I felt that I could find a unique and beneficial way for students to learn about taxicab geometry. Thus, rather than drawing it out or using graph paper I decided to use an actual map. Now for you future educators, I would use a map that of the county your school is in. That way students can relate to the surroundings better. Plus, I really enjoy the personal connection when making an activity.
​All you need as a teacher is a map such as the one posted to the left. You can photocopy the specific location or range you want your students to work with. I have attached a worksheet full of questions that a third grader could do.
​Fun fact: I tried this activity with the kids I nanny. They loved it! I even made it more interactive by actually driving in my car with the direction that tell me. Then have them count how many block we went by based on the streets we passed. They're quite smart kids!

​This website also has lots of ideas for how to teach taxicab geometry.
http://emat6000taxicab.weebly.com/teacher-resources.html

        We discussed briefly in class about taxicab geometry. I am amazed that I’ve gone 5 years being a math major and haven’t heard of about 70% of the things we talk about in this class. It’s a pretty neat thing to think about! Anyways, since I’ve never heard of this type of geometry I wanted to do some research on it. That way I can see if I could ever or would ever use this in my own classroom.
        Taxicab Geometry was created by a man named Hermann Minkowski in 19th century in Germany.  This type of geometry changes the Euclidean distance formula to the metric proposed by Hermann. He introduced the formula where distance between two points (x,y) and (w,z) is represented by the formula: d=|w-x| +|z-y|. The main idea behind the creation of this new distance formula was that the distance between those two points is not measured by a straight line but rather on a horizontal or vertical line as if traveling on an actual road. For example, in the picture provided above the red, blue, and yellow lines represent taxicab geometry.The parts that stay the same as in Euclidean geometry are the characteristics of points, lines, and angles.
        Throughout the making of taxicab geometry there were a couple other people that were mentioned when researching about taxicab geometry. In 1952, Karl Menger created a geometry exhibit in Chicago named Museum of Science and Industry. Menger also wrote a booklet that was the first to mention the word taxicab geometry in it. By 1975, the taxicab geometry was still not talked about. Until, a man by the name Eugene Krause published a book called “Taxicab Geometry: An Adventure in Non-Euclidean Geometry. Once the book was published taxicab geometry people started to use it.
        There are everyday uses that taxicab geometry relates too. For example, you can use taxicab geometry in chess. The distance between squares on the chessboard for rooks and bishops is measured in taxicab distance.
        I think about how I could use this in a classroom. I feel this type of geometry could be for all ages. For instance, for younger grades you could create a map and in order to get to the next “hint” you need to figure out a path and how many steps you need to take. For the older grades, you could have them plot points on a graph and then figure out the distance between those points by counting the squares.
I really enjoy how flexible this geometry is in relation to how a teacher can use it. I’m glad I came to class and learned about this geometry because it could be a segway into other concepts in math a student may face.

Work cited:
http://taxicabgeometry.altervista.org/general/index.html
***picture from googling taxicab geometry

          According to Wikipedia the definition of tessellation is done on a flat surface and is described as tilling of plane using one or more geometric shapes with no overlap or gaps. There are different types of tessellation: periodic tilling, or non-periodic tilling’s. A periodic tilling is tile patterns that repeat. A non-periodic tilling is tile patterns that lack repetition.
          Tessellations were created in Ancient Rome and in Islamic art in various palaces. The real mathematician is M.C. Escher which made tessellations a creation for artistic effects.
          When looking at various images in class, I realized that there can be either complex and simplistic patterns. Thus, I wanted to see what it took to create both kinds of tessellation. My tilling’s are shown below. I created a simplistic one and a more complex design.
          In the figure to the right, I started with the three pink triangles connected together at the tips. Then realized that I could create three smaller purple triangles which can look like a hexagon behind the pink triangles. Then I colored in the yellow to make a bigger triangle. I began to think about how to make sure this pattern will repeat. Thus, I colored around the shape created to form a hexagon. Then connected the hexagons together.
          Once I finished creating the repeated pattern I realized that by connecting them together I created an orange smaller hexagon. Therefore, I realized that if you were to do this as a profession, such as the Islamic art in palaces, it would take a lot of planning and thinking even though you are connecting shapes together.
          Next, I decided to try a simpler concept of tessellation but still incorporating repetition. (This figure is shown below. Thus, with this new tessellation I started with filling in a 5x5 square and then repeating the pattern. I then created a smaller square inside the 5x5 square and realized I could create four trapezoids and a square to fill that inner square. Then, as I thought about what to put around the inner square, I wanted to create a cross type shape (pink) and filled in the remaining space with the color blue. 
          I again sat back and noticed that once I connected the access blue it created another wider cross. As I stated before, I didn’t think by doing a simpler tessellation that I would be able to form another shape when connected. This is a neat concept to see that you wouldn’t notice by looking at someone else tessellation work.       
          This concept is really intriguing to me because as try and relate it to my future classroom. I feel that in order to get children used to shapes and their relation to other shapes this type of interaction would be a great activity. This activity would also focus on a couple Standard for Mathematical Practice. Them being able to reason and also to think abstractly.
          As I think about my own tessellation creations I begin to realize before started I pictured the whole idea in my mind before actually creating it. Thus, displaying the thinking abstractly mathematical practice. Then, having to discuss why and how I created would be the best for my future students on being able to communicate their mathematical ideas to others. ​

           ​People know the concepts of math but do they know the people who created these concepts? If you would ask a random person on the street to name a mathematician you would probably get one of two answers. One answer being that they don’t know any mathematicians by name but could tell you a concept of math. The second answer would most likely be someone mentioning the name Pythagoras. I believe this is due to how strongly enforced the Pythagorean Theorem is taught in the school system. However, there was a man that came before Pythagoras that has founded a couple key concepts of mathematics as well. The man’s name is Thales of Miletus.
          Before taking this class I have never even heard of this man’s name before; which I find really surprising seeing as I am a math major. Thus, if I were to ask myself that question I would fall into the answer category of only being able to name math concepts over mathematicians. Therefore, I found this man interesting because I didn’t know anybody other than Pythagoras that used the idea of triangles; thus, I decided to research into him further.
          Thales of Miletus came 30 years before Pythagoras did and 300 years before Euclid did. The work Thales did throughout his life has benefited both Pythagoras and Euclid’s discoveries. Yet, maybe 5% of the world population knows who this man is! (my estimated opinion).
          Thales whole outlook on mathematics was to use geometry to solve real-life problems. He understood similar triangles and right triangles and used that in practical ways. For instance, Thales could measure the height of the pyramid using the shadows. He would use the pyramid shadow and his own shadow at the moment when his own shadow was equal to his height. He used the fact that they made a right triangle because he created a scenario with two equal “legs” which created a 45-degree right triangle; which he knew were all similar. Then he used the fact that the pyramids shadow measured from the center of the pyramid was equal to its height. Thus, he used a triangle to figure out the height of pyramids. A cool fact about his finding is that the math he used in this problem is used in present-day trigonometry. This is crazy because I’m surprised his name wasn’t mentioned in any of my trigonometry classes.
          Thales is also credited for discovering a method of measuring the distance to a ship from the sea. The reason he decided to approach this was due to an enemy warship dropped anchor off the coast of Miletus, where Thales is from. Again, Thales used triangles, there similarities, and their angles created to figure out the distance. This benefited the country because they could tell whether a boat was coming to trade or to battle.
          These two findings struck me as interesting once I thought about them more. As I think back to my elementary school days and even substitute teaching in a classroom students come upon story problems all the time. Most often than not the students use the same strategy like how Thales solved the height of the triangle or the ship to shore. We were taught to create a right triangle to find the distance relationship or ratio between the objects given. For example, a common used math problem deals with either trying to find the height of a tree, its shadow, a wall, or the wall’s shadow using other information given. (A diagram is shown to the right).
 

          Thales’ biggest accomplishment in the math world would be his theorem, known as Thales’ Theorem. Thales’ Theorem is about geometry. It states that if three points are on a circle where the line between two of those points is the diameter, then the angle at the remaining point created a 90-degree angle. As a result of these findings Thales has been known as the first true mathematician and the first known person to make a discovery and get credit for it.
         There are many other math aspects that Thales has been attributed. For example, he discovered that a circle is bisected by any of its diameters, he found that the angles at the base of an isosceles triangle are equal, he knew that when two straight lines cut each other the vertically opposite angles are equal, and he figured out that two triangles are equal in all respects if they have two angles and one side respectively equal.
         Taking a look back at everything Thales accomplished in his mathematics life; the facts state he was just as successful as Pythagoras or Euclid. My opinion is that Thales set the foundation, in a sense, for Pythagoras and Euclid to discover their mathematical ideas. Now if anyone would ask me to describe a mathematician I can answer them stating this man’s name and his contributions.

Today we discussed many different components math can be seen as. When a person ask me what my major is, I respond stating I major in Math with an emphasis in Elementary Education. Everyone responds with the same question: "Why did you choose math? What is Math?" Math is a complex subject for people to grasp. People understand the purpose of Science, Social Studies, English, and Health majors because those are referred to commonly in society. However, Math tends to get a negative outlook. This can be because Math is subject that most people tend to either struggle or excel in. This may be because math has certain answer or solutions. However, what most people don't realize is that there are multiple ways to solve a particular problem. Every single person uses some type of math throughout each day. When people ask me 'What is Math?' I respond with 'when do you use math personally?' This takes people by surprise because when they think about it and respond they don't realize all the times math comes into play in their own lives. Therefore, my definition of math is using various strategies to problem solve by communicating, analyzing, and interpreting the situation given. This definition can be using in the various subtopic in the concept of Math.
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​Math wasn't just thought about overnight. There were many people and concepts that made the concept of mathematics the idea it is today. I will be listing my personal list of top 5 biggest discoveries or milestones in math history that have been used throughout the timeline of mathematics.
​One of my personal five milestones includes the Pythagorean Theorem. The Pythagorean Theorem is defined the relationship between the three sides of a right triangle. It states that the hypotenuse squared is equal to the sum of the squares of the other two sides. This was a milestone because it led to other characteristics of triangles such as SAS, AAS, etc. Thus, another milestone I believe to make math history is the Distance Formula. The Distance Formula is when you are given two points on a coordinate plane and can find the length or the distance between them. This formula can be performed on a coordinate plane or it can be useful to figure out lengths of triangles that are not right triangles.
​The next couple milestones I believe to be beneficial to helping create the future discoveries of mathematics. A man by the name of Albert Einstein with the creation of the formula that expresses the relationship between mass and energy (E=mc^2). This led way in both the math department and the science department advances. Another milestone deals with Euclidean Geometry. The acknowledgement of the fifth axiom allowed the creation of many different quadrilaterals. Finally, the development or creation of the various areas and perimeter of multiple shapes includes circle, square, triangle, cone, etc. There are plenty more findings in the mathematic subject that feed off each other which when you think about it is pretty neat fact.