Problem Statement: Circles can be cut in many different ways, but it can result in a different number of pieces each time. A circle with three cuts could result in six pieces or seven pieces. This data could be put into an In and Out table to show a pattern and to aid in the findings of an equation that can find the maximum amount of pieces for a certain number of cuts. Our job was to analyze the table already given to us and extend the table by looking for the maximum number of pieces that can be found with a certain amount of cuts.
Process: I drew out ten different circles in which I would draw lines according to the number of cuts required. In the first circle, I would cut it once, in the second circle, I would cut it twice and so on. I discovered that it was easy to miss pieces or mess up the lines which would result in incorrect data. I also learned that it was way harder to work with this problem on lines paper instead of white paper, however I was still able to complete the problem successfully. I received some help from a fellow student who helped me find an easier way to draw the lines without getting confused which can be seen in circles seven through ten. After cutting all the circles, I counted all the pieces created by those cuts and put them into an In and Out table. I found a pattern in the table to create an equation that will find the number of pieces for any number of cuts.
Solution: I did find a pattern in my table that correlates with the number of cuts and the number of pieces. The obvious pattern in the number of cuts was that the numbers go up by one, so it is a very consistent pattern. For the number of pieces, the pattern went up by two, three, four and so on. For example, if you add two and two it would equal four, four plus three is seven and seven plus four is eleven; you can see this pattern in the number of pieces, so in a way it is a consistent pattern. After trial and error plus some help from a student, I was able to get an equation that will tell us what the maximum number of pieces in the next circle is. The equation is F(n) = F(n-1) + n and I knew this was the correct equation because after testing it with numbers, I found that it worked.
For example:
F(10) = F(10-1) + 10
F(10) = F(9) + 10 < Here we know that 10 must be added to the number of pieces from 9 cuts to equal 56 pieces for 10 cuts.
Another example:
F(11) = F(11-1) + 11
F(11) = F(10) + 11 < Again we add 11 to the number of pieces from circle 10 to equal 67, so now we know that a circle with 11 cuts will
have a maximum of 67 pieces.
Process: I drew out ten different circles in which I would draw lines according to the number of cuts required. In the first circle, I would cut it once, in the second circle, I would cut it twice and so on. I discovered that it was easy to miss pieces or mess up the lines which would result in incorrect data. I also learned that it was way harder to work with this problem on lines paper instead of white paper, however I was still able to complete the problem successfully. I received some help from a fellow student who helped me find an easier way to draw the lines without getting confused which can be seen in circles seven through ten. After cutting all the circles, I counted all the pieces created by those cuts and put them into an In and Out table. I found a pattern in the table to create an equation that will find the number of pieces for any number of cuts.
Solution: I did find a pattern in my table that correlates with the number of cuts and the number of pieces. The obvious pattern in the number of cuts was that the numbers go up by one, so it is a very consistent pattern. For the number of pieces, the pattern went up by two, three, four and so on. For example, if you add two and two it would equal four, four plus three is seven and seven plus four is eleven; you can see this pattern in the number of pieces, so in a way it is a consistent pattern. After trial and error plus some help from a student, I was able to get an equation that will tell us what the maximum number of pieces in the next circle is. The equation is F(n) = F(n-1) + n and I knew this was the correct equation because after testing it with numbers, I found that it worked.
For example:
F(10) = F(10-1) + 10
F(10) = F(9) + 10 < Here we know that 10 must be added to the number of pieces from 9 cuts to equal 56 pieces for 10 cuts.
Another example:
F(11) = F(11-1) + 11
F(11) = F(10) + 11 < Again we add 11 to the number of pieces from circle 10 to equal 67, so now we know that a circle with 11 cuts will
have a maximum of 67 pieces.
Equation: F(n) = F(n-1) + n
|
# of Cuts (n)
1 2 3 4 5 6 7 8 9 10 |
# of Pieces F(n)
2 4 7 11 16 22 29 37 46 56 |
**Note: Captions in the images of the slideshow aren't accurate with my table due to my forgetfulness of changing the numbers. However, the numbers in the table are correct**
Self-Assessment: I learned that there are many different ways to cut a circle and get different amounts of pieces. It's very strategic in where cuts are placed, for example, if you are serving pie to seven guests and you can only cut it three times then you can cut it strategically to get a seventh piece! Of course this is a complicated way to serve pie, but all the same, it results in a different amount of pieces with the same amount of cuts. I think I would have to assign myself a 9 out of 10 because I feel I did the best I could've possibly have done, however, I wasn't very successful in getting an equation myself without help. I feel if I thought about it more, I could've found it on my own, but I did not. I did get accurate data which had a consistent pattern that was very helpful to use in the equation and I would always double check my circles to make sure I got all the pieces and cuts. I think a habit I used was Staying Organized because I kept my circles away from each other so it wouldn't be cluttered and I tried to draw my lines as cleanly as I could to make it easier on myself when it came to counting pieces and intersections. I also kept an organized table where I could distinctly find the data that went together and the patterns. Overall, this was a very interesting problem to look into and it really got my brain thinking about geometry while giving me a new perspective on how to cut circles.