Cutting the Pie POW
Problem Statement:
For this problem, we mainly worked with circles and figuring out the different ways on how to cut them, with only a certain amount of cuts. Our goal was to cut the circle 1 through 10 times to find the maximum amount of pieces, for the specific amount of cuts. However, our overall goal for this problem was to find an equation to answer an "x" amount of cuts on a single circle. The problem started out by telling us to begin with one cut and then two and three, and so on. There was two restraints with this problem: each line had to be straight, and each line had to go from either end of the circle
For this problem, we mainly worked with circles and figuring out the different ways on how to cut them, with only a certain amount of cuts. Our goal was to cut the circle 1 through 10 times to find the maximum amount of pieces, for the specific amount of cuts. However, our overall goal for this problem was to find an equation to answer an "x" amount of cuts on a single circle. The problem started out by telling us to begin with one cut and then two and three, and so on. There was two restraints with this problem: each line had to be straight, and each line had to go from either end of the circle
Process Description:
When I began this problem, I was somewhat confused on where to start. I went to a few of my classmates for help, and how they began. I knew that the table on the sheet of paper, already had the maximum of pieces for each line, starting with lines 1, 2, and 3. I then figured out that I should start by drawing a circle and making random cuts, but I also had to keep in mind that I am only aloud a certain amount of cuts. I soon found out that to get to a greater amount of cuts, I had to make a cut off-center to the previous cut (you can see this in the pictures below). The problem that I faced was that I didn't know when to stop with the amount of cuts on a circle. After I would complete a randomly cut circle, and count the pieces, I would then draw another one and just hope for a greater amount of pieces for that circle. A while after I had tried multiple ways on getting the maximum amount of pieces, I figured out that if I make cuts really close to each other, but not touching, that would make a tiny little piece by itself right there! As soon as I found that out, this problem became a lot clearer to me. I came up with a certain pattern of the cuts to try on the circle. I have to admit, I still tried random lines in the circles, because I was so determined to get to a specific amount of pieces. While using this method of learning, I proceeded onto 5 lines, then 6. I found that when I started on 7 cuts and up, it just got confusing from there, even when I did use my method. I would spend so much time trying to determine the maximum amount of cuts, but the numbers kept getting higher, so I just threw in the towel, and told myself that I would predict the rest of the numbers, according to the patterns I saw. When I de-briefed with a few of my classmates about the problem, I found out that my predictions were correct! After I reviewed all of my data, I began to try and create an equation that shows the specific amount of pieces you get based on how many cuts you are given. Finally, after so many failed attempts, I came up with an equation that fit this problem. You can find my solution below.
When I began this problem, I was somewhat confused on where to start. I went to a few of my classmates for help, and how they began. I knew that the table on the sheet of paper, already had the maximum of pieces for each line, starting with lines 1, 2, and 3. I then figured out that I should start by drawing a circle and making random cuts, but I also had to keep in mind that I am only aloud a certain amount of cuts. I soon found out that to get to a greater amount of cuts, I had to make a cut off-center to the previous cut (you can see this in the pictures below). The problem that I faced was that I didn't know when to stop with the amount of cuts on a circle. After I would complete a randomly cut circle, and count the pieces, I would then draw another one and just hope for a greater amount of pieces for that circle. A while after I had tried multiple ways on getting the maximum amount of pieces, I figured out that if I make cuts really close to each other, but not touching, that would make a tiny little piece by itself right there! As soon as I found that out, this problem became a lot clearer to me. I came up with a certain pattern of the cuts to try on the circle. I have to admit, I still tried random lines in the circles, because I was so determined to get to a specific amount of pieces. While using this method of learning, I proceeded onto 5 lines, then 6. I found that when I started on 7 cuts and up, it just got confusing from there, even when I did use my method. I would spend so much time trying to determine the maximum amount of cuts, but the numbers kept getting higher, so I just threw in the towel, and told myself that I would predict the rest of the numbers, according to the patterns I saw. When I de-briefed with a few of my classmates about the problem, I found out that my predictions were correct! After I reviewed all of my data, I began to try and create an equation that shows the specific amount of pieces you get based on how many cuts you are given. Finally, after so many failed attempts, I came up with an equation that fit this problem. You can find my solution below.
Here are the diagrams that I used when coming to a conclusion
Solution:
The equation that I came to by looking over my data is:
x (x + 1) + 1 = answer
2
x = number of cuts
Note: The "+ 1" at the end of the equation is adding 1 to the entire equation.
The equation that I came to by looking over my data is:
x (x + 1) + 1 = answer
2
x = number of cuts
Note: The "+ 1" at the end of the equation is adding 1 to the entire equation.
Self-Assessment and Reflection:
While working on this problem, I learned that there are patterns everywhere, even if you don't see them. Even in everyday life, there are patterns all around us. I found that out during this problem, because the main focus of this problem was trying to determine the specific patterns within our data. By using the in-and-out table was also a great way to find those patterns. Towards the end of this problem, I figured out that I didn't necessarily need to draw and cut circles 7 through 10, because I could find the answer to the next number, by looking at the previous number. I also learned different patterns of how to cut up a circle to get to a maximum amount of pieces. As I look back at my diagram of the 6th circle, I can see that if I just add another line that intersects all of the previous lines, then it would get me the maximum number of pieces... plus it looks cool. For this assignment, I think I would give myself a 10 out of 10, because I honestly tried really hard, and was determined to get to an answer. I also spent a few hours trying to find the perfect pattern to get to the maximum number of pieces. In this problem, I have definitely used the habit of a mathematician: 'looking for patterns.' I believe that I have conquered this habit because I constantly looked for patterns within my data to find a certain pattern that would help me understand my data more. Another habit of a mathematician I used in this assignment is: 'stay organized.' In this specific problem, you have to be organized, or else it will be difficult to find the patterns. I feel that I have done an adequate job in staying organized, as you can see in the pictures above.
While working on this problem, I learned that there are patterns everywhere, even if you don't see them. Even in everyday life, there are patterns all around us. I found that out during this problem, because the main focus of this problem was trying to determine the specific patterns within our data. By using the in-and-out table was also a great way to find those patterns. Towards the end of this problem, I figured out that I didn't necessarily need to draw and cut circles 7 through 10, because I could find the answer to the next number, by looking at the previous number. I also learned different patterns of how to cut up a circle to get to a maximum amount of pieces. As I look back at my diagram of the 6th circle, I can see that if I just add another line that intersects all of the previous lines, then it would get me the maximum number of pieces... plus it looks cool. For this assignment, I think I would give myself a 10 out of 10, because I honestly tried really hard, and was determined to get to an answer. I also spent a few hours trying to find the perfect pattern to get to the maximum number of pieces. In this problem, I have definitely used the habit of a mathematician: 'looking for patterns.' I believe that I have conquered this habit because I constantly looked for patterns within my data to find a certain pattern that would help me understand my data more. Another habit of a mathematician I used in this assignment is: 'stay organized.' In this specific problem, you have to be organized, or else it will be difficult to find the patterns. I feel that I have done an adequate job in staying organized, as you can see in the pictures above.