What was the purpose of this week?
Throughout the duration of this week, we studied the benefits of making mistakes, embracing the struggle, and support in group work. Via multiple maths activities as well as Stanford University's videos, we discovered how making mistakes causes your brain to grow, making connections that aren't otherwise formed when the answer is correct on the first try.
Week Overview
In small groups, we worked through four problems over the course of the week: Tiling a Rectangle, Squares to Stairs, Hailstone Sequences, and Painted Cubes.
Tiling a Rectangle: In this problem, we began with an 11x13 rectangle which we were then tasked to divide up, or tile, into the fewest squares possible.
Squares to Stairs: This problem was based off a visual of four figures that appeared to be growing in a pattern. The figures grew from one square, to three, to six, then to ten, and so on, in the configuration of a staircase. Our task was to find a pattern in the amount of squares in relation to the figure number, and then use that pattern to predict future figures.
Hailstone Sequences: Through this problem, we learned about a mathematical pattern known as a hailstone sequence. The rules for creating these sequences are very specific; if the number given is even, divide that number by two. If the number is odd, multiply by three then add one. An example of a hailstone sequence is: 6, 3, 10, 5, 16, 8, 4, 2, 1.
Painted Cubes: This problem required us to build our own visual from sugar cubes, then picture the following situation: A 3x3 cube is dipped in paint. How many of the individual 1x1 cubes that make up the larger cube will have three of their sides covered? How many will have two? How many cubes will have one side painted, and how many will have none? Based on our sugar cube model, we used an x/y table to display our results.
Tiling a Rectangle: In this problem, we began with an 11x13 rectangle which we were then tasked to divide up, or tile, into the fewest squares possible.
Squares to Stairs: This problem was based off a visual of four figures that appeared to be growing in a pattern. The figures grew from one square, to three, to six, then to ten, and so on, in the configuration of a staircase. Our task was to find a pattern in the amount of squares in relation to the figure number, and then use that pattern to predict future figures.
Hailstone Sequences: Through this problem, we learned about a mathematical pattern known as a hailstone sequence. The rules for creating these sequences are very specific; if the number given is even, divide that number by two. If the number is odd, multiply by three then add one. An example of a hailstone sequence is: 6, 3, 10, 5, 16, 8, 4, 2, 1.
Painted Cubes: This problem required us to build our own visual from sugar cubes, then picture the following situation: A 3x3 cube is dipped in paint. How many of the individual 1x1 cubes that make up the larger cube will have three of their sides covered? How many will have two? How many cubes will have one side painted, and how many will have none? Based on our sugar cube model, we used an x/y table to display our results.
Video Takeaways
In the video titled, "Ways We See Math," we learned that everyone sees a maths problem differently, and there never really is a "correct" solution. There may be a right answer, but anyone could potentially get to that answer a different way. This video also showed us that no one is born as a maths person, and that everyone is capable of completing problems and enjoying maths. In another video, "Mistakes are Powerful," we learned that when you make a mistake, especially in maths, synapses fire and cause our brain to actually grow, and then more synapses fire once we've realized that the mistake was made. Therefore, trial and error in maths class is critical for learning and brain development.
Problem Extension Write-Up
I chose to take a deeper look at the "Hailstone Sequences" problem. This problem was very intriguing to me, since no one in the class came to a consensus about finding and proving a pattern. For my extension of the problem, I decided to change the rules that apply to all odd numbers in the sequence. Instead of adding one after multiplying the number by three, I subtracted one and continued the pattern as such. This is my work and the results I received after attempting to create a sequence using the new rule.
I can conclude that the original hailstone sequence rules were certainly there for a reason; they made the most sense and resulted in a clear pattern that would eventually end up in the same place, no matter what number you started with. Using the new rules, the pattern results were inconclusive. Many of my attempted sequences began to repeat once they reached 2, unless I began with a multiple of 7 (7, 14, 21...) While the sequence beginning with 21 was never finished, I believe that it would have also began to repeat itself eventually. With this problem, I really struggled with seeing a pattern. This is a challenge that I never really overcame, it is still difficult for me to recognize and provide a proof for a pattern in any given sequence. I believe that the habit of a mathematician that I used most throughout this problem was conjecture and test. While that may seem like a pretty obvious answer, it was the only way for me to really work toward the goal of proving that a pattern existed. My tests, based on the conjecture related to the multiple of seven theory, was inconclusive, while still being beneficial to my overall understanding of the problem.
Reflection
Throughout this week, I tried my best to be very open-minded when it came to solving problems. Even though we mainly worked individually, I attempted to solve the problems in a way that either wasn't the easiest method, or one that I was very comfortable using. I stepped out of my comfort zone with this week, and I plan to continue to think critically through the rest of the year and try to embrace the struggle much more than I have in the past.