Problem of the Week #1
Imagine a Square
Problem Statement
For this Problem of the Week write-up, we were required to “imagine” a square and try to find a pattern in how it can be broken down into smaller squares. Some of the rules that this problem had was that once a square had been divided, the larger one could no longer be counted, we couldn’t count any overlapping squares. However, with that aside we were free to divide into as many smaller squares. Our main goal was to try and see how many numbers were possible and were not possible. For example you have a chart of numbers one through twenty and you need to identify what can happen. In order to reach the set goal and conclusion, we were required to at first work by ourselves to come up with a strategy to a solution. Than we were to work with the people at our table to fine tune the given strategies.
Process Description
Before attempting to try and solve the given problem we had to define what a square is. To me a square can be defined as a two dimensional image with four connecting lines. It is also the building block for many more images, objects, and more. Having figured out what I think a square is and how I see it, I got to hear what other people thought it defined as four equal sides, four 90* angles, perfect symmetry, and that it is a quadrilateral. Once we all had an idea of what everyone else thought of a square, we started to work out how a square is divided and if there was any pattern. First we worked on it individually and what I did was to divide a square into four parts, than into as many parts as I could. However, I think I failed to look for any kind of patterns during the individual round. Because of this I ended up with a square that had a total of 236 smaller squares, and no pattern. Luckily, we were also given time to work with the other people at our tables, because I was able to look at the square at a slower pace and then I started to see what worked and what couldn’t.
My Work
Solution
At first my solution was the assumption, that as long as you can physically show the work, than you can go as small as possible, to get an almost infinite amount of squares. However at first I had failed to look for any patterns before breaking it down, so it wasn’t till we started collaborating with our table members that when you break it down slowly and divide each square once. Then you start to see a total amount of numbers showing which number is possible, whereas before I thought only certain numbers were possible. I know that the solution that my table members is correct because I tested it, as well as the whole class and we were able to see that it was reliable for what we were working on. For the Imagine a Square, we found that the for each new block, three new squares are being added and the previous larger square cannot be counted.
Reflection
I think that this P.O.W taught me more about squares. It also taught me how to look for patterns and that visualization can be a huge benefit. The reason because, in life eventually you’re going to run into a situation where you are required to look for patterns and see if there is anything that can help you. I think that visualization can be super important as we progress into the year as we continue to get into more complex math problems. It will help organize everything, and it can also simplify the entire equation. At first view this didn’t look like something that wasn’t really going to challenge me, I still don’t think it was entirely challenging, but I got something out of it. I will say that it was slightly challenging, I would have appreciated something that was more my level. Having done this problem, I think that I deserve an 8/10, as it can be seen from the work, documentation, and how that I was able to connect multiple theories. Throughout this entire process, I think that I really showed collaboration and listening, as well as visualization. The reason being that if I came up with an equation that I feel works really good and possibly better than those at me table, I may have not been willing to listen and see what could have been better about mine.