A Week of Inspirational Math
This week, we completed math problems and watched videos about confidence in math. I believe the purpose of this was to help ease us into the new class and the work we'll be doing, and to encourage students to try their best. We experimented with many different problems and applied the practices of working slowly and embracing mistakes.
Day #1
We watched a video explaining brain growth and did a dot card activity. The teacher showed a slide with a number of dots on it for a couple of seconds, and then, without counting them, we guessed how many were on the slide. We discussed strategies for identifying the number of dots, and compared different approaches. |
Day #2
On the second day, we watched two videos highlighting the importance of keeping an open mind in math and not being afraid to make mistakes. We worked on a problem showing growth in a figure, and analyzed different patterns to help visualize it. Afterwards, we compared our ideas and found an equation to express the problem. |
Day #3
On day three, the video we watched talked about the importance of going at your own pace when completing a math problem. We learned about hailstone sequences and tested different strings of numbers to try to find a pattern. Students tested the problem with different numbers and rules, and came up with conjectures on the problem. The goal of the activity was not to solve the problem, but rather to practice recognizing patterns and learning how to make factually sound conjectures. |
Day #4
On the last day, we watched a video explaining the value in visualizing your work by drawing a diagram or making a 3D replica. We used sugar cubes to construct 3x3x3 cubes, and used the model to figure out the number of sides showing for each piece. For example, we concluded that a 3x3x3 cube would have eight pieces with three sides showing, twelve pieces with two sides showing, six pieces with one side showing, and one piece not showing at all. |
The videos that resonated with me most were the video on speed and the video on mistakes. Knowing that taking your time and making mistakes is okay, beneficial actually, to the work process really helps me feel more confident in math. In the past, I would always be worried about getting my work done quickly and efficiently, and I would get really stressed out when that didn't happen. I've never been very fast at completing math work, so watching these videos relieved some of that stress and made me feel more comfortable in math class.
Problem Extensions
The problem I chose to extend is "Tiling a 11x13 rectangle." The reason I chose this problem is that it struck me right away. For this problem we had to fill up a rectangle that is eleven units tall and thirteen units long, with squares. The goal was to use as few squares as possible. While doing the problem, I took note of the fact that the dimensions of the rectangle we were tiling were both prime numbers. I experimented with a few different arrangements until I came up with an answer, six. I talked with my classmates and we agreed that this is the smallest amount. But that's only for an 11x13 rectangle. When I realized there is very strict limit to the ways a rectangle with dimensions of eleven and thirteen can be tiled, I wanted to know if this is the same for other rectangles, specifically ones with dimensions of even numbers and other prime numbers. I decided to experiment with both and note any patterns I saw.
After drawing and tiling a couple more rectangles, I was unable to find any obvious patterns. The one thing I did noticed was that with each rectangle, the original and the two extra, the answer was always either six or five. I think that this is interesting and it made me wonder if all rectangles are like that. I also noticed that the rectangles, when using the configuration with the smallest amount of tiles, followed a slightly similar structure, with one large square taking up most of the space, two or three medium squares at the side, and a few small squares lining the others. I was frustrated at first with the idea that there really seemed to only be one way to solve the problem, so I tried different combinations to see if I could get the same answer with different sized squares. However, I was unsuccessful. I overcame this by connecting this fact to the pattern I saw, and was able to find somewhat of a correlation between each rectangle.
After drawing and tiling a couple more rectangles, I was unable to find any obvious patterns. The one thing I did noticed was that with each rectangle, the original and the two extra, the answer was always either six or five. I think that this is interesting and it made me wonder if all rectangles are like that. I also noticed that the rectangles, when using the configuration with the smallest amount of tiles, followed a slightly similar structure, with one large square taking up most of the space, two or three medium squares at the side, and a few small squares lining the others. I was frustrated at first with the idea that there really seemed to only be one way to solve the problem, so I tried different combinations to see if I could get the same answer with different sized squares. However, I was unsuccessful. I overcame this by connecting this fact to the pattern I saw, and was able to find somewhat of a correlation between each rectangle.
Reflection
The assignments we did during the "Week of Inspirational Math" were really interesting and rather fun. I enjoyed looking for patterns in each problem and being able to share and hear other students ideas and answers. It was really helpful to be able to do these assignments in class and work with my peers. Most of them weren't that hard for me, though some where a bit confusing at first. I worked hard at solving these problems and believe that they are a really good way to challenge and stimulate the mind.