## Week of Inspirational Math

**Purpose:**

We've spent the last week working on four different activities which both expanded our thinking as well as connected to the Habits of a Mathematician. Although the tasks appeared simple, as we worked on them it became more challenging as well as more complex. We had to take time to assess the situation and find a way, our way, to solve the problems given to us. We not only had to share our answer, but also had to challenge ourselves and our ways of thinking compared to our classmates. The videos were also able to teach us the Habits of a Mathematician and different experiences others also went through, whether it be a tough time understanding a problem or making a lot of mistakes.

**Activities:**

**Building Shapes**

In the first activity, table groups would work on creating different shapes out of rope with the help of everyone involved. The teachers would then play the role of the skeptic and ask critical questions to the table group so that we all could, individually or as a group, explain why we believed our shape was correct and accurate. At first in this activity, I didn't completely understand how we were supposed to describe each shape and prove its accuracy, and it seemed at the beginning as a simple and easy task, not only to create the shape using rope, which is so easy to draw, but also to confidently and persuasively explain the accuracy and reasoning behind the shape and process of obtaining the shape. As the skeptic continued to ask clarifying questions, I was able to understand where we needed more detail to explain, such as the angles of the shape and how we could prove the correct angles, or how all sides were equal and somehow had to fold the rope so that every time there would be equality between each side. As we continued through more complex shapes, it became harder to understand how to do it myself, which then made it harder to explain and prove with evidence if we did get to that point. I realized that through this activity we used all of the Habits of a Mathematician, between collaborating with our group, explaining our answer to the skeptic, and presenting to the full class.

**Number Visual Pennies**

The second activity was focused on finding patterns and using those patterns to figure out how everything could fit together. We were given 100 pennies and a sheet with different shapes made out of smaller circles. You can find the sheet at the bottom of the page along with all other documentation. Once given the sheet, we had to somehow figure out a way to get all 100 pennies in the smaller circles of each shape so that none were left out. Also, each shape had to have the same amount of pennies in each smaller circle. For example, if there were 4 circles that formed the points of a square, each point would have to have the same amount of pennies, maybe 10 in each, so that in each circle was ten pennies and in total there would 40 pennies in that square. The fun thing about this activity is that depending on how you thought about and saw the problem, it was harder or easier for you to come up with a quick solution, or maybe you could think of a different way to find that solution.

**One Cut Geometry**

For the third activity, we were given patty paper, a ruler, a pencil, and a set of shapes which we had to then draw and cut out. We had to use the patty paper and, without using the edges of the paper, draw the different shapes provided. We were then told we would have to cut out the shape. The catch to this was that we had to be able to cut the shape out with one straight cut. No turns, no second cuts. Of course, everyone in the class decided that folding their paper until all lines were lined up and able to be cut out was a reasonable way to create the different shapes. This became more and more difficult as the shapes became more complex, whether that be opposing angles or more angles and sides altogether. There was a lot of teamwork and group talk going on at this point, and many mistakes were being made. Honestly though, that was part of the activity. To make mistakes and hopefully learn from them.

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**Square Mania**

For the fourth and last activity, we were given a worksheet with different sized blocks in which we had to verify a certain amount of line segments and squares. At first it seemed like a simple and easy task, but as I continued through it, it became more complicated and confusing with no clear answer to the larger questions. It also became more confusing to understand which squares I had already found and marked because it was such a compact diagram. It was interesting to hear the discussion at our table which started with, "What colors should I use?" to "Have any of you figured this one out yet?" At a certain point, I got stuck and it became hard for me to continue or try to solve for an equation or certain amount of total squares. It was interesting to hear what others observed though, and how they perceived the question and found their answer. Some people tried to understand the problems through equations, whereas others used visual patterns found to explain the correlation of one problem to another.

**Videos:**

**Video 1: Strategies For Learning Mathematics**

In the first video, it talked about different strategies for solving math problems and gave a well thought out list of tips for both breaking down and completing the problems given. The first tip was to draw out the problem first so that you can visualize and understand it. This is an important and overlooked tip because many people think of only numbers when solving, but most of the time starting out with a drawing can help visualize and comprehend the problem, as well as imagine solutions and view the problem in different ways. It would open up a new area of possibilities of what the problem could expect or mean as well as help find a way to solve the problem if numbers could not give way to a clear solution. The second tip was teamwork so that we can hear each other's ideas and collaborate with each other to view the problem from multiple views. If we only look at the problem in our way, we have the possibility of missing part of the problem or part of the answer. When listening to others we are able to hear their perspectives and understand where they are coming from, and based on that we are able to review the problem and understand it in a new way. The third tip was experimentation to find a way to figure out the problem, not worrying about if its the wrong way, but instead looking at it as a way that could lead to a dead end or possibly another path, nonetheless leading to something else. I have done this multiple times out of either stubbornness or curiosity, and all times it has had some type of positive influence on my solving of the problem. It has steered me another way whether that be continuing through part of the problem or starting from scratch, either way knowing that was not the right way to solve the problem, and that there is another one, possibly closer than before. The fourth tip was to find new resources, such as Youtube, Google, or a book containing information. This would help us look from more than one perspective, especially if we don't have classmates to collaborate with. It also is another way to gain information when needing to do research or understand a topic or problem. The fifth and last tip was to start with a smaller case. Some would say looking at the big picture is a good idea, but in some problems the larger picture can be daunting and seemingly impossible to complete. This is why it is important to start with a smaller and less complex task, you will be able to understand the problem as well as look for a solution easier, and then go back to the bigger picture to complete the problem in whole. These five tips are very important to me when it comes to completing problems because they help me take a step back and usefully complete it.

**Video 2: Speed is not Important**

The second video was all about how speed isn't the most important part of solving a math problem ,although when growing up, it is believed to be. Ever done a multiplication table? They test you on how capable you are of math solely by how fast you can complete the problems, and can I just say, those stressed me out as a kid. In the video they talk about how your successes in math actually have nothing at all to do with your speed. They also point out that many great mathematicians were slow and deep thinkers, and took time on the problems they were given. It is most important that you understand the problem, not that you can do it in under a minute. A lot of people including myself struggle with this, and I think it's important to have a reminder that every travels at their own speeds, just like with walking or running. Sometimes instead of aimlessly sprinting to he location because you need to get there before others ruins your journey there, and instead of soaking everything in, you see only a blur around you. While walking, you can look around and take in what you see while still moving forward and at your own pace. It isn't a race, and you are allowed to understand what you see and take your time, basing your progress off of no one else but yourself. This is important to me because I always thought speed was the most important part, but now I know I can take my time and think deeper, and that understanding a math problem shouldn't have a timer on it.

**Video 3: Brains Grow and Change**

In the third video, they talk about how our brains grow and change, and how no one is ever really born a math person or a not-math person. We are all given the same capabilities to learn math more or less, and the more we learn the more we know. We now that when we learn something new or work on a topic we were struggling on, we create and strengthen pathways in our brain. The reason some people seem more like a math person is because they have had more positive messages or learned more than another. The more effort you put into learning something, the more your brain will develop new pathways. This has helped me understand that I am always learning something, and brain will continue to grow if I allow it. I now understand why it is important to keep your brain busy with problems and new topics, so that it has the opportunity to grow and learn as I please.

**Video 4: Believe in Yourself**

In the fourth video, I learned you have to believe in yourself to be able to complete the "impossible". When not believing in yourself, you are shutting your brain off to being able to fully understand and take in the problem and be able to come up with new ways to solve it, even if it might not be the right way at first. When you don't believe, it makes it a lot more difficult to pursue as well as be successful in math. This also opens up the door to the growth mindset, an idea we have here at High Tech that contributes to the main idea of this video. The meaning of a growth mindset is that when you are learning a new subject or topic, and things become difficult and seemingly impossible, instead of giving up and believing you aren't capable of solving it, you believe that you are capable, in fact more so, and be willing to see growth. This is opposite of the closed mindset, which is a mindset that prevents further learning by halting your process of believing in yourself. As cheesy as it may sound, it really is true. The moment you stop believing you're capable of completing a problem is the moment you become incapable. This helped me realize that it is possible for me to do a problem that seems impossible, but I just have to get the help I need and believe that I can conquer the problem.

**Video 5: Mistakes are Powerful**

For the fifth video, they talked about how mistakes are actually a very powerful and important part of the overall learning process. Although it may not seem like it, mistakes are a huge part of understanding and completing problems. Every time you make a mistake, synapses fire in your brain, so while you believe you made a dumb mistake, you're brain is actually learning and making connections, and helping to overall understand the problem. If you didn't challenge yourself and make mistakes, you wouldn't learn much and your brain would probably become less active than if you did make mistakes. You almost learn more while making mistakes than completing the problem. We are taught in school mistakes are bad and unnecessary, but it is actually a critical part of the learning experience. This is important for me to know since I think I'm always such a perfectionist, and to know mistakes are a good thing makes me feel more free to try a math problem, even if I won't succeed.

**Write-Up:**

I decided to write about the second activity we did over the week, Visual Number Pennies. For this activity we were given 100 pennies and different patterns in which we would have to stack the pennies. For example, one pattern was three stacks of pennies in the shape of a triangle, the stacks making the vertices. We then had to stack all 100 pennies into groups so that each shape had some pennies on their vertices. The catch was that each shape had to have the same amount of pennies in each stack, meaning that every vertice of the triangle could have five pennies while the pentagon could have three in each stack. There then became the challenge of how many in a stack per shape. I chose this problem because I feel that I used all of the Habits of a Mathematician and understood this problem the best as well as how it was solved. At first, I thought visualization could be good, but as I talked it over with my table group, I realized it would be more effective to instead use trial and error with numbers, hoping that it would all come together after a few tries. What I did was write out first the amount of vertices in each shape. once that had been done, I then placed in random numbers that could be reasonable for how many pennies in each stack for the amount of stacks in that shape. Then once all of the numbers had been filled in, I multiplied the stacks by how many in a stack and got the amount of pennies that would be in each shape. Once those numbers were found, I added them all together with hope of getting 100 as the sum. Instead came out 106, a pretty close guess. As I looked through the numbers I chose, I realized there was a shape with six vertices, and that if I took away one penny out of each stack in that shape, I could subtract six pennies, leaving us with 100 pennies. We then as a table group decided to try and see if this math work was correct by matching the pennies up and seeing if it properly used all 100, which it did. One challenge I faced was figuring out how to effectively solve this. There seemed to be different ideas of approaches at my table, and honestly I was just doing trial and error on a whim of possibility of success. I feel I've always been told trial and error isn't all that useful, but I was also told to try different ways and make mistakes, so I decided why not. It turned out to be the best thing I could have done to find an efficient solution to the answer. One Habit of a Mathematician I used while working on this problem is t start with a smaller case. Instead of diving right in to moving the pennies around, I looked at the numbers behind it and tried to find a solution through them so that I wouldn't have to constantly re-sort them and basically be getting myself nowhere.

**Reflection:**

I think during the Week of Inspirational Math, I learned to have fun while learning math, and to take all I could get from it, whether that be new ideas from both myself and others, or just making a bunch of mistakes to better understand the problem. I had a few challenges. Sometimes I didn't understand what the problem was asking or exactly what the solution was, but those challenges were overcome by teamwork, collaboration, problem solving skills, mistakes, and lots of determination. I also had a lot of successes, like being able comfortable to share my progress out loud, finding different ways to solve a problem, asking for help, and creating new solutions. This past week will effect my effort and participation positively for the rest of the year because Iw became comfortable sharing my failures and successes in front of the class and my group, got to know and learn about both my classmates and their ways of thinking, and put my full amount of effort into trying to solve and understand every activity. I feel that these habits will only continue to shine positively as we continue throughout the year with new topics and new problems to solve.