Problem Statement
Shown below is a pattern of 'growing' squares made from toothpicks.
It is pretty easy to count to the number of toothpicks in these first three patterns. How many toothpicks are in the 100th pattern? How many toothpicks are in the nth pattern?
What this problem is asking is what is the equation to find out the number of toothpicks it needs to find 100 squares, or any number of squares in general.
Process Description
When I first saw this pattern, I immediately saw that each square pattern was that of a perfect square, also known as a number squared. I made a mental note to myself to keep that in mind. I then tried to find the pattern to find the amount of toothpicks, which was no easy feat. The first method I tried to incorporate into the mathematic situation was to multiply the number of sides by four, because there are four sides in a square, and to divide the number of squares inside the previous square (when I say this, I mean that if it is the second square in the pattern we are looking for, we must look at the first square and see how many squares are inside the outline of the square [which is one]) by two. After this was done, you would need multiply the larger amount of half by two and add it to the sides multiplied. For example, to find the fourth square (as in the number of toothpicks in the fourth square) I would multiply four by four (sides), divide nine by two (round up so it is four and five), multiply five by four, and add four (other number from when we divided). The number of toothpicks would be forty, four multiplied by four is sixteen, five multiplied by four would be twenty, and add four to it overall would be forty.
However, this method does not work, as soon as I tried amalgamate this technique with the fifth sequence, it had no effect and I had received a false answer. After this one of my classmates helped me to understand the true equation that will find out how many toothpicks will be in each perfect square. Although she helped me, I still had a difficult time understanding the fashion of it. It was not until recently, that I understood what she was implying. The system for doing this would be 2n(n+1) = # of toothpicks.
Solution
With the equation, 2n(n+1) = # of toothpicks, I would be able to find the number of toothpicks for any sequence in the pattern even the 100th one. The number of toothpicks for the 100th one would be 20,200. I know this because inserting 100 where n is would give me the equation 2(100)[(100)+1] and when solved through it is the same as writing 20,200. Two multiplied by one hundred would be two hundred, one hundred added to one would be one hundred and one, and two hundred multiplied by one hundred and one would e twenty thousand, two hundred toothpicks.
Reflection
When doing this problem, I had a difficult time. This would be the most difficult open-ended problem I have done yet. At first glance I thought it would be easy like the others, but this one just was't right. I continued to have problems with it when I realized the method I used didn't work. Even then I continued to pursue this problem, through much difficulty. It wasn't until my classmate helped me did it began to make even a hint of sense. But overall, I never gave up, this is why I believe I deserve ten points.
Significance - This is a Habit of Mind that represented me during this because I understand the importance of projects and activities. The point of doing these kinds of problems is to understand how we think when given a problem and how we can improve or enhance the way we think. This is important for everyday life because we have problems everyday, some not as direct as the problem shown above, but we need to know how to solve it correctly and efficiently because we may not be the only ones affected by the outcome. Such as a doctor, if you were suddenly told that you didn't have the proper size of a material and only a size bigger and smaller than the required size, you would need to know how to fix the situation. This is why I think I used the Habit of Mind in this open-ended problem.
Shown below is a pattern of 'growing' squares made from toothpicks.
It is pretty easy to count to the number of toothpicks in these first three patterns. How many toothpicks are in the 100th pattern? How many toothpicks are in the nth pattern?
What this problem is asking is what is the equation to find out the number of toothpicks it needs to find 100 squares, or any number of squares in general.
Process Description
When I first saw this pattern, I immediately saw that each square pattern was that of a perfect square, also known as a number squared. I made a mental note to myself to keep that in mind. I then tried to find the pattern to find the amount of toothpicks, which was no easy feat. The first method I tried to incorporate into the mathematic situation was to multiply the number of sides by four, because there are four sides in a square, and to divide the number of squares inside the previous square (when I say this, I mean that if it is the second square in the pattern we are looking for, we must look at the first square and see how many squares are inside the outline of the square [which is one]) by two. After this was done, you would need multiply the larger amount of half by two and add it to the sides multiplied. For example, to find the fourth square (as in the number of toothpicks in the fourth square) I would multiply four by four (sides), divide nine by two (round up so it is four and five), multiply five by four, and add four (other number from when we divided). The number of toothpicks would be forty, four multiplied by four is sixteen, five multiplied by four would be twenty, and add four to it overall would be forty.
However, this method does not work, as soon as I tried amalgamate this technique with the fifth sequence, it had no effect and I had received a false answer. After this one of my classmates helped me to understand the true equation that will find out how many toothpicks will be in each perfect square. Although she helped me, I still had a difficult time understanding the fashion of it. It was not until recently, that I understood what she was implying. The system for doing this would be 2n(n+1) = # of toothpicks.
Solution
With the equation, 2n(n+1) = # of toothpicks, I would be able to find the number of toothpicks for any sequence in the pattern even the 100th one. The number of toothpicks for the 100th one would be 20,200. I know this because inserting 100 where n is would give me the equation 2(100)[(100)+1] and when solved through it is the same as writing 20,200. Two multiplied by one hundred would be two hundred, one hundred added to one would be one hundred and one, and two hundred multiplied by one hundred and one would e twenty thousand, two hundred toothpicks.
Reflection
When doing this problem, I had a difficult time. This would be the most difficult open-ended problem I have done yet. At first glance I thought it would be easy like the others, but this one just was't right. I continued to have problems with it when I realized the method I used didn't work. Even then I continued to pursue this problem, through much difficulty. It wasn't until my classmate helped me did it began to make even a hint of sense. But overall, I never gave up, this is why I believe I deserve ten points.
Significance - This is a Habit of Mind that represented me during this because I understand the importance of projects and activities. The point of doing these kinds of problems is to understand how we think when given a problem and how we can improve or enhance the way we think. This is important for everyday life because we have problems everyday, some not as direct as the problem shown above, but we need to know how to solve it correctly and efficiently because we may not be the only ones affected by the outcome. Such as a doctor, if you were suddenly told that you didn't have the proper size of a material and only a size bigger and smaller than the required size, you would need to know how to fix the situation. This is why I think I used the Habit of Mind in this open-ended problem.