Increasing Squares
1. Problem Statement
The problem that I will be explaining in this write-up is called Increasing Squares. The problem was a visual open-ended problem about squares. It was a series of visuals, like a series of numbers.The first case starts out as just one square, and every new case number extends another square up/north, left/west, and right/east.
The purpose of this problem was to find the pattern or equation for this problem. There were two questions that we answered to go with this problem: How do you see the shape growing?, and How many blocks were in case 100?
The first question can be answered in many ways and is more of a question to get your thinking process started, while the next question is aimed for you to take the brainstorm you already did and transform it into an equation.
2. Process Description
What I did for this problem was I first looked for all the patterns that I could find between the visuals. I found out that you add three blocks each time, that the case number can be added to a growing number, or the case number could be multiplied by two and then added to a growing number, each would get you the total number of squares in each case.
Next what I did for this problem was that I took what I knew about the patterns and tried to make an equation. It took me awhile to figure out an equation to fit from my patterns that I had noticed. After discovering my own patterns, I worked with my table partners to figure out an equation and get more ideas. Then I came up with my own equation.
After everyone had tried the problem by themselves and with the help of their table, we shared out all of our equations with each other. I found out that there were a lot of things I missed and I found out more equations to the problem.
3. Solution
The equations that worked for this problem were:
x y x = the case # y = the total # of blocks
1 1
2 4
3 7
4 10
5 13
6 16
7 19
8 22
9 25
10 27
You can use this table to look at the relationships between the numbers. This table proves that my equation works if you want to take the time to experiment with all of the numbers. This equation answers the first question, and also the second question if you solve it. The shape is growing by this equation and you can solve for the 100th case with this equation. The answer would be (100*2) + (100-2) which when solved equals 298 blocks. This answer can also be verified by the first equation I had as well. (99*3) + 1 which would also equal 298 blocks in the 100th case.
4. Self-Assessment and Reflection
This problem has taught me to look at the problem in different ways. It also taught me a little on how to work with others at my table group: to share your answers with them and get feedback too. I also found out not only how to find patterns, but how to put those specific patterns into an equation. My equation isn’t a very good example of that though. The only challenging part of this problem for me would be relating my equation back to the squares. I don’t know how that would work, since I used a table to find it and didn’t look at the squares and ask where it was represented on the image. I think that I deserve maybe a 10/10 for this work because I tried pretty hard, collaborated with my group, found an equation/solution to this problem, and also challenged myself to think of how the equation that I found related to the square images.
The Habits of a Mathematician that I used when solving this problem was Collaboration, Looking for patterns, and Seek why and Prove. Collaboration is easy to explain: I had help from my table partners and also gave help and everyone did well because we were asking each other questions. Also Looking for Patterns is easy to understand because the question that we were answering was asking what is the pattern to the growth on the squares. I also used Seek why and Prove because I made a table showing how my equation works and proof of it. What I would have liked to use more next time would be Stay Organized because my work is a little messy and not ‘beautiful work’. I hope to use this more in the future with open-ended problems. It is significant to use this habit because if your work is not organized then no one can read it and learn from it and/or see what I did.
The problem that I will be explaining in this write-up is called Increasing Squares. The problem was a visual open-ended problem about squares. It was a series of visuals, like a series of numbers.The first case starts out as just one square, and every new case number extends another square up/north, left/west, and right/east.
The purpose of this problem was to find the pattern or equation for this problem. There were two questions that we answered to go with this problem: How do you see the shape growing?, and How many blocks were in case 100?
The first question can be answered in many ways and is more of a question to get your thinking process started, while the next question is aimed for you to take the brainstorm you already did and transform it into an equation.
2. Process Description
What I did for this problem was I first looked for all the patterns that I could find between the visuals. I found out that you add three blocks each time, that the case number can be added to a growing number, or the case number could be multiplied by two and then added to a growing number, each would get you the total number of squares in each case.
- Case #2: 2+2, Case #3: 3+4, Case#4: 4+6 ...etc.
- Case #2: (2*2) + 0, Case #3: (3*2) +1, Case #4: (4*2) +2 ...etc.
Next what I did for this problem was that I took what I knew about the patterns and tried to make an equation. It took me awhile to figure out an equation to fit from my patterns that I had noticed. After discovering my own patterns, I worked with my table partners to figure out an equation and get more ideas. Then I came up with my own equation.
After everyone had tried the problem by themselves and with the help of their table, we shared out all of our equations with each other. I found out that there were a lot of things I missed and I found out more equations to the problem.
3. Solution
The equations that worked for this problem were:
- ([C - 1] *3) + 1 = total # of blocks
- (C *2) +(C - 2) = total # of blocks
x y x = the case # y = the total # of blocks
1 1
2 4
3 7
4 10
5 13
6 16
7 19
8 22
9 25
10 27
You can use this table to look at the relationships between the numbers. This table proves that my equation works if you want to take the time to experiment with all of the numbers. This equation answers the first question, and also the second question if you solve it. The shape is growing by this equation and you can solve for the 100th case with this equation. The answer would be (100*2) + (100-2) which when solved equals 298 blocks. This answer can also be verified by the first equation I had as well. (99*3) + 1 which would also equal 298 blocks in the 100th case.
4. Self-Assessment and Reflection
This problem has taught me to look at the problem in different ways. It also taught me a little on how to work with others at my table group: to share your answers with them and get feedback too. I also found out not only how to find patterns, but how to put those specific patterns into an equation. My equation isn’t a very good example of that though. The only challenging part of this problem for me would be relating my equation back to the squares. I don’t know how that would work, since I used a table to find it and didn’t look at the squares and ask where it was represented on the image. I think that I deserve maybe a 10/10 for this work because I tried pretty hard, collaborated with my group, found an equation/solution to this problem, and also challenged myself to think of how the equation that I found related to the square images.
The Habits of a Mathematician that I used when solving this problem was Collaboration, Looking for patterns, and Seek why and Prove. Collaboration is easy to explain: I had help from my table partners and also gave help and everyone did well because we were asking each other questions. Also Looking for Patterns is easy to understand because the question that we were answering was asking what is the pattern to the growth on the squares. I also used Seek why and Prove because I made a table showing how my equation works and proof of it. What I would have liked to use more next time would be Stay Organized because my work is a little messy and not ‘beautiful work’. I hope to use this more in the future with open-ended problems. It is significant to use this habit because if your work is not organized then no one can read it and learn from it and/or see what I did.