## Geometry POL Prep - Semester 1 Reflection

## Geometry Skills

**Describe one topic you feel you have mastered this semester.**

**Describe the activities that helped you learn this activity.**

After this first semester at Animas High School, I can look back and gladly say that I have learned so much. In Geometry, I have learned all about Fractals, Polygons, POWs(problem of the week), Explorations, and so much more. While doing Geometry I learned many concepts and topics, but personally I feel that I have mastered the Pythagorean Theorem, which allows me to find the missing side lengths, angles, and area. I have employed the Pythagorean Theorem in many, many projects, but most noticeably was POW 6, where I needed to find the area of a square, but the graph shown was made up of triangles. I used the Theorem to find the lengths of the hypotenuses of the smaller triangles and to find the area as well, but I also used it to find the side lengths of the square because it was made up from the triangles. After finding one side length I was able to find the area of the square, and the problem was solved. In the Process part of the write-up for POW 6, I was explaining how I used the Theorem to find the lengths of the graph and the sides of the square, and after doing this in multiple problems over and over, I eventually got better. While I used the Pythagorean Theorem for almost all of my projects in math this semester, some problems I cannot find the answer with the Theorem, such as the extra credit POW we were given. In the extra credit POW, I tried to find the side lengths of the square and the triangle and I succeeded, but I was still missing the solution to the part of the POW that I needed, the top of the triangle, and even with mastering the Pythagorean Theorem, I was unable to use this concept for this project.

But personally I feel that I have learned the most through Explorations, which are about 1 - 2 pages of math problems, and these push us to learn new ideas and to use our brains to figure them out with the knowledge that we have gained. The Explorations, for me, have allowed me to practice math that I need to learn without the pressure that is sometimes present in class. The Explorations for me have pushed me to try to expand the way I look at a mathematical problem, and have motivated me to notice how math is used in real life. In Exploration 10/13 there was a lot of word problems and questions that needed to be addressed, and because of this, it made me manage my time more efficiently and made me think about every single possible solution. Exploration 10/13 even motivated me to think more about symmetry and shapes.

**How have the Explorations (and group discussions of Explorations) changed your experience as a math student this year?**

I feel that doing the Explorations, as well as the group discussions, is a lot different than in my previous years of doing math, and that because it is so different, I am having to push myself to grow outside of my comfort zone. I think that because the Explorations push me outside my comfort zone, that I learn a lot more than I normally would, and that they are giving me an opportunity to experience what college level classes my be like. The group discussions of the Explorations are a valuable way to learn because one person may have done the problem a certain way, while another may have done the opposite, and the group discussions give everyone the chance to learn something new each time. I think that the group discussions are a good way to learn about how other people solve problems and how I can implement these ways and skills later in the school year in more POWs and Explorations.

**In what ways have they made math class more challenging, and in what ways have the Explorations and discussions helped you learn?**

For me, I think that having Explorations each week has made math class more challenging because they are getting us to think about problems we already know but in different scenarios, and while it is the same mathematical methods they we have been learning it makes the Explorations interesting. They are also challenging not just mathematically, but mentally because we often have to think through the problem multiple times, and sometimes the problem requires more than a quick look to understand. I personally feel that the Explorations have definitely helped everyone learn the important concepts of math, and I also think that the Explorations are also trying to prepare us for tougher, future assignments that will be given to us in harder math classes such as Algebra 2 and Pre-calc. The Explorations became more difficult as the year went on, and to me they kept building upon the foundation that everyone already knew, and is strengthening the math skills that we have. I also think that they are challenging because of the content and questions present.

**Describe a moment when you had a breakthrough in understanding a challenging concept. What nurtured your breakthrough? How did this empower you in moving forward? What did you learn from that experience?**

I think that the biggest breakthrough that I had with a challenging concept was with Exploration 12/8, where we had to take logic based questions and put them together to create a statement. And with this Exploration (12/8), logic was such a new concept to me, even though we started to do it early in the year, and to me putting logic based hypothesis' and conclusions together to create a true/false statement was so much more different than most schools that I was shocked at what we were learning. But I feel that I had a breakthrough in understanding the concept when I was putting P and Q, and -P and -Q, statements together, and even if some of the statements didn't make real sense at the time, I was able to comprehend the concept of what to do. I think what nurtured my breakthrough was when I simply realized that I was going to need this knowledge later in the school year and this has empowered me to push forward in all of my subjects, not just math. I learned a lot from this logic worksheet and from the breakthrough I had, and I feel that because I have pushed the boundaries of what I know in one subject, that I can implement this in others.

## Problem-Solving Skills

**Which Habit of a Mathematician do you feel you have the most mastery over?**

In math class we have these skills called the Habits of a Mathematician, which we employ in order to solve problems. While mastering a Habit takes a lot of practice, I feel that I am the best at Generating Ideas in order to approach a problem. The Habit of Generating Ideas is to formulate a plan, find more than one way to approach a problem, and to identify and apply necessary mathematical tools. For me having a plan is always my top priority for everything I do in school to outside of school. When I'm doing a POW or a hard Exploration I try to generate an idea of what I need to do, which for me implies drawing out what I already know on graph paper, or sometimes saying the question out loud, and this usually allows me to figure out how to solve the particular problem. Such as in the 10/13 Exploration (above), I was having trouble answering the first questions so I would draw the numbers mentioned in the problem out and fold them in half horizontally or vertically to see if they fit the solution I was looking for. I also used this Habit to approach certain problems, which allowed me to anticipate whether I was going to need a protractor, ruler, graph paper, or such, and if I needed to go back through my notes and find a certain equation. And while I usually had one idea for most problems, I would often try to find multiple ways to solve it, such as drawing the shape if necessary instead of using an equation, or getting another student's point of view. I used the Habit of Generating Ideas to analyze a problem to see what I would need to use in order to solve it, if the problem asked to plot points then I would draw a graph, if it was asking for a certain shape then I would draw specific lines to match, but mostly I used this Habit of a Mathematician to formulate a plan that I could use to measure out the time I spent on each problem or assignment.

**For which Habit(s) do you feel you have the most room to stretch? What do you think held you back from improving in this skill this semester?**

I think that I could improve upon the Habit of Communicating thinking in a clear and accessible way, which I often find troubling to do. Sometimes I am trying to help someone with a problem that we had been given, but often when I tried to explain it to them, it would make them more confused. I think that I have room to stretch with this Habit because I often spend extra time on useless information in my spare time when I could be improving on what already had. While I feel that I have a better hold on this Habit now that I have gone through the school semester and done a few POW write-ups, I still think that what may have held me back from improving radically on this Habit was because I was so packed with work from other classes. Because I had so much work for other classes, I would often try to do my math homework as fast as possible, sometimes skimping on what I did do. In the future I hope to improve this Habit, along with all of my other ones, and now that the semester is over I feel like I can create a plan to master all of them by improving my communication.