## Semester One Reflection

One topic that I believe I have mastered this semester is finding a missing side of a right triangle using the pythagorean theorem and also finding the area of a triangle. I believe that I mastered this topic faster than others because there was algebra involved and I am very good with algebra. I understood it very well last year and I still do and by incorporating it to geometry it helped me also master triangles. For example, POW 3 really helped me begin to understand the pythagorean theorem and use that equation to find the area for other shapes besides triangles. In POW 3 we were asked to find the side length of a line that represented a tarp on top of a tent. I used my algebra skills to split the entire shape up into right triangles so I could begin finding the side lengths of the rest of the shape. This POW was pretty challenging for me because I hadn't quite grasped a full understanding of polygons but once I solved it I had grown a lot through just one POW. Finding the area and using the pythagorean theorem became very simple and I felt very comfortable solving problems related to that without any help. After I had finished that POW and got a 10/10 I felt very confident that I had mastered triangles and most polygons. Our class then took a test that reviewed all the topics we had learned so far about polygons and the pythagorean theorem. I wasn't too worried about it because I was pretty sure that I could ace the test with no problem. When the time came to take the test I looked through it and was surprised to see that there were some answers regarding t=triangles that I wasn't sure that I could solve. For example, there was a question asking whether or not the three side lengths that were given could actually be side lengths for any kind of triangle. I was very nervous because I had totally forgot how to do it because I was more focused on finding the area and using the pythagorean theorem. I tried my very best on the test and ended up getting an 88%. This grade wasn't bad but I knew I could've done better so I worked hard on the things that I felt that I was lacking and I now think that I could take that quiz again and get a 100% on it easily. Im very glad that we took that test because it showed me what I needed to work on so that my knowledge could continue to grow. The evidence that I have to show that proves my growth is POW 4. When I first read this POW I didn't quite understand how I could solve it but then I quickly realized that I could incorporate triangles into it which made me quite happy due to the fact that I felt very confident in that topic. In this POW Maya and Jason are biking away from the same destination at 1:15pm but Maya is riding north and Jason is riding east. Maya is traveling at the speed of 32 mph while Jason is traveling at the speed of 24 mph. The goal is to determine at what time will Jason and Maya be 130 miles apart and how many miles have they both traveled from the starting destination. I incorporated right triangles by making the distance that the two were apart the hypotenuse of a right triangle and the distance that they were away from the starting point would be the other two sides. This made the problem very simple and I applied the knowledge that I had acquired from other activities and the triangle test to master it. To conclude, I have learned so much this semester and I am excited to learn more but the one topic that I have mastered has to be triangles and the pythagorean theorem.

The explorations have been quite the challenge for me for they have helped me grow. They can be a bit stressful for me because we usually have one day to complete them but it really helps me understand the topic we are learning very fast. Also, if I don't understand a problem our class goes over them the day they are due so I always gain new knowledge from them. They have also helped me become a better listener when Aliza is giving us a lecture. This is because the exploration will most likely be related to what Aliza is teaching us at the time and if you are not listening or if you are not engaged then the exploration will be very difficult and you probably won't finish it. This happened to me before I have learned my lesson to stay focused when Aliza is transitioning into a new topic. As the semester is coming to a close and I look back at all the explorations we have done I learned at least one new thing from each one and it has significantly improved my listening skills and learning ability.

There wasn't just one problem that I encountered during this year there were many related to just one topic and that topic is symmetry. For example, on the exploration 10/13 all the questions were related to symmetry in some way. I really didn't understand symmetry except for reflection but besides that I was very confused on a lot of the questions because they were way too difficult for my understanding. I tried my hardest to solve them but I just didn't know how so I ended up not finishing the exploration and got a 0 on it. This is really helped me understand that I needed to listen more and apply myself to the lectures Aliza gives us when we transition into a new topic. After this occurrence I tried my hardest to stay engaged during class discussions so I wouldn't be so confused and lost on another exploration. To conclude, I have encountered difficult problems throughout this semester but I have worked through them but I have learned if I don't listen or get engaged then the problems become a lot more difficult which can cause my grade to go down. In the next semester I will apply myself 100% in geometry and all my other class as well.

The Habit of a Mathematician that I have mastered this semester has to be generating ideas. I have always been pretty good at seeing a problem in math and in real life and being able to try and tackle this problem in different ways in order to solve it. I demonstrate this skills in every single POW that I have done. I read the problem without knowing anything about it or how to solve it and then I begin to closely analyze to see any kind of way to solve it. For example, for POW 8 I read the question then immediately started writing down the students names who could possibly win a prize. I didn't even know how to solve the problem yet but I had the idea to write them down with the matching prizes so it would be easier to understand. Once I did this I started to begin my process of solving it and you can read the process in the POW (It is posted below). Another example of how I generated ideas in order to solve a problem more efficiently was when we did the pumpkin carving contest. When my group first started this project we split up so we could each sketch the face for our pumpkin. The face had to include straight lines, polygons, and fractals. I quickly started my sketch and was very proud of the outcome. I had a great idea in my head, sketched it one paper, and my group ended up using my idea for the pumpkin face. The face met all the requirements and looked pretty cool. I used what I had learned in geometry and generated ideas to apply to my group’s project. I can't wait to see how I will grow even more in the upcoming semester and hopefully I can improve some of the Habits of a Mathematician.

One big Habit of a Mathematician that I can improve on is Communicating my thinking in a clear and accessible way. When I am solving a problem and begin to understand it, I have a difficult time when it comes to explain how I solved it and what I am thinking at the time. This weakness has made it kind of difficult to write my process for a POW because I just don't know how to put my thinking into words. This causes me to spend a lot of time writing the process which can make POWs very stressful for me. As the months have passed I have noticed that the more I write my POWs that I do, the better I get at Communicating my thinking. So I am hoping that in the next semester we continue to do POWS so I can continue to improve this Habit of a Mathematician.