Purpose of this week's investigations/video watching:
I believe that the purpose of this week of investigations and video-watching was first of all, to clear up what actually happens when you make mistakes and realize that you made a mistake, even if some of us were already aware. Some people, when approaching a problem are hesitant because they think that by troubleshooting and trying various methods, no growth will be made at all because they aren’t necessarily finding reliable methods for solving. This being said, I feel like this week’s videos were to re-introduce the fact that brain growth occurs even when reliable problem solving methods aren’t being found at the moment.
Activities/videos overview:
For our first task, we started off with the 11x13 rectangle problem, where we used troubleshooting/patterns to discover what the smallest number of squares possible you could find in the rectangle was. What me and my table group came up with was that the smallest amount of squares able to be found was 6, unless you divided cubic units and had squares without all whole numbers. The next activity we worked on was the “squares to stairs” problem, where we were shown the beginning of a pattern, shown as cubic units in the form of stairs. The first image shows one cubic unit, then three cubic units in the form of stairs, then six cubic units as a slightly larger form of stairs. We were given different questions, such as “how many units will the tenth figure have?” and “Can a figure be made up of 190 units?”. We were able to find numerous patterns, one being that for each advancement of a figure, it would increase by whatever number figure it was. For reference, you can look at the image above showing my work process/methods. In the third activity, we went over a mysterious mathematical pattern known as hailstone sequences. Even though these patterns, also shown above, may seem arbitrary or unpredictable, they follow a particular form. You may start off your sequence with any random number of your choice. Next, if the number is even, you divide it in half. If the number is odd, you multiply by three and add one to keep the pattern going. If the number remains even, you continue dividing in half. If the number reaches a point of no longer being even, you continue to multiply by three and add one. While exploring this pattern, we thought of hypothetical questions such as what would happen if you started a sequence with a negative number. The final math activity we worked on during the week was the painted cube problem, where we found out how many sides of the individual units of a 3 by 3 cube would be covered in paint if the entire cube was covered in paint. This problem was an example of a visual-related problem where you need to think by using mainly visuals.
Two messages I got from the daily videos:
Every day during the week, we would watch a new video based on brain-growth and what happens when you learn new concepts. One message that I was definitely able to take away from the videos was that no one is born as a "math person" or not. Your level of mathematical capabilities is entirely dependent on your will and desire to learn, and your belief in your own abilities. I think that this message had an impact on me because I am usually not one to believe in my math abilities, and after seeing that this limits your brain growth even more, I now have more of a desire to go into a problem with an open mind. Another message I got from the five videos we watched was that believing in yourself as soon as you start a problem increases your chances of finding a reliable method.
Hailstone Sequences: Reflection on the week of inspirational math:
Hailstone sequences are sequences of numbers, also shown above, that may seem arbitrary or unpredictable, but they follow a particular form. You may start off your sequence with any random number of your choice. Next, if the number is even, you divide it in half. If the number is odd, you multiply by three and add one to keep the pattern going. If the number remains even, you continue dividing in half. If the number reaches a point of no longer being even, you continue to multiply by three and add one.
I chose to expand on this 'problem' because I had a strong curiosity as to what would happen if you started your sequence with a negative number. Even if we asked this hypothetical question after experimenting with this problem, I needed to find out what would happen, and this was the perfect opportunity to do so.
I decided to create five different sequences all starting out with a negative number, and made sure that each number differed a resonable amount. The sequences I came up with are shown in the image below. A definite difference between starting off with positive numbers and starting with negative numbers is that when you have an odd number that needs to be multiplied by three and have one added to it, the negative number appears to have had one number subtracted from it instead of being added because a negative number plus one is just one closer to being a positive number. The most prominent thing that I noticed after the first couple of sequences was that not after a long time, the pattern starts repeating itself over and over. In the first example, I started off with a larger number (-4) and it took no time for the pattern to start repeating itself. In the last example where I started off with -37 though, it took longer to make the full circle back to -37. Every time I created a new sequence with a negative number, though, eventually the numbers would just start repeating until it was pointless to continue.
One challenge I faced while working on this problem was seeking why and proving because I'm almost sure that there are many meanings behind the patterns within these sequences, but I don't feel confident in this area, and finding out what these patterns mean could lead me to important conclusions about a problem. A habit of mathematicians that I used in working on this problem was finding patterns because this was the main element that made these sequences interesting. The main thing that I was doing during the course of this problem was looking for sequences.
I believe that the purpose of this week of investigations and video-watching was first of all, to clear up what actually happens when you make mistakes and realize that you made a mistake, even if some of us were already aware. Some people, when approaching a problem are hesitant because they think that by troubleshooting and trying various methods, no growth will be made at all because they aren’t necessarily finding reliable methods for solving. This being said, I feel like this week’s videos were to re-introduce the fact that brain growth occurs even when reliable problem solving methods aren’t being found at the moment.
Activities/videos overview:
For our first task, we started off with the 11x13 rectangle problem, where we used troubleshooting/patterns to discover what the smallest number of squares possible you could find in the rectangle was. What me and my table group came up with was that the smallest amount of squares able to be found was 6, unless you divided cubic units and had squares without all whole numbers. The next activity we worked on was the “squares to stairs” problem, where we were shown the beginning of a pattern, shown as cubic units in the form of stairs. The first image shows one cubic unit, then three cubic units in the form of stairs, then six cubic units as a slightly larger form of stairs. We were given different questions, such as “how many units will the tenth figure have?” and “Can a figure be made up of 190 units?”. We were able to find numerous patterns, one being that for each advancement of a figure, it would increase by whatever number figure it was. For reference, you can look at the image above showing my work process/methods. In the third activity, we went over a mysterious mathematical pattern known as hailstone sequences. Even though these patterns, also shown above, may seem arbitrary or unpredictable, they follow a particular form. You may start off your sequence with any random number of your choice. Next, if the number is even, you divide it in half. If the number is odd, you multiply by three and add one to keep the pattern going. If the number remains even, you continue dividing in half. If the number reaches a point of no longer being even, you continue to multiply by three and add one. While exploring this pattern, we thought of hypothetical questions such as what would happen if you started a sequence with a negative number. The final math activity we worked on during the week was the painted cube problem, where we found out how many sides of the individual units of a 3 by 3 cube would be covered in paint if the entire cube was covered in paint. This problem was an example of a visual-related problem where you need to think by using mainly visuals.
Two messages I got from the daily videos:
Every day during the week, we would watch a new video based on brain-growth and what happens when you learn new concepts. One message that I was definitely able to take away from the videos was that no one is born as a "math person" or not. Your level of mathematical capabilities is entirely dependent on your will and desire to learn, and your belief in your own abilities. I think that this message had an impact on me because I am usually not one to believe in my math abilities, and after seeing that this limits your brain growth even more, I now have more of a desire to go into a problem with an open mind. Another message I got from the five videos we watched was that believing in yourself as soon as you start a problem increases your chances of finding a reliable method.
Hailstone Sequences: Reflection on the week of inspirational math:
Hailstone sequences are sequences of numbers, also shown above, that may seem arbitrary or unpredictable, but they follow a particular form. You may start off your sequence with any random number of your choice. Next, if the number is even, you divide it in half. If the number is odd, you multiply by three and add one to keep the pattern going. If the number remains even, you continue dividing in half. If the number reaches a point of no longer being even, you continue to multiply by three and add one.
I chose to expand on this 'problem' because I had a strong curiosity as to what would happen if you started your sequence with a negative number. Even if we asked this hypothetical question after experimenting with this problem, I needed to find out what would happen, and this was the perfect opportunity to do so.
I decided to create five different sequences all starting out with a negative number, and made sure that each number differed a resonable amount. The sequences I came up with are shown in the image below. A definite difference between starting off with positive numbers and starting with negative numbers is that when you have an odd number that needs to be multiplied by three and have one added to it, the negative number appears to have had one number subtracted from it instead of being added because a negative number plus one is just one closer to being a positive number. The most prominent thing that I noticed after the first couple of sequences was that not after a long time, the pattern starts repeating itself over and over. In the first example, I started off with a larger number (-4) and it took no time for the pattern to start repeating itself. In the last example where I started off with -37 though, it took longer to make the full circle back to -37. Every time I created a new sequence with a negative number, though, eventually the numbers would just start repeating until it was pointless to continue.
One challenge I faced while working on this problem was seeking why and proving because I'm almost sure that there are many meanings behind the patterns within these sequences, but I don't feel confident in this area, and finding out what these patterns mean could lead me to important conclusions about a problem. A habit of mathematicians that I used in working on this problem was finding patterns because this was the main element that made these sequences interesting. The main thing that I was doing during the course of this problem was looking for sequences.
Reflection on the week of inspirational math:
I think that starting off the year with this one week of focused, concentrated, problem solving was beneficial towards warming up our mathematical mind set. The videos we watched during the week were also a helpful refresher because they encouraged me to have a more positive outlook on math. I think that overall this week, my performance wasn't the best that it could've been, but I did succeed on thinking deeply about and completing each math concept. I definitely could've participated more by volunteering to share what I had (there were times that I had something to share and chose not to) or raising my hand, but I feel like now that I've gotten a little warmed up to the year, I'll actively try to participate more when I know that I have something worth sharing.
I think that starting off the year with this one week of focused, concentrated, problem solving was beneficial towards warming up our mathematical mind set. The videos we watched during the week were also a helpful refresher because they encouraged me to have a more positive outlook on math. I think that overall this week, my performance wasn't the best that it could've been, but I did succeed on thinking deeply about and completing each math concept. I definitely could've participated more by volunteering to share what I had (there were times that I had something to share and chose not to) or raising my hand, but I feel like now that I've gotten a little warmed up to the year, I'll actively try to participate more when I know that I have something worth sharing.