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

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

taxicab_geometry_worksheet.docx |

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

]]>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:

***picture from googling taxicab geometry

That being said, I wanted to research some woman that played a role in the history of mathematics. I also wanted to see what struggles they had to deal with when presenting or even collaborating with other mathematicians. This idea of how they had to do math in that time period. This is because I know a lot of my future career deals with being able to collaborate and bounce ideas off of my colleagues.

A woman by the name of Maria Agnesi was an Italian mathematician, philosopher, philanthropist from 1718-1799. She was a child prodigy, even though she was the oldest of 21 other siblings. She studied language and mathematics. One of her famous works was when you wrote a textbook to explain math to her younger brothers, which is a noted math textbook to this day. Here’s a side note thought for you. I found it fascinating that she wrote the book for her brothers, what about her sisters? I found out that this was because her father told her that was her responsibility was to educate her brothers. Anyways, she was also the first woman to be appointed a university professor of mathematics. She also brought many ideas from contemporary mathematical thinkers due to her knowing various languages. Once she was released from the duty of educating her brothers she chose to work with the less fortunate.

Another woman by the name of Sophia Germain was a French mathematician in 1776-1831. We talked about her the most in class. I found her story to be quite interesting. She was self-taught from books and lecture notes about mathematics. Her most famous work was accomplishing a limited proof of Fermat’s Last Theorem, which stated that for any prime under 100 where certain conditions were met. However, she had to use a male name, M. LeBlanc, in order to attend or even talk about her work with other mathematicians. It wasn’t until Gauss came along that Germain got recognized for being a woman in mathematics. He wrote a letter about how her strength and how she faced problems was noble, especially in this type of field. Another fun fact about her was that she studied geometry to escape boredom during the French Revolution when she was trapped in her room.

The last women I am going to mention is Mary Somerville. She was a Scottish and British mathematician in 1780-1872. At age 15, she noticed some algebraic formulas used as decoration in a fashion magazine, and from that she began to study (on her own) algebra to make sense of what she saw. However, when she began this journey her family opposed it. She even got a hold of Euclid’s Elements of Geometry without her parent’s permission. Her second husband, Dr. William Somerville, approved of her study of mathematics. Throughout her life she has accomplished a lot mathematically. She was one of the first two women admitted into the Royal Astronomical Society, she has a college (Somerville College) named after her, and was named “Queen of Nineteenth Century Science”.

Even though I only touched on a few women in mathematics; there experiences and triumphs are incredible. I really enjoyed Maria because she displayed a different type of math, teaching, that as I mentioned earlier in the lower levels in mainly a women dominant career. I enjoyed learning about how in order for Sophia to even talk to other about her work she had to change her name to sound like a man until Gauss finally recognized her work even though she’s a woman. Finally, when reading about Mary I took a second to think about if I had the strength to go against my families wishes to study a subject such as math. That determination from a woman when at that time weren’t highly credited in the field of mathematics.

In conclusion, I give credit to the women in mathematics. Their strength and tenacity is something that I want to strive for in my future career. Obviously I may not face as big of challenges as they did but I will have challenges that I will need to face. I do notice when I say I’m a math major a majority of the people say it’s mainly a male major. Now knowing the struggle women had to go through just to get their work noticed is so incredible.

Work Cited

Frenkel starts off not liking math until a man by the name Evgeny Evgenievich converts him to focus on mathematics. This began his journey and love of mathematics. Throughout his life Frenkel has dealt with struggles and triumph moments. For instance, when he was applying to go to college he was discriminated against due to the fact that he was Jewish. (Note that his childhood life took place in Russia). He went through an extensive application process to get into the most mathematical prestigious college, Moscow State University. The application took the average mathematics lover around an hour but since Frenkel was Jewish they questioned him for 5 hours or so and he did not pass (even though he answered all their questions correctly). Who would want to go through that process to know that there’s a good probability you won’t pass?

However, this did not discourage Frenkel completely in following his dreams to becoming and learning more about mathematics. He got accepted to Moscow Institute of Oil and Gas which had an applied mathematics program. Throughout the four years there, Frenkel met multiple mathematicians that helped him achieve a goal of his. This was to solve a problem or conjecture most people couldn’t. Thus, with the help of Dmitry Borisovich Fuchs, Frenkel published his work on braided groups. Although, this was not the end of Frenkels math journey. He later met Borya Feigin. Frenkel credits him as the best advisor he has had because he ‘turbocharged’ his mathematical career. However, little did Frenkel know that Borya would play such an important role in his life until later.

Towards the end of college, Frenkel’s mathematical career brought him to the United States, Harvard University in specific. Where he met another beneficial mathematician, known as Vladimir Drinfeld. While working with Drinfeld for multiple years they published a mathematical finding dealing with the Lie groups. This program is known as Langlands Program. However, to top his mathematical contributions Frenkel, along with others, created a grant to support their research about how Langland Program and electromagnetic duality are related. Thus, Frenkel has had an amazing mathematical career considering he didn’t even like math at first.

As I reflect on this book, I think about my mindset while reading. I went into reading this book trying to find an aspect I could use or learn in order to benefit my elementary career. That being said, I took a couple themes out of this book that my students can relate to. The first being this idea of collaboration. Throughout Frenkels math journey he came across multiple mathematicians that he learned and shared his ideas with. In return they founded and created multiple mathematical findings. This is a good concept for students to grasp because some problems you can’t do it all by yourself. It’s okay to ask for help. Frenkel made plenty of mistakes but it was how he asked questions and collaborated with his “mentors” when he was wrong. Also, working with multiple people Frenkel could make hidden connections between different math concepts he has learned; which ultimately this is the goal of an educator. The second concept I will take away from this is the commitment and drive Frenkel had to succeed. As stated earlier, Frenkel had been through some trials in his life that he was not used too. He had to overcome those using perseverance in order to get closer to accomplish his dreams. This is a great theme for students to know. As I said earlier, Frenkel wasn’t perfect. He made mistakes but he used those mistakes and asked the right questions to point him in the proper direction. I would love for my future students to find something to commit too even though at first they might not succeed but will learn to try again. Finally, the last concept is love and appreciation for the process. Frenkel didn’t love math right away but he gave it a chance and eventually appreciated the subject enough to devote all his time into learning as much as he could. This is another goal of an educator. You want your students to want to learn or at least want to try to learn because once they understand it they begin to “love” it.

I want to end with one of my favorite quotes from this book because I feel like as a future educator we are sometimes overlooked. On page 129, Frenkel is giving credit to his advisor Borya on an epiphany moment of how he effected his life. He states “it’s hard work being a teacher! I guess in many ways it’s like having children. You have to sacrifice a lot, not asking for anything in return. Of course, the rewards can also be tremendous. But how do you decide in which direction to point students, when to give them a helping hand and when to throw them in the deep waters and let them learn to swim on their own? This is art. No one can teach you how to do this.”]]>

Jen Silverman has been studying the trisection of a cube. She found a Chinese mathematician by the name of Liu Hui. Lui Hui focused his attention mainly on the Pythagorean theorem. Thus, Jen wanted to find a way for her students to figure out a formula for the volume of a cone or pyramid using Lui Hui solids. After hard work and determination, she created a three 3-D figure involving different shapes that create a cube when put together. Her diagram/construction outline is shown to the left.

The most important aspect in this diagram is the relationship these three figures have with each other. She has stated that Bienao is one-sixth of the cube, Yangma is one-third of the cube, and Giandu is one-half of the cube. However, that is not all the relationship aspect she has created. She also states that the Bienao is one-third of the Giandu and one-half of the Yangma, and the Yangma is two-thirds of Giandu. Thus, you put them together to create a perfect cube.

I think this topic of finding hidden shapes within shapes is so interesting; especially Jen’s thought process behind this creation. We got a chance to look at the cube that she created and tried to draw an outline in order to make the certain cubes. I thought doing this activity backwords (seeing the figure and then creating its shape) was interesting and challenging. My final work at the end of this blog.

I realized that in order to create the outline I needed to figure out common shapes that the three cubes shared. I found that each section either has a square or a triangle that is connected to another shape. After that I eye balled the image in which a certain shape is connected to that shared shaped. Overall, it was a guess and check mentality throughout finding these images. I thought it was also interesting that my drawing were different then her's but still made a cube in the end. Thus, it opens up the discussion with students about multiple approaches to figure out a problem. I would love to figure out multiple ways to approach this. I love that Jen approached it in a sense of finding a formula. Thus, making connections between geometry and algebra in an interactive way. If anyone can find her work or method I'd be interested in reading it. I've tried searching for it online but was running up short everytime.

If I were to incorporate this activity into a classroom, depending on the grade level, I would work backwards with the older children. With the younger elementary levels, I would see if from the diagram they can find common shapes and how these shapes are connected. Overall, I really liked how there were multiple relationship connections throughout this activity. I am very biased because I find it more entertaining to have hands on activities for students to learn by doing. Therefore, this activity fits my mentality down to a tee!

]]>The most important aspect in this diagram is the relationship these three figures have with each other. She has stated that Bienao is one-sixth of the cube, Yangma is one-third of the cube, and Giandu is one-half of the cube. However, that is not all the relationship aspect she has created. She also states that the Bienao is one-third of the Giandu and one-half of the Yangma, and the Yangma is two-thirds of Giandu. Thus, you put them together to create a perfect cube.

I think this topic of finding hidden shapes within shapes is so interesting; especially Jen’s thought process behind this creation. We got a chance to look at the cube that she created and tried to draw an outline in order to make the certain cubes. I thought doing this activity backwords (seeing the figure and then creating its shape) was interesting and challenging. My final work at the end of this blog.

I realized that in order to create the outline I needed to figure out common shapes that the three cubes shared. I found that each section either has a square or a triangle that is connected to another shape. After that I eye balled the image in which a certain shape is connected to that shared shaped. Overall, it was a guess and check mentality throughout finding these images. I thought it was also interesting that my drawing were different then her's but still made a cube in the end. Thus, it opens up the discussion with students about multiple approaches to figure out a problem. I would love to figure out multiple ways to approach this. I love that Jen approached it in a sense of finding a formula. Thus, making connections between geometry and algebra in an interactive way. If anyone can find her work or method I'd be interested in reading it. I've tried searching for it online but was running up short everytime.

If I were to incorporate this activity into a classroom, depending on the grade level, I would work backwards with the older children. With the younger elementary levels, I would see if from the diagram they can find common shapes and how these shapes are connected. Overall, I really liked how there were multiple relationship connections throughout this activity. I am very biased because I find it more entertaining to have hands on activities for students to learn by doing. Therefore, this activity fits my mentality down to a tee!

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.

]]>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.

]]>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.

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.]]>