.
In Figure 1 (lower left), I kept track of my geoboard sketched in a systematic manner, and recorded the results that I found in a table. My goal was to identify one general formula that I could apply to all aspects of the restrictions that Freddie had initially set, so I tried to find commonalities within the data. As demonstrated in Figure 1 as a whole, there were many variations of the same rule, and numerous polygons applied to the same rule. I knew that Freddie’s formula would also partially assist me in finding Frashy’s superformula, so I made sure that my process was smooth and methodical as I searched for patterns. By balancing visuals with numbers, I could more effectively see the relationships between values, which was extremely helpful as I moved forward with the task at hand. Below, you will find the data that I recorded for Freddie’s formulas, and you may refer to Figure 1A for the specific examples that I established:
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After reviewing the figures in the table, I noticed a strong connection between the In values, and the fact that the Out values were close to half of their counterparts, but not quite. As a result, I developed an equation after realizing that the Out values were just one less than half of each In value:
From here, I moved onto the next component of Freddie’s problem, which was to repeat the same process but with figures that had one peg on the interior. This was a slight difference from finding polygons with no pegs on the interior, so I expected a small change in the formula following the data collection step. Once again, I drew some shapes that followed the pattern on the geoboard, and documented my findings in a table. (See Figure 1B on previous page for the examples I developed).
The most significant difference between this table and the one prior was the fact that the input values started at four. Because I had to allow space for one peg on each shape’s interior, the most minimal input value I could utilize was 4. Comparing the data with the set before, I immediately saw that my observation about the In and Out values being related by being close to half of the input was still correct. However, the output or area was exactly half of the input in this scenario. Therefore, I used my answer from question 1a to guide me in creating a new formula:
At this point, I was gliding along in the process of finding a formula for Freddie’s various requirements. For the next part of his shortcut, I had to organize and display data that corresponded to polygons with two pegs on the interior. I imitated the same procedure as I had used, and my results can be found below. (See Figure 1C on previous page for the examples I developed):
Seeing that the formula for zero pegs on the interior was y = x/2 – 1 and the formula for one peg on the interior was y = x/2, I found that the increase by one for the output would most likely carry over to the next question, too. The last stage of testing Freddie’s shortcut was to include one more data set. To accomplish this, I drew polygons that had three pegs on the interior, and jotted the inputs and outputs down. (See Figure 1D on pg. 2 for the figures I developed.)
As I expected, the area that corresponded to each input increased by 1, which meant that my thoughts on the transformation of the general equation were correct. The new formula was:
Now I had to somehow combine all three equations into one overall formula that could fit any of Freddie’s requirements. To make the process easier on myself, I replaced x and y with the more fitting variables of b and A: b for boundary pegs and A for the polygon’s area. In addition, I included the variable of i to represent interior pegs and which was also existent in each of the equations that I had developed before. I knew that I would still keep the “y = x/2” portion (or A = b/2), because that was the basis of how I went about solving the problem. I also considered the fact that when there were no pegs on the interior of a polygon, the area decreased by 1, but when a peg was added, that output was neutralized. Because of this, I developed “Freddie’s Formula”, which could be used accordingly for any one of his scenarios:
To prove that this formula does in fact work, I plugged a set of input and output values from question 1a:
At this stage in the exercise, I could move on to Sally’s formula because I had just clearly defended my reasoning for Freddie’s equation.
Using the information given for Sally’s formula, the input values would correspond with the number of pegs on the interior for a polygon with four pegs on the boundary, and the output values would correspond with the appropriate areas. Unlike Freddie’s example, I started my In-Out table with 0 for my first “x” value, since it coincided with the number of pegs on the interior of a polygon. I then planned to include figures down the line with one peg on the interior, two pegs on the interior, etc. In Figure 2A (left), I drew the polygons on the geoboard and formatted my work in the same way that I had for Freddie, which really helped me keep a level head as I was breaking down the problems. By maintaining my work habits, I could more easily compare and contrast the two formulas in the end in order to find Frashy’s superformula. My initial task was to set up a table that matched the first round of polygons, and below, you will find the data that I recorded for Sally’s formulas, You can look to Figure 2A for the specific examples that I established: |
Right away, this pattern was glaring at me. The area of Sally’s polygons were 1 more than the pegs on their interiors, so I found:
After this step, I moved onto the next part of Sally’s problem, which was to replicate the same process but with figures that had five pegs on the boundary. Like the process I followed to find Freddie’s formula, there was only a small tweak to the structure of the equation. Once again, I drew some shapes that followed the pattern on the geoboard, and documented my findings in a table.
At first, I believed that the equation that applied to the last problem might have been simple out of luck, but this one followed that same trend. The area was 1.5 more than the number of pegs on the boundary of a five-sided figure, so I took note of that in a formula for the second part of Sally’s quest to finding a true equation:
It was at this point that I spotted another resemblance to my approach for Freddie’s problem. I recognized the increase by .5 between the two equations, and thought that might continue for the next set. Sure enough, it did. (See Figure 2C on the previous page for the figures I developed.)
With a pattern in mind and my thoughts collected, all that I needed to do was manipulate the data I had found into a generalized equation, like I had for Freddie. I understood that the formula needed to include the number of interior pegs + a relationship between the boundary pegs and the area. It made sense to me to revert back to Freddie’s Formula and see if I could identify any connections between the structure of that sequence and the direction that Sally’s shortcut was going, so I tested it out using values from Sally’s data:
I was shocked. It worked for Sally, too! This would lead me to an easier time finding Frashy’s superformula, and I was glad that I could link the two more specialized equations together.
Part 3: Solution
Finally, it was time to bring all three pieces of the problem together. In the last leg of the exercise and as stated in the directions, Frashy claimed that she had a superformula in which one could give her the number of pegs in the interior and the number of pegs on the boundary, and she could dish out the corresponding area for those values. To try and prove Frashy’s declaration, I grabbed three examples from both Freddie and Sally’s problems. This would create a diverse group in which my statement could be made about whether my previous work was true. I also thought it was a smart way to tie all of the aspects of the problem as a whole together, and I hoped that everything I had done up to this point would pay off in the end.
Part 3: Solution
Finally, it was time to bring all three pieces of the problem together. In the last leg of the exercise and as stated in the directions, Frashy claimed that she had a superformula in which one could give her the number of pegs in the interior and the number of pegs on the boundary, and she could dish out the corresponding area for those values. To try and prove Frashy’s declaration, I grabbed three examples from both Freddie and Sally’s problems. This would create a diverse group in which my statement could be made about whether my previous work was true. I also thought it was a smart way to tie all of the aspects of the problem as a whole together, and I hoped that everything I had done up to this point would pay off in the end.
Using the three figures (above), I created a table with two inputs and an output, the inputs being the number of pegs on the polygon’s interior and the number of pegs on its boundary, and the output being the area:
Then I took the values and plugged them into the general equation that I had developed for both Freddie and Sally:
All three examples fit the parameters, so I knew that I was correct.
Frashy’s superformula was: A = i + b/2 – 1
Part 4: Extensions
Reflection
The “Just Count the Pegs” problem was crucial in my ability to understand concepts surrounding making mathematical connections, translating data, and finding area. There were multiple questions surrounding one problem, and each had to be handled delicately in order for me to find the correct answer. Because each individual presented in the problem had his or her own unique way of generating the area of any polygon, I had to identify the differences between each method, and use those contrasts to find similarities. I also gained more knowledge about the application of a formula, and how to manipulate it properly in order to satisfy the restraints given in the problem, or a problem in general.
I honestly believe that I deserve a 10 out of 10 for my work on this problem because of the creativity I used in founding my ideas and logic for each component. I treated each part separately, but I was not shy to connect my findings after I was confident in my answers, and I think that it is important when it comes to math. I took risks in the way that I approached the problem – and while it may have been unconventional at times – it was the right way to tackle the information I was initially given. I also put a lot of work into producing my diagrams, which was also a crucial part of the process. The visuals helped me to walk the reader through my thinking in a step-by-step manner, and it actually was beneficial to my own understanding of the problem, too.
The most prominent Habit of a Mathematician shown in my work was Conjecture & Test. Early on, I recognized the fact that I had three components to deal with, so I needed to find a way to associate each one with the other two. My “eureka” moment was when I used the formula I had established for Freddie in Sally’s problem, and it worked. If I had not been looking to strike connections between each aspect of my work, I would not have made as much progress as I did. I also used this habit to construct polygons on the geoboard, and how their corresponding values translated to the tables. Some shapes were impossible to create with the given circumstances, so I had to work my way around that obstacle in order to be successful in the end.
Frashy’s superformula was: A = i + b/2 – 1
Part 4: Extensions
- Simple: Using the superformula, find the area of a polygon with four pegs on the boundary and two interior pegs.
- Moderate: Using the superformula, find the area of a polygon with five pegs on the boundary and three interior pegs.
- Difficult: Using the superformula, find the area of a polygon with nineteen pegs on the boundary and nine interior pegs.
Reflection
The “Just Count the Pegs” problem was crucial in my ability to understand concepts surrounding making mathematical connections, translating data, and finding area. There were multiple questions surrounding one problem, and each had to be handled delicately in order for me to find the correct answer. Because each individual presented in the problem had his or her own unique way of generating the area of any polygon, I had to identify the differences between each method, and use those contrasts to find similarities. I also gained more knowledge about the application of a formula, and how to manipulate it properly in order to satisfy the restraints given in the problem, or a problem in general.
I honestly believe that I deserve a 10 out of 10 for my work on this problem because of the creativity I used in founding my ideas and logic for each component. I treated each part separately, but I was not shy to connect my findings after I was confident in my answers, and I think that it is important when it comes to math. I took risks in the way that I approached the problem – and while it may have been unconventional at times – it was the right way to tackle the information I was initially given. I also put a lot of work into producing my diagrams, which was also a crucial part of the process. The visuals helped me to walk the reader through my thinking in a step-by-step manner, and it actually was beneficial to my own understanding of the problem, too.
The most prominent Habit of a Mathematician shown in my work was Conjecture & Test. Early on, I recognized the fact that I had three components to deal with, so I needed to find a way to associate each one with the other two. My “eureka” moment was when I used the formula I had established for Freddie in Sally’s problem, and it worked. If I had not been looking to strike connections between each aspect of my work, I would not have made as much progress as I did. I also used this habit to construct polygons on the geoboard, and how their corresponding values translated to the tables. Some shapes were impossible to create with the given circumstances, so I had to work my way around that obstacle in order to be successful in the end.