Pascal's Problem: Write-up
1. Problem Statement:
In Pascal’s Problem you are given the same triangle tower that I have above. It briefly tells you what is happening in the triangle which is, you get the number by the sum of the two spots above it. This pattern will continue infinitely. The questions/tasks we were asked to solve were:
- What is the largest number you can get in the 10th, 15th, and 20th row
- Write an equation that will allow you to find the largest number of any given row
- What would the first 5 rows look like if you were asked to multiple instead of add
2. Process Description:
At first my group assumed that the problem wouldn’t be that hard, but that was before we figured out we had to write our own equation. For the part asking for the largest number in row ten and fifteen we just made a larger triangle tower and filled it out all the way to the fifteenth row. From that point on our group tried a number of different equations that we thought might work. Here are some that we tried:
Solving for the multiplication came from a single group member who came in excited to share that the Pascal’s triangle tower would infinitely be one just because one times one will always be one and it will never go over that.
Group Work: The group I was in was a group of four, all eager to solve this problem. I think that everyone in the group contributed whether they were trying different possible equations and never giving up or actually solving a key part of the triangle. As most groups have, we had a leader but not one that was overpowering. She encouraged us all to keep trying even we were annoyed and thinking of giving up, and she brought enthusiasm into solving this problem. In the end when we had accepted we didn’t find the answer we had a great time working together making our poster appealing to the eye and also understandable.
- # of row -1 4
- x-1/4
- # of row squared
- biggest # divided by row #
Solving for the multiplication came from a single group member who came in excited to share that the Pascal’s triangle tower would infinitely be one just because one times one will always be one and it will never go over that.
Group Work: The group I was in was a group of four, all eager to solve this problem. I think that everyone in the group contributed whether they were trying different possible equations and never giving up or actually solving a key part of the triangle. As most groups have, we had a leader but not one that was overpowering. She encouraged us all to keep trying even we were annoyed and thinking of giving up, and she brought enthusiasm into solving this problem. In the end when we had accepted we didn’t find the answer we had a great time working together making our poster appealing to the eye and also understandable.
3. Solution:
Our group, unfortunately, was not able to find a solution/ equation for Pascal’s problem of addition. We did find the largest number in the tenth row, which was 126, and also the largest number in the fifteenth row, which was 6435. The info that we did find, for the diagonal columns, was that the outside rows will always be one, f(x)= 1. In the second diagonal column the numbers would always go up by one, f(x)= x-1. To find the numbers in the third diagonal column you would use, f(x)= (x-1) [1+ .5(x-4)]. The last thing we came up with is that you will always be solving for the median of the row.
The solution for the multiplication Pascal’s problem was easy because we found out you will never get a bigger number than one because one times one will always equal one. The answer is that the triangle would be an infinite tower of ones. One being the largest number in each row.
The solution for the multiplication Pascal’s problem was easy because we found out you will never get a bigger number than one because one times one will always equal one. The answer is that the triangle would be an infinite tower of ones. One being the largest number in each row.
4. Self-Assessment and Reflection:
What I learned most for this problem, since we didn’t come up with a final answer, was mostly just how to guess and check. A lot of trying to solve this was being creative in coming up with different equations that may or may not work. The extent of all the things I tried were crazy and this problem allowed me to get creative in my thinking. Relating to group work I learn about participation. It’s always better when everyone helps out and also just reassuring other group members of the progress that they are making. As a grade I’d give myself an 8 out of 10 just because I probably could have contributed a little more to the solutions. Even though I tried multiple things I was the only person that didn’t come up with something semi helpful. I contributed a lot into the creation of the poster though. For the mathematical practices and expectations I believe I used “make sense of problems and persevere through solving them”. In the case of solving Pascal’s problem I think my group did a really great job of persevering through it. We never gave up trying even when we wanted to and in the end we did discover some sort of information. I wouldn’t really save that we “made sense of the problem” because it was very confusing so I would say I also used “developing successful mathematical habits of work”. We have been told multiple times that group work is extremely important with being successful. Being able to work in groups teaches you how to be collaborative and getting ideas from other members of the group cause spark an idea you never would have had in the first place.