The first book I read this semester was The Math Book by Clifford A. Pickover. The second book that I read was Mathematician's Delight by W.W. Sawyer. I did not get to read the whole book, but I was able to skim through the entire book. On W.W. Sawyer's website the book is explained like this, "In this book Sawyer criticizes the teaching of math without context. The best way to learn geometry is to follow the road which the human race originally followed: Do things, makes things, notice things, arrange things, and only then reason about things." This book seemed interesting to me because it starts with simple arithmetic and algebra and proceeds by gradual steps through graphs, logarithms, and trigonometry to calculus and the world of numbers. This book is very thin and easy to read. It W.W. Sawyer knows how to teach and he shows that throughout this book. This book is a great place to start exploring mathematics! I would recommend this book to anyone that would like to explore mathematics, or would even like to know more about mathematics!
A magic square is an NxN matrix in which every row, column, and diagonal add up to the same number. Srinivasa Ramanujan was an Indian mathematician. He had almost no formal training in pure mathematics, but made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. When he was 12 years old, someone gave him a trigonometry book, and he taught himself. At 16 years old, a friend of his family gave him a math encyclopedia with over 6,000 theorems. That was all the math training he had. At age 23, he generated a formula that would calculate all primes up to 100,000,000. He then was invited to move to Cambridge and there he proved or conjectured over 3,000 results, including the best algorithms we have to this day for generating the digits of pi. He died at age 32. Ramanujan created a super magic square. The top row is his birthdate (December 22, 1887). This is a super magic square because not only do the rows, columns, and diagonals add up to the same number, but the four corners, the four middle squares (17, 9, 24, 89), the first and last rows two middle numbers (12, 18, 86, 23), and the first and last columns two middle numbers (88, 10, 25, 16) all add up to the sum of 139. Check out his magic square below.
Seeing his magic square made me interested into making my own birthday magic square. My birthday is August 14, 1992, so that was my first row. Then I started looking at Ramanujan's magic square and subtracted or added the difference to each number in his magic square. I ended up with this...
I realized this did not work because not all the rows, columns, and diagonals add up to the same number. Not only was this not a super magic square like Ramanujan's, but this was not a magic square at all. I then started playing around with the numbers row by row and I was then able to get every row and column to add to the same number (133), but not the diagonals. Here is what my new magic square looked like...
Next, I messed around with these numbers to try and get the diagonals to add to 133, but I could not get it. Then I found this website that showed you how to do a Ramanujan's magic square for any special occasion date. I was finally able to solve my birthday special magic square! My number that everything needs to add to is 14+8+19+92=133. Here is my birthday magic square!
I wanted to see if I could find another birthday using this same formula, so I decided to try my nieces birthday August 30, 2011. Her number that everything needs to add to is 30+8+20+11=69. Here is here birthday magic square!
Having googled the answer to finally figure out the solution, I noticed a pattern within the way to solve a birthday magic square. I noticed two boxes together always equals a different two boxes added together. Then those four boxes are then apart of another equation to find out the next boxes. For example the second step is b+c=m+p. Well then the b, c, m, and p are all apart of a different set of equations. B is apart of b+n=g+k, and so on, but once a letter is used twice, it is not used again.The mathematics behind the birthday magic square is rich! It may look like simple addition, but the patterns within the addition is what makes it rich!
I think this is a fun activity to try. You can do this with any date! First, I would try it on your own to see if you can create your birthday magic square without the help of the website, just grab a pencil, paper, and calculator. I came close to figuring mine out without the help of the website! Then after spending some time and having different magic squares that do not work, go to the website here and try to see where you went wrong within your birthday magic square!
In mathematics everyone talks about famous men mathematicians. I have been studying mathematics for four years and I have rarely heard of female mathematicians. I did some research and found 5 female mathematicians that everyone should know about!
1.) Hypatia (ca. 350-415)
Hypatia was one of the earliest female mathematicians. She was the daughter of Theon. She followed in her father's footsteps in the study of astronomy and math and Hypatia was a philosopher. "It is thought that Book III of Theon’s version of Ptolemy’s Almagest—the treatise that established the Earth-centric model for the universe that wouldn’t be overturned until the time of Copernicus and Galileo—was actually the work of Hypatia." To learn more about Hypatia please click here.
2.) Sophie Germain (1776-1831)
Sophie Germain studied mathematics and geometry. She taught herself Latin and Greek. Since she was female she was unable to study at École Polytechnique. Joseph Lagrange was a faculty member there and Sophie created a false name so she could attend. When Joseph found out Sophie was a woman, he became her mentor. She was the first woman to win a prize from the French Academy of Science. Her prize work was on a theory of elasticity. To learn more about Sophie Germain please click here.
3.) Ada Lovelace (1815-1852)
Ada Lovelace never knew her father. Her mother was very protective of her. Her mother wanted her to study science and mathematics. Lovelace began to work with Charles Babbage, who was an inventor and mathematician. She was going to help translate an Italian mathematician's memoir. She went beyond completing the translation. She wrote her own set of notes about the machine and included a method for calculating a sequence of Bernoulli numbers. This is acknowledged as the world's first computer program. To learn more about Ada Lovelace please click here.
4.) Sofia Kovalevskaya (1850-1891)
Sofia was Russian. She was not allowed to attend a university because she was a Russian woman. She married Vladimir Kovalevsky and moved to Germany. In Germany she was not attend university lectures, but she was privately tutored. After writing treatises on partial differential equations, Abelian integrals and Saturn's rings she received her doctorate. She eventually was able to teach lecture in math at the University of Stockholm. Then she became the first woman in that region of Europe to receive a full professorship. She won the Prix Bordin, the French Academy of Sciences in 1888, and a prize from the Swedish Academy of Sciences. Her prize work was focused on the rotation of a solid body. To learn more about Sofia Kovalevskaya please click here.
5.) Emmy Noether (1882-1935)
Albert Einstein refereed to Emmy Noether as “the most significant creative mathematical genius thus far produced since the higher education of women began.” With rules against woman at universities in Germany, Emmy's mathematic education was delayed. She obtained her PhD and was still unable to obtain a position as a professor. She had the title of unofficial associate professor at the University of Göttingen. In 1933, she lost that position because she was Jewish. After losing her position at the University of Göttingen, she moved to America and did research and lectured at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. In America she made significant advances in the field of algebra. To learn more about Emmy Noether please click here.
There are many important woman in the field of mathematics. These are five that I enjoyed learning about. I learned about these five women through The Smithsonian Magazine here. The Smithsonian Magazine has wonderful articles about woman in mathematics. To search for articles on the Smithsonian Magazine click here. Many think that mathematics is a man field, but without some of these important woman, math may not have developed the same. Men and Women both have helped shape mathematics as we know it today! These women have influenced my mathematical career. Without them standing up for themselves, the number of women mathematicians would have decreased. In a report by Catherine Hobbs & Esmyr Koomen from the Department of Mathematical Sciences, School of Technology, Oxford Brookes
University, Wheatley Campus, in 2006 they have shown the percentage of women mathematicians increasing. This graph below is one of many from their report.
To read the full report please click here. These women helped me be able to study mathematics. Without them, mathematics may have been shaped differently and I may not have been able to be a mathematics major. One way things would have been different without these women is the first computer program was created by Ada Lovelace. I have used many different computer programs in the past years. If she did not invent the first computer program, we would not be able to know if someone else would have created one and life would be completely different. I use a computer for everything! These women changed not only mathematics, but they changed my life too!
I just finished reading The Math Book by Clifford A. Pickover. I would highly recommend this book to anyone! This book was very informative, and a variety of people could relate to the book.
The layout of this book was very simple. It was in chronological order and provided the major milestones that have created mathematics. There were 250 milestones in the history of mathematics. These milestones include the greatest mathematicians. The author starts with a milestone in c. 150 Million B.C. and ends in the year 2007. Some of the milestones include history behind items that people do everyday, like play tic-tac-toe! The book was written in 2009. Each milestone is one page (left side) and then on the right side there is a picture to go along with the milestone. It is a very quick and easy read.
I have picked a few examples from the book that were my favorite to share. The ones I have picked to share I think most people would enjoy learning about even if they do not like mathematics.
· Ant Odometer (c. 150 Million B.C.) : This was the first milestone listed in this book. German and Swiss scientists discovered that ants “count” steps to judge distance. After ants had reached their destination, the scientists would add stilts or shorten their legs to see how the ants traveled back to their original destination.
· Dice (c. 3000 B.C.) : Can you imagine a world without dice and random numbers? Dice were originally made from anklebones of hoofed animals. Dice are used to teach probability. Now dice are used for many things people do everyday.
· Rubik’s Cube (1974) : Have you ever messed around with a rubik’s cube and got frustrated that you could not finish it? I have! After reading about the invention of the Rubik’s Cube I started playing around with one again because I was determined to figure it out. In this one page summary of the Rubik’s Cube, you find out that no configuration requires more than 20 moves to solve. That is just crazy to think about for me, since I have worked with Rubik’s Cubes for hours and could not figure them out!
Those are just three of my favorites that I thought most people could relate to! This book is a quick, easy, and enjoyable read. It is just fascinating to read about the major milestones in mathematics that you would have never thought was a major milestone for mathematics. I would recommend everyone to read this book to see what you can relate to, and think about how your life would be different if these were never created within mathematics!
In class as a group we had to create a bungee cord that would hold our action figure Thor. The object was to create the best bungee cord that made you come closest to the ground, but not to hit the ground. We were only allowed 10 rubber bands, and we had to base a prediction off of our 10 rubber bands. Then we were able to make our bungee cord with as many rubber bands as we predicted. Then as a class we let our characters bungee jump!
Why did we do this and how is it related to Mathematics you ask? Well, it is related through Galileo. He was an Italian physicist, mathematician, engineer, astronomer, and philosopher. Galileo produced some mathematics. These were called Galileo's Paradox and it showed that "there are as many perfect squares as there are whole numbers, even though most numbers are not perfect squares."
Another thing Galileo explored was gravitational pull on an object. Galileo wanted to see what would happen if he had two of the same objects, but different weights, when released from his hands, what would reach the ground first. With how far science and math has come, we now know that the two objects would hit the ground at the same time because of gravitational forces, but Galileo did not know that then, so he collected data and made his assumption. You can explore Galileo's Experiments here.
This is where the bungee jump came into play with our math class. Before our final predication of many things, we collected data. We tied our 10 rubber bands together and wrapped one rubber band completely around our figures legs, therefore leaving us with 9 rubber bands in our bungee cord. We then grabbed a meter stick and started measuring how far Thor fell with 2 rubber bands, 4 rubber bands, 6 rubber bands, 8 rubber bands, and 9 rubber bands. We saw an increase in length, but a decrease in length between each rubber bands. What this means is with 2 rubber bands we saw a fall of 85cm, 4 rubber bands we saw a fall of 105cm, and 6 rubber bands we saw a fall of 115cm. As you can see the total fall is increasing, but the distance between is decreasing. We could only experiment and get data with those 9 rubber bands, and then we had to make a prediction. We knew the ground was 505cm. We used a linear regression and graphed our data to make a prediction. The graph we used told us that we needed anywhere from 40 to 55 rubber bands, which we thought was high. We then used a linear regression and found we need 35 rubber bands. We wanted to be on the safe side so we used 34 rubber bands. This meant that we had 33 rubber bands in our bungee cord and one was completely tied around Thor's feet. You can watch how close it was below.
As you can see our first attempt was very close. Thor did not hit the ground, but he was about 2-4cm from the ground. I would say that was a successful bungee jump!
In the next video you can see another bungee jump (Hippo) and then Thor's bungee jump at a different angle, and in slow motion.
I think this is a great lesson for students to try. Within this lesson there is history, science, mathematics, and it is hands on! Students get to make a hypothesis and try it to see how close they were. I think students would enjoy this lesson. I know I enjoyed it!
You should try it yourself. Find a ledge to do the bungee jump with an action figure or Barbie, grab some rubber bands, and TRY IT!!
I have worked with nets of a shape in a lot of my math classes before. A net is a pattern that you can cut and fold to make a model of a solid shape. In MTH 495 this summer, we started working to form a cube, but with three different shapes. The three shapes that formed together to form the cube were a square pyramid, triangular prism, and triangular pyramid. This is not as easy as it sounds! Take a look at the original nets that make the 3D shape below.
Now, the nets below are the nets for the three shapes that once built, form together to make a cube.
As you can see there is a difference between the "typical" net for the shape, and the net needed to complete this task of having the three shapes form together to make a cube. Nets are fun to work with. I have found a fun website that allows you to move 3D shapes, see their nets, count the number of edges, and just lets you explore the shape. On this website you can even make your own net to see what type of shape it makes.
Tessellations are defined as a pattern of shapes that fit perfectly together, and is when you cover a surface with a pattern of flat shapes so that there are no overlaps or gaps. A regular tessellation is a pattern made by repeating a regular polygon. I have worked with tessellations in a lot of different classes. I think tessellations are a lot of fun. They let you be creative and you see different patterns within the original pattern. In MTH 495 I started working with pattern blocks to create the pattern below.
Pictured Above: Pattern created with pattern blocks in MTH 495.
Then, Professor Golden saw a different pattern within my pattern and started working with the same pattern on isometric dot paper. He kept my same pattern, but outlined it different. His pattern is shown below.
Pictured Above: Professor Golden's view of my pattern from the original pattern I created.
Then when I went home, I started working with the pattern I created in class on isometric dot paper. With the pattern I saw 2 potential patterns people could see. In the picture below you will see three patterns. The first is how the pattern blocks were colored. The second part is the same pattern just black and white. The third pattern is one that people could see if they ignore the rhombus and triangles drawn on the outside of the hexagon. On the top part of the paper, you will see the pattern formed together to see how it would look. On the bottom part of the paper you will see one big pattern block from each to see what the inside looks like.
Pictured Above: Original pattern shown 3 different ways.
This is just one tessellation that I worked with. What makes this tessellation work? What makes this tessellation work is the smooth sides. you could fill the large outside hexagon ring with triangles, rhombus, or trapezoids. The length of the edges in the different shapes always equals at least one other edge on a different shape. That is why this tessellation works! My Professor started working with this same tessellation. See his blogpost here.
I think tessellations are a lot of fun. Also, I think working with tessellations with students would be fun because they have a very creative imagination. I found a website that I would use to introduce tessellations to my students. It tells students what tessellations are and has examples of some. The website I found that would be helpful to students is called Cool Math 4 Kids.
Cool Math 4 Kids
I have worked a lot with the Pythagorean Theorem. I used this theorem to solve problems in elementary school, middle school, high school, and college. It was not until college that I actually learned how it formed and why it worked. In two of my math classes this previous year I had to write proofs for the Pythagorean Theorem. I found a video online, that I find the easiest way prove the Pythagorean Theorem. In this video the guy shows how to arrive at the equation. I looked back onto one of my proofs I wrote for Euclidean Geometry and I wrote it using the same method. I think this would be a great video to show in high school or even middle school to students using the Pythagorean Theorem. It is the most basic proof of this theorem and it would give the students a background of why the equation/theorem works!
Check out the video here.
Euclid is a greek mathematician. He is known as the "Father of Geometry". He wrote the Elements. This book consisted of his five postulates. He based all of his theorems off of these postulates. This book became one of the most famous books in mathematics. It is still used today to teach and learn about mathematics. The Elements have been studied for 24 centuries in different languages. Euclid wrote a proof of the Pythagorean Theorem that is different then above. This just shows that there is more then one way to solve the Pythagorean Theorem. I have included below Euclid's proof of the Pythagorean Theorem, and an interactive site to explore Euclid's Elements if you'd like to learn more about them!
Euclid's Proof of the Pythagorean Theorem
What exactly is Math?
This question could be answered in many different ways, depending on who is asked. I do not think there is an exact definition to the word math. Since I am a mathematics major, math means more to me then some. To me math is more then just a subject learned in school, calculations, geometry, shapes, numbers, and formulas. I have found that mathematics includes explanation, critical thinking, logic, problem solving, and daily activities. Everyone uses mathematics everyday in different ways. Before college I thought mathematics was what I learned in high school. After taking multiple math classes throughout my college years, I have learned math is much more then what is on the surface. I have learned the deeper understanding of math, like why things work the way they do when math is involved. Math is an amazing topic. Nobody could ever know every single thing there is about math because it keeps growing!
Without doing research about mathematics, I do not know many big moments that have happened. I have not had the opportunity to research mathematicians. I mainly learn about the content in my classes. As a future educator, the biggest moments in mathematics are going to differ then others. The top 5 moments I am going to talk about, would change how math is taught if they were never discovered.
(1) A big discovery that most people know about is pi. Pi is a famous number and some people celebrate pi day on March 14th every year.
(2) Another big moment that has helped shape math is the Pythagorean Theorem. This theorem helps us in triangles and finding missing side lengths. In one of my math classes I had to prove the Pythagorean Theorem and I found that most beneficial because I was able to see how it worked.
(3) A huge discovery in math is the basic operations. These basic operations include addition, subtraction, multiplication, and division. People use these operations everyday. In my eyes they are the most important discovery in mathematics.
(4) NUMBERS! What would math be if there were no numbers? It would be very hard to do math without any numbers.
(5) In grade school most of us had to learn the quadratic formula. If I did not have to learn the quadratic formula I would not have been able to solve some math problems. I think the quadratic formula is powerful tool used in algebra. It helps us solve quadratic equations.