Growing Tiles
1. Problem Statement
Question: How do you see the shape growing?
Our first official assignment of 10th grade math was a project called "Growing Tiles". We started out working on in by ourselves then advancing to working within our table groups to solve the problem. For the problem, there was 4 cases of tiles. Case 1 had 1 tile (or square). Case 2 had that same tile with 1 tile on each side- the left, right and top of the middle tile, with 4 tiles in total.
Question: How do you see the shape growing?
Our first official assignment of 10th grade math was a project called "Growing Tiles". We started out working on in by ourselves then advancing to working within our table groups to solve the problem. For the problem, there was 4 cases of tiles. Case 1 had 1 tile (or square). Case 2 had that same tile with 1 tile on each side- the left, right and top of the middle tile, with 4 tiles in total.
Case 3 had the same tiles as Case 2, with one more tile on the left, right and top of the original tiles. This continues for the next Cases. The question we needed to answer was"How do you see the shape growing?" How did Case 2 change from Case 1? What is the pattern these cases follow? Is there an equation that best defines how the tiles grow?
The second part of the problem was to answer the question: How many tiles are there in Case 100?
2. Process Description
I started by noticing that each time there was a new case, 3 more were added on. They were added onto the right, left and top of the tiles (pic 2). So basically Case 1 + 3 = Case 2 (pic 1). Case 1 has one tiles, then add 3 tiles, you get 4 tiles which is the amount of tiles Case 2 has. This will always hold true, and this answered the first question of the problem.
The second part of the problem was to answer the question: How many tiles are there in Case 100?
2. Process Description
I started by noticing that each time there was a new case, 3 more were added on. They were added onto the right, left and top of the tiles (pic 2). So basically Case 1 + 3 = Case 2 (pic 1). Case 1 has one tiles, then add 3 tiles, you get 4 tiles which is the amount of tiles Case 2 has. This will always hold true, and this answered the first question of the problem.
I then I had to answer the second part of the problem. I realized that it would be very difficult to answer the second question by going with the equation Case (x) + 3 = Case (z). I would have to figure out the amount of tiles in each Case up until Case 100. I had to find a quicker way.
Now I noticed that we had two categories: the Case number and the amount of tiles. I started to compare the two categories. I labeled the Case number (x) and the amount of tiles (y). I created a chart to find the difference between x and y, which is shown below (pic 3). How did we get from x to y? I had the first column as the Case number or x. The second column was the number of tiles or y. The last column was how we got from x to y, the amount we added, labeled as (w).
I then noticed that the last column (w) was evenly consecutive: 0, 2, 4, 6, 8 (pic 6). Then, if you scoot the c column up one or in other words add 2 to each number you have a pattern. The pattern is the case number (x) times 2 will equal the difference between x and y, or the last column, (w) (pic 5).
Now I noticed that we had two categories: the Case number and the amount of tiles. I started to compare the two categories. I labeled the Case number (x) and the amount of tiles (y). I created a chart to find the difference between x and y, which is shown below (pic 3). How did we get from x to y? I had the first column as the Case number or x. The second column was the number of tiles or y. The last column was how we got from x to y, the amount we added, labeled as (w).
I then noticed that the last column (w) was evenly consecutive: 0, 2, 4, 6, 8 (pic 6). Then, if you scoot the c column up one or in other words add 2 to each number you have a pattern. The pattern is the case number (x) times 2 will equal the difference between x and y, or the last column, (w) (pic 5).
When working with my group, I was the first one to come up with an equation. We didn't really throw out any other ideas on how to solve the problem. We all just went along with my equation. I took my group through my process and help them understand what I had done.
3. Solution
After going through that lengthy process of trying to find an equation, I finally made a breakthrough.
I then got the equation: the case number (plus 2 of the case number minus 2) equals the amount of tiles in the case. Or in simpler terms: x+(2x-2)=y (pic 4). (The word after "of", in the picture, is the world "blocks"). I got this by comparing the case number (x) to the amount of tiles (y), creating a chart with that information and the difference between the two categories (w). I then saw that if you add 2 to the w value, x times 2 would equal w. But, in order to get the correct number of tiles for the specific case, you have to subtract 2. This is where I got the 2x-2 part. But now we just have the difference between x and y. We must add back the x to equation to get y. Therefore we have: x=(2x-2)=y.
To make absolute sure that this actually works, plug in a number for the x value. Let's try 3, since we know the answer to it.
3+(2(3)-2)=y.
3+(6-2)=y.
3+(4)=y.
7=y.
Now, lets check the answer in the chart. Yep, it's seven. Let's try one more, the number 5=x.
5+(2(5)-2)=y.
5+(10-2)=y.
5+(8)=y.
13=y.
This is true, just look at the chart.
Then I had to answer the question "How many blocks are in case 100?" Because I had the equation, the problem would be easy to solve: just plug 100 into "x".
100+(2(100)-2)=y.
100+(200-2)=y.
100+(198)+y.
298=y.
3. Solution
After going through that lengthy process of trying to find an equation, I finally made a breakthrough.
I then got the equation: the case number (plus 2 of the case number minus 2) equals the amount of tiles in the case. Or in simpler terms: x+(2x-2)=y (pic 4). (The word after "of", in the picture, is the world "blocks"). I got this by comparing the case number (x) to the amount of tiles (y), creating a chart with that information and the difference between the two categories (w). I then saw that if you add 2 to the w value, x times 2 would equal w. But, in order to get the correct number of tiles for the specific case, you have to subtract 2. This is where I got the 2x-2 part. But now we just have the difference between x and y. We must add back the x to equation to get y. Therefore we have: x=(2x-2)=y.
To make absolute sure that this actually works, plug in a number for the x value. Let's try 3, since we know the answer to it.
3+(2(3)-2)=y.
3+(6-2)=y.
3+(4)=y.
7=y.
Now, lets check the answer in the chart. Yep, it's seven. Let's try one more, the number 5=x.
5+(2(5)-2)=y.
5+(10-2)=y.
5+(8)=y.
13=y.
This is true, just look at the chart.
Then I had to answer the question "How many blocks are in case 100?" Because I had the equation, the problem would be easy to solve: just plug 100 into "x".
100+(2(100)-2)=y.
100+(200-2)=y.
100+(198)+y.
298=y.
The last part of the problem our teacher wanted us to solve was how to relate our equation back to the diagram. This was very tricky for me. I didn't really use the diagram to solve it. I used my head and charts. I worked a lot with my group members to figure it out, it took many tries. Finally, we came to a solution.
Take the diagram we were given (pic 7). Look at case number 2. Add 1 tile on both the left and the right of Case 2 (the tile with the red x in it). The middle column has 2 tiles in it, just like the Case number. This middle column going vertically will be the Case number or the x in the equation (the tiles inside the green oval) (pic 8). With one tile added on the left and right, the left and right sides have 2 tiles in it, just like the middle column or x. The left and right columns represent the 2x in the equation. Where does the minus two come in? Now, we have to subtract the tile on the left and right to get the diagram to its original form, in other words this is the negative 2.
Take the diagram we were given (pic 7). Look at case number 2. Add 1 tile on both the left and the right of Case 2 (the tile with the red x in it). The middle column has 2 tiles in it, just like the Case number. This middle column going vertically will be the Case number or the x in the equation (the tiles inside the green oval) (pic 8). With one tile added on the left and right, the left and right sides have 2 tiles in it, just like the middle column or x. The left and right columns represent the 2x in the equation. Where does the minus two come in? Now, we have to subtract the tile on the left and right to get the diagram to its original form, in other words this is the negative 2.
4. Self-Assessment and Reflection
I have learned that a big part of mathematics is truly thinking outside the box. Many times people don't associate those two terms together, but you'd be surprised how often they co-exist. I learned this last year through combining math and arts, but this was more along the lines of creative thinking, not necessarily creative doing.
Why I say that my thinking in this problem was outside the box was the fact that no one in our class or team came up with this equation. Our teacher, himself said that no one had this equation. Most people would use pictures to guide them to an equation or solution for this problem. I, however didn't, I used charts instead. Also, I saw that the w value was evenly consecutive. I saw that if you added 2 to the w value, it would make 2x=w true, which I feel that my class didn't really understand.
Another reason I say that is that when my group and I were trying to solve the part about connecting the equation back to the diagram, we had a really hard time. What I finally figured out what was that you had to add imaginary tiles to the left and right of the case. I really had to stretch my thinking to solve this, to really go "out there". i just had to think.
I'd assign myself a 9.5 or 10. I started out the problem thinking it was simple- you just add 3 to the amount of tiles in the case to get the amount of tiles for the next case. I realized that it'd take forever to find the amount of tiles in case 100. Then I started the process to find an equation. With some time and outside the box thinking I devised an equation which i described above. I got this equation fairly quickly and easily. I then assisted my group members in helping them understand the equation I figured out. Though I may not always like math I feel that i"m pretty good at it. The tricky part came next: relating the equation back to the diagram. For this part, I did come up with the final answer to explain the equation through visuals, but I didn't get their myself. My group members and I pushed and helped each other, throwing out ideas and suggestions to finally get our long awaited solution. It took many tries, all which didn't work, but got us continually closer to the answer. I'm very proud that our group kept going and never stopped. We didn't give up, though it seemed like we couldn't get the solution. We persisted and did find the answer.
One of my habits of a mathematician was Looking for Patterns. I feel as though I embodied this, because really when you're looking for an equation, what else are you doing? As I mentioned above, I compared x to y and looked for patterns that way. I did find patterns and continued on path until I got my final answer. I found that the difference between and y value was evenly consecutive and that if you add two to the w value 2x=w. Patterns definitely helped me solve this problem.
Another habit I think I use in solving the problem was Staying Organized, which is almost always against my nature. But, in this case I used it by creating charts to help me understand the process I was doing. This, in turn, not only helped me but others as well. I made a chart when I compared x to y and when I added 2 to the w value, so I could see the process. This really helped me understand what I was doing so I could follow well.
I have learned that a big part of mathematics is truly thinking outside the box. Many times people don't associate those two terms together, but you'd be surprised how often they co-exist. I learned this last year through combining math and arts, but this was more along the lines of creative thinking, not necessarily creative doing.
Why I say that my thinking in this problem was outside the box was the fact that no one in our class or team came up with this equation. Our teacher, himself said that no one had this equation. Most people would use pictures to guide them to an equation or solution for this problem. I, however didn't, I used charts instead. Also, I saw that the w value was evenly consecutive. I saw that if you added 2 to the w value, it would make 2x=w true, which I feel that my class didn't really understand.
Another reason I say that is that when my group and I were trying to solve the part about connecting the equation back to the diagram, we had a really hard time. What I finally figured out what was that you had to add imaginary tiles to the left and right of the case. I really had to stretch my thinking to solve this, to really go "out there". i just had to think.
I'd assign myself a 9.5 or 10. I started out the problem thinking it was simple- you just add 3 to the amount of tiles in the case to get the amount of tiles for the next case. I realized that it'd take forever to find the amount of tiles in case 100. Then I started the process to find an equation. With some time and outside the box thinking I devised an equation which i described above. I got this equation fairly quickly and easily. I then assisted my group members in helping them understand the equation I figured out. Though I may not always like math I feel that i"m pretty good at it. The tricky part came next: relating the equation back to the diagram. For this part, I did come up with the final answer to explain the equation through visuals, but I didn't get their myself. My group members and I pushed and helped each other, throwing out ideas and suggestions to finally get our long awaited solution. It took many tries, all which didn't work, but got us continually closer to the answer. I'm very proud that our group kept going and never stopped. We didn't give up, though it seemed like we couldn't get the solution. We persisted and did find the answer.
One of my habits of a mathematician was Looking for Patterns. I feel as though I embodied this, because really when you're looking for an equation, what else are you doing? As I mentioned above, I compared x to y and looked for patterns that way. I did find patterns and continued on path until I got my final answer. I found that the difference between and y value was evenly consecutive and that if you add two to the w value 2x=w. Patterns definitely helped me solve this problem.
Another habit I think I use in solving the problem was Staying Organized, which is almost always against my nature. But, in this case I used it by creating charts to help me understand the process I was doing. This, in turn, not only helped me but others as well. I made a chart when I compared x to y and when I added 2 to the w value, so I could see the process. This really helped me understand what I was doing so I could follow well.