POW #3: Dave Bakes a Pie
For the “Dave Bakes a Pie” problem of the week, I used my skills in identifying patterns and the visualization of mathematics to help me find an answer. It was essential that I looked for sensible and logical conclusions within the data I gathered, but the drawings I created were just as important. Without a way of seeing the problem in front of me, I would have been lost and I doubt I would have been as successful in finding the solution. Math is usually thought of as being a variety of concepts entailing simply numbers and equations. However, diagrams and images can be just as helpful, if not more beneficial to finding the correct answer to a question. This assignment was a unique way of approaching a problem for me – illustrating the information – but it was very effective for my understanding of the material involved.
Part 1: Problem Statement
In this problem, Dave baked a pie and decided to cut it in different ways. He first cut it one time, making two pieces. He then sliced two cuts into the pie, making four separate pieces. These cuts were not symmetrical and did not always go through the middle, but they were all straight and went all the way across the pie. After the first two attempts, Dave made three cuts in two different pies, resulting in varied outcomes. The first pie yielded six pieces with three cuts, while the second pie was divided into seven pieces with three cuts. Was seven the largest number of pieces that three cuts into a pie could create? More importantly, the focus question I had to answer was, What is the largest number of pieces that can be produced from a given number of cuts? If I was to be given a certain number of cuts that I could make in a pie, how could I be certain that the number of pieces I had created was the maximum? I had to find out.
Part 2: Process
There were three steps I had to take in order to solve for the answer(s). I first had to find the largest number of pieces that I could get from four and five cuts respectively. The second portion of the problem was to find a pattern in the table (below), and to use that information to find the largest number of pieces that could be generated from ten cuts. In addition, I had to explain why that pattern was occurring, and the reasoning behind it. Finally, I had to use what I had collected above to find a rule for an In-Out table, and to write the output in terms of a variable for the input.
To start off the problem, I took a look at the table I was given:
Part 1: Problem Statement
In this problem, Dave baked a pie and decided to cut it in different ways. He first cut it one time, making two pieces. He then sliced two cuts into the pie, making four separate pieces. These cuts were not symmetrical and did not always go through the middle, but they were all straight and went all the way across the pie. After the first two attempts, Dave made three cuts in two different pies, resulting in varied outcomes. The first pie yielded six pieces with three cuts, while the second pie was divided into seven pieces with three cuts. Was seven the largest number of pieces that three cuts into a pie could create? More importantly, the focus question I had to answer was, What is the largest number of pieces that can be produced from a given number of cuts? If I was to be given a certain number of cuts that I could make in a pie, how could I be certain that the number of pieces I had created was the maximum? I had to find out.
Part 2: Process
There were three steps I had to take in order to solve for the answer(s). I first had to find the largest number of pieces that I could get from four and five cuts respectively. The second portion of the problem was to find a pattern in the table (below), and to use that information to find the largest number of pieces that could be generated from ten cuts. In addition, I had to explain why that pattern was occurring, and the reasoning behind it. Finally, I had to use what I had collected above to find a rule for an In-Out table, and to write the output in terms of a variable for the input.
To start off the problem, I took a look at the table I was given:
With the table, I also had access to the first three input values, so I added those into the table to help me visualize the problem.
It was here that I started to discover some patterns in the data. Since I needed to convince myself that a maximum of seven pieces could be made with three cuts, I started to make connections between numbers. Literally. If I drew a line “zig-zagging” from left to right down the table, each number would match up in terms of addition. If you will refer to the first table below, you will see that starting in the second column with the number 2, that digit could be added to the next “Number of Cuts” value to get the next “Maximum Number of Pieces” value. This would also be considered the solution of the previous addition problem.
By following this pattern, the logical conclusion for the maximum number of pieces with four cuts would be eleven. I had the data; now all I had to do was prove it using a diagram. Sure enough – after much trial and error – I found that the fitting circle for four cuts had the maximum of eleven pieces that I was looking for (see right). Finding that eleven was the corresponding solution to the number four also helped me to locate the maximum number of pieces for a pie with five cuts in it. This solution was sixteen. Using the information that I had just collected, I updated my table so that I could stay on track with my progress, and verified those conclusions through two more diagrams.
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I had identified similarities within the table, and now I could find the maximum number of pieces from 10 cuts, as shown below.
Part 3: Solution
My answer was a maximum of 55 pieces for 10 cuts, but I did not have an equation. With the method that I used to solve this problem, there was not a clear and concise way to describe the process in the form of a function. Any function or equation that I developed would be incoherent with the work I completed, so I decided to look at the steps I took to finding the solution in a different way. I did not need the data anymore; I had already passed that stage of the process. It was now time to put that data into action, and to validate my solution in a formula that made sense. While I solved the problem, I had only considered the “Number of Cuts” and the “Maximum Number of Pieces” for each input and output value. However, if I was to add “Maximum Number of Intersections” to the table, it could change the game, so to speak. I went ahead and made a new section on my table (below). By calculating the number of intersections for each input value, I hoped that I could find a potential pattern within the data that could lead me to a function.
I decided I would go about determining the number of intersections in each pie by backtracking to the diagrams I had earlier, and using those to help me.
I highlighted each of the intersections in the diagrams with yellow stars, and at this point I could once again update my main table.
It was here when I was able to identify a pattern in the data, and create a function. Assigning the variables c, p and i to the “Number of Cuts”, “Maximum Number of Pieces”, and “Maximum Number of Intersections” columns respectively, I was able to generate the function below:
f(p) = [(c + i ) + 1]
Now, all that was left was to test my function, and confirm that it worked with the data I had collected.
f(p) = [(c + i ) + 1]
Now, all that was left was to test my function, and confirm that it worked with the data I had collected.
All of the answers checked with the function I established, so I knew I was right. It was really tough to create the visuals for the pies after 5 cuts, but I knew I could be confident in my answer for 10 cuts because I had found the same exact solution through two alternate methods. Since I took the steps to confirm that I was on the right track by implementing the “Number of Intersections” component in the table, I could conclude that my answer was correct. By constructing a function using variables to three parts of the problem, I could form a clear and concise way to express the initial information and solution.
Part 4: Extensions
Reflection
In this problem, I learned how to prove data with visualizations, identify patterns, and approach a problem through multiple methods. It was crucial for me to create diagrams that represented the tables and patterns I had, and use them to seek a solution. Without any way of conceptualizing the work I had done, I would not have been able to stay organized, and the process I took to find an answer would not have been nearly as effective. In addition, I learned to assign variables in order to create a function or an equation, and how to utilize them properly. I set up the relationship between the variables as a function of p, or the “Maximum Number of Pieces” from the table, which helped me stay on track. Once I had established that the goal was to solve for p, I knew I was on the right track for the rest of the variables. All I had to do was put together my thoughts in the form of a function, and make sure that my work was represented in a sensible manner.
For this assignment, I truly believe that I deserve a 10 out of 10 for my thorough explanation of the process I took to find a solution, and for my multiple approaches to the problem. I expanded on the basic patterns I saw, and included why I was seeing those values. I also included relevant and interesting visuals throughout my work and commentary, which enhanced the quality of my paper. After I solved the problem and found an answer, I could have gone with that and jumped straight to a conclusion. Instead, I recognized that I could not develop a pertinent function, so I continued on and looked at the problem a different way. By taking the extra steps to find a more convincing answer, I demonstrated my complete knowledge of the problem. Finally, I created my own variables in the function I formed, which exemplified the creativity and inventiveness I used to find the right answer.
The most prominent Habit of a Mathematician I displayed in my work was Seek Why & Prove. I initially found a solution with the first method I took, but I had neither an efficient nor reasonable way to explain my process. An answer itself does not cut it with this type of problem, and I knew I had to provide an explanation and a function for my work. By exploring another route with the “Number of Intersections” of each value, I confirmed my solution, and really was able to understand the problem on another level. I am satisfied with the work I did in the “Dave Bakes a Pie” problem of the week, and I have no regrets about the steps I took.
Part 4: Extensions
- Simple: What is the maximum number of pieces that could be made from 8 cuts?
- Moderate: How many intersections would a pie with 16 cuts have?
- Difficult: What would the equation look like if it was a function of the variable c or i?
Reflection
In this problem, I learned how to prove data with visualizations, identify patterns, and approach a problem through multiple methods. It was crucial for me to create diagrams that represented the tables and patterns I had, and use them to seek a solution. Without any way of conceptualizing the work I had done, I would not have been able to stay organized, and the process I took to find an answer would not have been nearly as effective. In addition, I learned to assign variables in order to create a function or an equation, and how to utilize them properly. I set up the relationship between the variables as a function of p, or the “Maximum Number of Pieces” from the table, which helped me stay on track. Once I had established that the goal was to solve for p, I knew I was on the right track for the rest of the variables. All I had to do was put together my thoughts in the form of a function, and make sure that my work was represented in a sensible manner.
For this assignment, I truly believe that I deserve a 10 out of 10 for my thorough explanation of the process I took to find a solution, and for my multiple approaches to the problem. I expanded on the basic patterns I saw, and included why I was seeing those values. I also included relevant and interesting visuals throughout my work and commentary, which enhanced the quality of my paper. After I solved the problem and found an answer, I could have gone with that and jumped straight to a conclusion. Instead, I recognized that I could not develop a pertinent function, so I continued on and looked at the problem a different way. By taking the extra steps to find a more convincing answer, I demonstrated my complete knowledge of the problem. Finally, I created my own variables in the function I formed, which exemplified the creativity and inventiveness I used to find the right answer.
The most prominent Habit of a Mathematician I displayed in my work was Seek Why & Prove. I initially found a solution with the first method I took, but I had neither an efficient nor reasonable way to explain my process. An answer itself does not cut it with this type of problem, and I knew I had to provide an explanation and a function for my work. By exploring another route with the “Number of Intersections” of each value, I confirmed my solution, and really was able to understand the problem on another level. I am satisfied with the work I did in the “Dave Bakes a Pie” problem of the week, and I have no regrets about the steps I took.