Tiling a 11*13 Rectangle
This week within our Math class we had an introduction to what we will be doing throughout the school year. Already we have learned how we will be presenting our work and what resources we have available in order to keep up with our in class work.
The purpose of this week activities was to mainly, give us a an idea of what our work ethic should look like. Within the problems that we worked on, we focused on working with our groups in a successful manner, and getting into the proper math mind set. For example, our first tasks involved using team effort to get answers to our prompts. This would allow us to finish everything much quicker and showed us how to properly communicate and cooperate for the future. The videos that we saw served as proof that everyone is capable of being successful in math and that mistakes will be made, but shouldn’t be considered as bad. Mistakes are part of learning and it’s just something that will happen constantly.
The first activity that we did was called: Tiling a 11*13 Rectangle. The goal of this activity was to divide the rectangle into squares in the least amount possible. I worked on this problem to various degrees of success, and the lowest amount of squares that I could divide the rectangle in was 6. The second activity that we completed was called Squares to Stairs in which we would find out how many squares it would take to make a stair, and the linear increment that it had. With this information we could find how many squares are in any size stairs. The third activity that we worked on this week was called Hailstone Sequences, which a set of rules that creates a sequence out of any number, usually terminating in 4,2, and 1. The final activity that we looked at was called, Painted Cube. This activity had us determine what sides would get get painted on a 3 by 3 cube, and the individual statistics. The videos were mainly about how the brain works and the different effects that mistakes have. The videos also covered topics such as how timed work has an effect on the brain.
One message that I got out of the videos is that the best kind of learning comes from mistakes and this message has inspired me since I heard it in 9th grade. I first thought that mistakes equaled failure, but they are really just paths to success. Without failures, we can never succeed, we can never improve. And I feel that what we are trying to focus on is improvement. That is why this quote has really stuck out to me because I connected with it and has given me a new mindset. The second message that I took out of those videos is that by believing that you can, it is possible to learn much more. And although it does sound cliched, it is true and is something that has proved itself to me before, I never really understood how it occurred. But now I understand that it all relies on how much you want to learn and this has an immediate correlation with what you actually learn.
The purpose of this week activities was to mainly, give us a an idea of what our work ethic should look like. Within the problems that we worked on, we focused on working with our groups in a successful manner, and getting into the proper math mind set. For example, our first tasks involved using team effort to get answers to our prompts. This would allow us to finish everything much quicker and showed us how to properly communicate and cooperate for the future. The videos that we saw served as proof that everyone is capable of being successful in math and that mistakes will be made, but shouldn’t be considered as bad. Mistakes are part of learning and it’s just something that will happen constantly.
The first activity that we did was called: Tiling a 11*13 Rectangle. The goal of this activity was to divide the rectangle into squares in the least amount possible. I worked on this problem to various degrees of success, and the lowest amount of squares that I could divide the rectangle in was 6. The second activity that we completed was called Squares to Stairs in which we would find out how many squares it would take to make a stair, and the linear increment that it had. With this information we could find how many squares are in any size stairs. The third activity that we worked on this week was called Hailstone Sequences, which a set of rules that creates a sequence out of any number, usually terminating in 4,2, and 1. The final activity that we looked at was called, Painted Cube. This activity had us determine what sides would get get painted on a 3 by 3 cube, and the individual statistics. The videos were mainly about how the brain works and the different effects that mistakes have. The videos also covered topics such as how timed work has an effect on the brain.
One message that I got out of the videos is that the best kind of learning comes from mistakes and this message has inspired me since I heard it in 9th grade. I first thought that mistakes equaled failure, but they are really just paths to success. Without failures, we can never succeed, we can never improve. And I feel that what we are trying to focus on is improvement. That is why this quote has really stuck out to me because I connected with it and has given me a new mindset. The second message that I took out of those videos is that by believing that you can, it is possible to learn much more. And although it does sound cliched, it is true and is something that has proved itself to me before, I never really understood how it occurred. But now I understand that it all relies on how much you want to learn and this has an immediate correlation with what you actually learn.
The problem that I chose to make my write on, is called: Tiling a 11*13 rectangle. What the task asks to do, is for you to cut the rectangle into squares. The goal of the problem is to cut the rectangle in the least amount of squares. When I tried to solve the problem, the lowest amount that I managed to get was 6. This was by cutting the rectangle into one 7*7 square, one 6*6 square, one 5*5 square, two 4*4 squares, and one 1*1 square. Although I felt that I had a pretty solid solution I felt that this could be done in fewer squares if the rectangle were to be even. Therefor my extension for what I would like to do with this problem is to see whether or not we could divide this rectangle into less squares if the dimensions were to be 10*12. The reason I chose to do this is because it does seem plausible to cut the rectangle into fewer squares.
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When I began working on this problem I decided that my best chance at trying to cut the rectangle into less than 6 squares was to use the method of conjecturing and testing. I first began by trying out a couple of techniques that worked out in the original problem. And because I was using the same techniques, I was getting the same results. From there I decided that it would be best if I started trying out using one large square and filling in the rest of the empty space. This only gave me the result of one rectangle divided into 6 squares, so I continued to work on the problem and see if I could get fewer. By my sixth attempt, I managed to break the rectangle into 5 smaller squares. I achieved this by cutting the rectangle into two 6*6 squares, and three 4*4 squares. Although this was a simple looking task, it did require quite a bit of work and had me stumped for a little while. The biggest challenge that I faced was trying to achieve the 5 squares goal, because for a couple of minutes, I actually believed that it wouldn’t be possible. But after thinking thoroughly about it, I realized that it should be possible and so I kept on working and got to my goal in the end. As for my Habits of a Mathematician, I mainly used conjecturing and testing, in order to find a solution to my extension.
From what I saw this week in my math class, I feel that I am ready for what will be coming in the next year and I am prepared to stay in a healthy mindset that will allow me to learn and succeed. I know that a big part of what we have to do involves teamwork and participation, so I am willing to work with my fellow peers and be active within our class community. Through these efforts I will be able to work my way through the year and finish off strong. |