I believe the purpose of watching the week of investigations and video watching was to help us become more confident in making mistakes and taking our time. There were other messages from the videos but I feel like these were the two main messages. The different problems we did in class helped us to find different ways of solving problems and seeing how others saw the problem. The different activities we did were Tiling a 11x13 Rectangle, Squares to Stairs, Hailstone Sequences, and the Painted Cube. For the Tiling a 11x13 Rectangle, we drew a 11x13 rectangle on a piece of graph paper and had to draw squares on the inside of it and fill up the whole rectangle with as few squares as possible. Then for the Squares to Stairs, we had to look at four different figures that were growing outwards and creating a stair like figure. Our goal was to explain how it was growing and to see if we could predict figure 55. For the Hailstone Sequences, we had to take a number and and if it was even, divide it by two. If it was odd, we would multiply it by three and add one. We would do this until we got to one and we would look for patterns. Lastly for the Painted Cube, we created a 3x3x3 cube with sugar cubes and pretended to dunk it in a bucket of paint. We marked all of the sides with a marker and had to figure out how any cubes had 3 sides painted, how many had 2, how many had 1, and how many had 0 sides painted. We also watched videos before starting each of these activities. Most of them talked about how it was okay to make mistakes because that's when our brain grows the most. I think these activities and videos were meant to help us understand that and feel less embarrassed about making a mistake. It also helped to push us towards having a growth mind set. A couple other messages from the videos were 1. No matter what people may think, it is actually okay to count with your fingers because it can be very helpful to you. You don't only count with your fingers but you also use your fingers to represent certain things. 2. It is okay to take your time on a problem. It doesn't mean you're slow or not as smart as others. It means that you are thinking more deeply about the problem. Mathematicians spend years on problems because they are thinking deeply about it and are pushing the problem's boundaries.
Hailstone Sequences:
Up above is a slide show of he different photos of a few of the many Hailstone Sequences I did. There are also two photos of how I "extended" the problem by adding a rule change. I kept the first rule about dividing the number by two if it was even. I did however change the second rule slightly. Instead of it being multiply the number by three and add one if the number is odd, I changed it to multiply by three and subtract if the number is odd. It wasn't a crazy rule change but I was curious to see what might happen. I didn't really face any challenges when solving these problems. The only difficult part was getting used to the rule change when typing it into my calculator. It was also a little frustrating when a few of the sequences felt like they went on forever. For example, I had one sequence that took up at least as quarter of a page in my notebook. I feel that one habit of a Mathematician was that I chose to test out many different numbers and extend the problem to see how it would differ or if it would stay the same. I was looking for patterns in all of my sequences. One pattern I did find in the first part of the problem was that when each sequence was coming to an end, they all finished with 4-2-1. With the extended problem, the few sequences I did try ended the same as well. The all repeated the last few digits 14-7-20-10-5-7 etc. It was interesting to see how one slight change throws off the whole pattern. I chose this problem to talk about because this was the one I found to be the most interesting and I had a lot of fun working on it.
In conclusion, I felt that I could have tested out more sequences with the rule change. They may have turned out the same but it would have been better to double check. I only tested out two sequence. After doing these activities and watching these videos, I feel that I'll be able to go throughout this year and be okay with making mistakes ans feeling less self conscious about them. I also feel that I will be more open to sharing and participating in class more than I have in any other year.