Week of inspirational maths
Purpose of inspirational maths week: |
Throughout our week of inspirational maths, we learned from visuals such as group activities and videos, all geared towards brain growth and finding solutions from group work. In on of the first videos we watched, we tackled the topic of brain growth, and how synapses fire as we struggle as well as when we recognize our mistakes. After this week, the biggest takeaway for me is that there are many ways to get to one answer and that the work can be even more important than the answer.
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Week overview: |
As a class we engaged in investigations such as "Hailstone sequence", "squares to stairs", and "fewest squares." Through these team building exercises as well as individual work, we became comfortable sharing our ideas and collaborating with our peers. In "squares to stairs", we were given a series of growing cubes that formed a stair case structure. As a class we discussed different ways to discover patterns as well as how to create shortcuts when working with large sets of numbers or data. In "hailstone sequence" we were given a series of numbers, and after identifying a pattern (if the number was even, you divide by two. If the number was odd, you multiply by three and add one) we began to conjecture and test, testing if we could come to an overall conclusion about the pattern. The first activity we did was "fewest squares", where we had to find the fewest amount of perfect squares that would fit in a 11x13 rectangle. And finally, in "painted cubes", we took a 3x3 cube and created a visual to showcase the amount of cubes would have three sides covered in painted if the cube was coated in it. We then expanded on the problem and created a in and out table to figure out how many cubes had two and one side(s) covered. In the videos we watched, we grew a better understanding of maths misconceptions, such as the myth that some people are born math people, while others are not. These simplistic, educational videos informed us in a engaging way.
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Video takeaways: |
In the video titled "mistakes are powerful", we learned that when we make mistakes, synapses fire and our brains grow, then again when we realize our mistakes. This really struck me because throughout my elementary education, I had always thought that mistakes were bad and shouldn't be celebrated. this really changed the way I viewed making mistakes and celebrating the recognition of those mistakes. Another video that had a big takeaway for me was " brains think of maths visually" where we learned about how you shouldn't be discouraged from using your hands to count or drawing visuals to understand a concept. As a hands on and visual learner, this was a big takeaway from this video because I have always been a vivid note taker as well as having visual aids when solving problems, and to have the video confirm that it's good to do that was a big relief. Now when I'm in the classroom I will count on my fingers with pride.
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Extended problem write up: |
For my extended problem, I chose to dig deeper into the "hailstone sequence" activity. In this activity, we were given the number sequence of 20,10,5,16,8,4,2,1. After trying to discover patterns, we concluded that the common pattern for this number set was that if the number was even, then you divide by two, where of it was an odd number, you multiply by three and add one to the sum. Then in groups we began to find a conjecture: a rule that would apply to the pattern no matter what the numbers. After starting with numbers like 44, 160, and 90, our group realized that eventually, all sequences would end with the rotation of 4, 2,1,4,2,1…. and so on. It was included that if you ended up at 16 in your sequence, then naturally the next one would be 8, then 4, causing the loop to begin. I chose this problem to extend on because I'm amazed by how numbers can fall into patterns and can have rules to them that generalizes for that specific pattern. When deciding what problem I wanted to expand on, I figured it would be interesting to experiment with number patterns and different rules, given theres millions of potential combinations. The first approach I took to solving the expanded problem was to create a new rule and summaries the pattern. My first attempt was to simply double the existing pattern, so the new rule would be that if your number was even, you divide by 4 and if it was odd, you multiply by 6 and add 2 to the sum. I also decided to start with the first number as the original sequence: 20. As a result, the number were almost a complete match, ending with a loop just like the first sequence, except it looked like 8, 2, 0.5, 8, 2, 0.5… looping over and over.I repeated this rule two more times, starting with 80 and 64. My conclusion was that if you started with an even number, your sequence would end up looping. I reminded myself that I could find a pattern with all whole numbers, and started with a new rule. The new rule was a little different, and I hoped that by branching out I might strike gold. If the number was even, you multiply by 3, if the number is odd, you divide by 3 and minus 1 from the sum. The results of this were interesting. With my first sequence this round starting at 100, it quickly became clear that I had made too much "wiggle room" for the even numbers meaning that when you multiply them by three, the range goes up and up, while dividing them by three and subtracting one didn't have a bigger impact which meant instead of the numbers getting smaller and smaller until they loop, they get bigger and bigger until I finally decided to stop at 209,952. Feeling a little defeated, i decided to try the rule once more starting with 62. I'm sure you can already predict the outcome: 62, 186, 558, 1674, 5022, 15066,…. and so on, each number getting bigger and bigger. My final conclusion was that if you don't give the numbers enough "room" to shrink, meaning not dividing them by bigger numbers, then they can't shrink into smaller numbers that will end up looping themselves.
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Challenge and HOAMM: |
One challenge I faced while creating new rules for this problem as well as finding a sequence that worked was trying to find a balance of what numbers to start with and how much to divide the numbers by. If you cut down the numbers too much, they became too small like 0.5, 0.1, etc. But if you didn't divide them by enough, you soon had a sequence of big whole numbers such as 79,343. To find the right amount to keep the pattern going/ finding a pattern was challenging when there were so many possibilities. I overcame this challenge by reminding myself that mathmaticians like to find patterns and solve equations for the sake of solving and finding patterns, and that I have the ability to conduct work much like that too. One habit of a mathematician I used when solving this problem was conjecture and test, since with a plethora of numbers at my disposal, I had to test different scenarios as well as test what numbers worked with the rule and which ones didn't. not to mention trying to figure out why those numbers wouldn't create a pattern.
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reflection: |
during this week of inspirational maths, I saw myself look at maths in a whole new way. I found myself realizing that I could be a maths person, and that my fear of maths could be all physiological due to teachers telling me some people are born maths people and some are not when I was younger. During group work I felt right at home since I have always been a visual, hands on learner. Looking into the rest of the year, I want to put all my effort into each problem I do and use the group roles as a guide for asking my peers for clarifying questions and help. Whenever I see a challenging problem, my mind will now know that time and speed doesn't matter, and that if I do make a mistake, that when I realize I made one I will have brain growth and a chance to correct my problem. I think this week I put forth my best effort in problems as well as a desire to create beautiful work.
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