Growing Tiles
We were presented with an open ended problem called Growing Tiles. For this problem, we were given a diagram of four different stages of a pattern, case one, case two, case three, and case four. We were also presented with the question of how do you see the shape growing, and how many blocks would be in case 100. The first case is a picture of a single square block. The second case is a picture of three blocks wide and one block going up in the middle. The third case is five blocks for the width and two blocks going up in the middle. The fourth case has a total of seven on the bottom and three going up from the middle block. The pattern grew with each stage, and there seems to be a constant growth of the pattern, one block growing from the top, the right side, and the left side each stage. Here is a picture of the diagram, as well as my work on the problem, which includes tables, patterns, and of course my equations.
When first presented with this problem, I saw the growth almost immediately. It was obvious that the pattern was growing one block from the top, right, and left sides. We were then asked how many blocks would be in the 100th case. I first tried just drawing out more stages of the pattern, but that wasn’t really getting me anywhere, and it also took up a lot of time. My classmates and I then tried to put everything into an in and out table, but my thought process didn’t fully understand the table. So then I decided to come up with a simple equation that fit the problem. The equation I came up with was case n x 3 - 2 = y. So I plugged in 100 as the case number, multiplied it by three (the number of growing legs in the pattern) and subtracted it by 2 for the two constants that would not be used (there is only one constant block). The next part of the problem was figuring out an equation that could fit for figuring out how many blocks were in x pattern, and since I already had an equation that could fit any case number, I challenged myself to figure out a different equation that would also fit the problem. This was also when we were able to work with the rest of our group members, and we came up with the equation (x-1) x 3 +1= y. This was very similar to my original equation. X represents the case number, and you subtract 1 which represents the constant. You then multiply it by 3 to represent the three legs of the pattern, and finally you add one to represent adding the constant back into the equation. Out of all the methods I liked coming up with the equation the best because it was simple, easy, and could work with any case number without much effort.
The solution ended up being our two final equations, case n x 3 - 2 = y and (x-1) x 3 +1= y, you could use either one to solve the problem. You could prove that it is correct by switching out any of the three variables, (case number, number of legs, or the number of constant blocks). If you were to use case number 2, have four constant blocks, and six legs, the equation would still work. So the equation would look like this: 2 (case number) x 6 (number of legs) - 3 (constants) = 9 (total number of blocks). If you were to draw out case number 2 with these different variables, and counted the number of blocks, you would get the same answer. So any numbers you use would always give you the correct answer.
I never really realized how much open ended problems make you really think about all aspects of a problem, and I had to open my mind up to a lot of different ways of completing this problem. I learned that there are many ways you can change the variables in a problem to change the numbers in an equation completely, but it will still get you the correct answer. We also learned about growing and constant variables, as well as patterns. I would give myself a 8/10 for this problem, because I feel like I fully understand the problem and I could explain it to a peer, but I also could've done neater and nicer work to make the problem really clear and I also could have explored even more ways of figuring out the problem. The two habits of a mathematician I used during this POW were being systematic and visualizing. During the process, i changed a lot of the variables of the original problem out just to see how it would affect the answer and equation. By doing this it gave me a better understanding of the problem and gave me a more open mind about other problems like this one. I used visualizing by making more stages of the pattern to get a better representation of what I was dealing with as well as making charts and tables to organize and display the information I collected in a clean and neat fashion. So all in all I felt like I learned a lot from this problem about a variety of different things, and I hope to become better at solving open ended problems in the future.
The solution ended up being our two final equations, case n x 3 - 2 = y and (x-1) x 3 +1= y, you could use either one to solve the problem. You could prove that it is correct by switching out any of the three variables, (case number, number of legs, or the number of constant blocks). If you were to use case number 2, have four constant blocks, and six legs, the equation would still work. So the equation would look like this: 2 (case number) x 6 (number of legs) - 3 (constants) = 9 (total number of blocks). If you were to draw out case number 2 with these different variables, and counted the number of blocks, you would get the same answer. So any numbers you use would always give you the correct answer.
I never really realized how much open ended problems make you really think about all aspects of a problem, and I had to open my mind up to a lot of different ways of completing this problem. I learned that there are many ways you can change the variables in a problem to change the numbers in an equation completely, but it will still get you the correct answer. We also learned about growing and constant variables, as well as patterns. I would give myself a 8/10 for this problem, because I feel like I fully understand the problem and I could explain it to a peer, but I also could've done neater and nicer work to make the problem really clear and I also could have explored even more ways of figuring out the problem. The two habits of a mathematician I used during this POW were being systematic and visualizing. During the process, i changed a lot of the variables of the original problem out just to see how it would affect the answer and equation. By doing this it gave me a better understanding of the problem and gave me a more open mind about other problems like this one. I used visualizing by making more stages of the pattern to get a better representation of what I was dealing with as well as making charts and tables to organize and display the information I collected in a clean and neat fashion. So all in all I felt like I learned a lot from this problem about a variety of different things, and I hope to become better at solving open ended problems in the future.