Helpful+Links+for+MATH

__[|Marcus du Sautoy: Symmetry, reality's riddle]Link (use the Insert Link button in the Editor toolbar, choose "External Link.")__ This will be followed by a **proper MLA citation**. Use of online generators such as citationmachine.net is acceptable.
 * Your reason for including this link. You might consider the following questions: How did you find it? Why did you find it helpful? What concerns might you have about this link? (Your name goes in parenthesis at the end of the bullet point.) (With name)
 * A review from one other student. (With name)

Also available: My PowerPoint that I used to introduce certain "knowledge issues" in Math:
 * [[file:TOK Math various topics.ppt]]

= = HELPFUL LINKS FOR MATHEMATICS "Using math to feed your dog!" youtube video August 29 2010. http://www.youtube.com/watch?v=5C67-mPqm4M&feature=related I found this video (link is above) to be very helpful. The video shows the problem that any dog owner is presented with, feeding your dog properly. Using basic equations and arithmetic, the owner was able to find the answer. This reinforces the idea of math being more of a way of knowledge rather than raw knowledge. The video is somewhat 'cheesy' but it presents a everyday problem, involving math, in other words, the blend of the physical, using concise concepts. (Filippo)

[|link] "Nature By Numbers." //Beauty Of Nature And Fibonacci Theory: The Digital Presentation//. Web. 6 Feb 2011. .

* I decided to post this link up here because i found it really cool how all these mathimatical formulas and concepts show up in nature around us. it shows some concepts that where mentioned today in class such as the golden ratio. Also it raised the question again for me wether math is invented or discovered. From this video, i think that math is a way of explaining to each other concepts and phenomena that we find around us in nature. It is something that surrounds us and we only catch and realize glimpses of it as we observe our surroundings. I think that math is discovered. ( JP Kwak)

"Numbers at Play." Video Channel on TED.com. TED Conferences, LLC, n.d. Web. 26 Jan 2011. .
 * I love the TED videos-- they are always thought-provoking and short enough that it won't make your brain hurt for *too* long. The one we saw in class was "Arthur Benjamin Does Mathemagic" on page 2. If you check it out, try to read through the comments, too, for different perspectives of the various viewers over the years. (Wall)
 * I was reading through the comments section and there were a load of haters and doubters. Despite this, I read a comment made by someone that I really liked. Arthur Benjamin is a "Mathemagician", and regardless if the Math is correct or not, he got me and probably many others watching the video to actually be wow'd and amazed by the usually boring subject of Math! (Sean Williams)

"Naturally Occurring fractals." 1.1 n. pag. Web. 17 Feb 2011. []
 * I found this link by searching for math in nature, after watching many different and interesting documentaries about how mathematical theorems appear in nature. This link gives a very good explanation of fractals with a further link, and provides us with many thought-provoking pictures, which showcase the fractals in nature. I found that using the knowledge of fractals and their appearance in nature, I drew the conclusion that math must be discovered, and not invented. There are other problems with this link, however. No real author is cited, and no other author's work is cited either. Also, the site calls itself "your favorite source of random info", so it should not be taken too seriously. (Gerard Belmans)

"Golden Ratio." //Wikipedia// 1.1 (2009): 1. Web. 17 Feb 2011. <[]>
 * After some further research of mathematical properties in nature, I remembered the golden ratio, something considered very important in all forms of art, and even in beauty. I thought it was very interesting that even human ideas of beauty itself can be expressed as a ratio, and that is the golden ratio. The link contains a good amount of information of obtaining the ratio, its history and all its applications discovered so far. I found it by searching up the golden ratio. Since the website is Wikipedia, there are many concerns associated with this link. It can be edited by anyone, and although it is all properly cited, It isn't always accurate. (Gerard Belmans)

"The Math Used in Professional Soccer." Math Worksheet Center. Math Worksheet Center, n.d. Web. 29 Jan 2011. .
 * Me, being a sportsman, decided to look into how math is used in sports. This page tells us the many ways math is used in a sport like soccer. The main part I liked was the part where it said that "Fantasy Sports Teams" are a good way to apply math to sports. I myself play Fantasy Football(Soccer), and I do realize that a lot of math is used to calculate the cost of players and predicting how many points they will get you. In a way it is applying math and certain aspects of Economics to the art of Sports. (Sean Williams)
 * Good site, but I found some of the concepts explained in this site to be rather rudimentary. The title states that "Math [is] Used in Professional Soccer," but that can be said about any sport; the speed and velocity of a traveling ball can be applied to every sport. However, this does show that mathematics surrounds us. Is it safe to say that without mathematics, sports aren't possible? Is it safe to say that without mathematics, nothing is possible? Moreover, another question arises: Is mathematics, thus, "real"??? (Harry O'Sullivan)
 * I found this site pretty interesting and I believe that applying math to professional soccer actually works. By using math, one can predict who will win and by how much. I watched a program that predicted the winner of 2010 world cup. The mathematicians predicted Spain as the winner calculating how much they score every match, their match strategy, and match records. The mathematicians said that this statics could be false since there could be unexpected lucky shots or mistakes by goalkeepers. However, they said that ideasitically Spain willl win the world cup using math and Spain did win. So math can actually be used in this world. As Author Benjamin says in ted.com ([]), statistics of math can be applied in our lives effectively. (Chae Young Moon)
 * I found it intriguing how math concepts could be taught to students in a unique way like this - by relating them to sports. I think that this is a good way of teaching kids who may not find math appealing to them at first, as owning a fantasy soccer team is arguably much more interesting than solving math problems for homework. However, I also agree with Harry's comment on how math could relate to any sport, and how nothing is possible without math. I think it can be said that almost anything we do is related to math in some way, which is why it is such a vital area of knowledge. (Edward Cannell)

<span style="color: #333333; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">Terr, David. "Math and Sports." Math Lessons. Math Amazement, 2009. Web. 29 Jan 2011. <http://www.mathamazement.com/Math_&_Society/Math-and-Sports.html>.
 * <span style="color: #333333; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">I have always wondered why American sports seem to be obsessed of stats and numbers (elements of math). For example.. "Kobe Bryant's lifetime freethrow percentage is 83%" I don't understand the need to stress someone's mathematical stats because in Sports, different situations have different circumstances.. therefore a number does not determine if a shot will go in or not. David Terr says in one line that these numbers "distinguish players". I dont think math and its numbers should be used to determine someone. (Sean Williams)
 * <span style="color: #333333; font-family: Tahoma,Geneva,sans-serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">I think many people are obsessed with numbers and stats of characters in online games....Those people play games to raise their character's power, speed, stamina, and etcetera so that they can stand above others. Actually, boys usually play a lot of Pokemon and try their best to raise Pokemons' level. Perhaps numbers are easy to compare, so people might be using those stats because it makes people feel "smart". (Shawn)

<span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"The Story of Math." //The Story of Math: Language of Universe//. BBC: Television. 3 Feb 2011. <[]>.
 * =<span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">This is a BBC documentary about Math. As the title says, the whole documentary talks about why we created the math and what math is all about. I came to find this video because I was thinking about what the real quality of math is. When we say math, the first things that come into our mind are probably addition, subtraction, multiplication, and division. However, this video presents us different aspects of math. It talks about how math is used in different times of history for not only purpose of calculating but for communicating with the others. Also it talks about how different people of different periods approached math differently. However, the thing that I found most interesting in this video is that it shows how these mathematics are in our lives without us realizing it and how we use it unconciously. I think if one watches this video, one can watch math in different perspective. People will be able to perceive math as language used by all people to describe innumerable objects rather than boring addition, subtraction, division, or multiplication of numbers. (Chae Young Moon) =
 * When we think about mathematics, we often first think about additions and subtractions, the "boring" parts of math. However, these "boring" parts of math is only a visual aid that helps us understand theories with larger concepts and ideas. This video really helped me in understanding that math helps us find patterns in the nature that seems unexplainable, but can be explained through using mathematics. It also helped me understand that humans are not the only animal in this world that has a concept about mathematics. Even small animals, such as spiders can have a mathematical concept, in the form of sense of distance. (Jangho Seo)

"Marcus du Sautoy: Symmetry, reality's riddle." //TED//. Web. 7 Feb 2011. <http://www.ted.com/talks/marcus_du_sautoy_symmetry_reality_s_riddle.html>. __[|Marcus du Sautoy: Symmetry, reality's riddle]__
 * "...but contained inside those documents, was a new language, a language to understand the fundamental concepts of science." Upon hearing those words, I found myself thinking back to our class discussion. Without a language, can maths be expressed? Anyhow, I came across this clip whilst watching a few TED videos. As Mister Wall said, TED videos are intriguing and yet relatively short, so I thought this video would useful and appropriate. The speaker goes on to explain how we can see elements of mathematics in our everyday lives, and how symmetry is what defines a lot of shapes and figures. Also, he goes on to talk about how the number zero was "invented" by an Indian mathematician. "Everything has symmetry," the speaker states. I am not sure to what extent this is true, but I found the video interesting and relevant. (Harry O'Sullivan)
 * I noticed that this is the same guy that came our on the BBC documentary that we watched during class. This video got me into a new concept of mathematics. Although I don't pay much attention of symmetry, now I know how symmetry can play an important role in our lives. Also, this video taught me about symmetry more in depth. The speaker claims that everything has symmetry, which I think is a generalization, but at the same time I think its an interesting concept that we can explore to find out more about mathematics. Also, while watching this video, I kept on asking myself, was symmetry invented or discovered? (Jangho Seo)
 * This video is a spectacular video that shows the hidden patterns in the nature. There are symmetries all over the places in our lives and they are used daily by numerous organisms (?) or hidden everywhere. The another great thing about this video is that, as Harry mentioned, the fact that zero was invented by an Indian mathematician was very interesting. Also the speaker claims that everything has a symmetry within it. I think this is true because, everything in our world has some kind of patterns and without those patterns this world cannot exist and I think symmetry is one of those patterns. (Chae Young Moon)

<span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"MATHEMATICS IS REAL: WHY AND HOW?." //The Way to Truth//. The Way to Truth, 26 Oct 2000. Web. 7 Feb 2011. [|Mathematics is Real] <span style="font-family: Arial,Helvetica,sans-serif;">[|Jamieson, Anne. "Do Humans have an Innate Capacity for Mathematics?."][|//Science Clarified//][|. The Way to Truth, n.d. Web. 7 Feb 2011.][|Do Humans have an Innate Capacity for Mathematics?]
 * <span style="font-family: Arial,Helvetica,sans-serif;">I thought this website based on ideas of Islam would be interesting to look at. The author here explains the origin of math, and the two schools of thought. The Formalists claim that all equations are based on certain postulates. What I found astonishing was a Turkish way to check the addition." Let us say, we are going to add 154 to 275, for which we get the answer 429. Adding the digits of each of the first two numbers, we get 1+5+4 = 10 and 2+7+5 = 14. The next step is to subtract 9 from each of these two sums, giving us 1 and 5 respectively. The third step is to add these two results together, 1+5 = 6. Now we do the same thing with the digits of the answer we are wanting to check, namely 429, and again subtract 9: 4+2+9 = 15, 15-9 = 6. The fact that we end up with the same number (i.e. 6) means that our addition was correct. This way of checking an addition exists independently of us. We did not create it, we discovered it." This substantiates the notion that math is DISCOVERED, not invented. This website, in short, states that "the universe has a mathematical order or mathematics is the branch of science studying the miraculous order of the universe, the order which the Absolute Orderer and Determiner, One Who determines a certain measure for everything, has established." We may refer to the various examples shown here to support our opinions for the discussion onward. (Shawn)
 * <span style="font-family: Arial,Helvetica,sans-serif;">An interesting link for sure. Covers fibonnaci's famous series of numbers, as well as their appearances in nature. Also provides excellent points in favor of math existing before we "invented" it, and also of Formalists, those who think humans invented the numbers when we learned to count. A good link if you need to brush up on arguments of either side of the 'Is math invented' debate. (Gerard)
 * <span style="font-family: Arial,Helvetica,sans-serif;">This website organized plentiful information and arguments of people who believe in an innate ability of math and those who don't. It helps us examine what these statements are all based on, and because this website refers to many of these scientific researches, it is an interesting website to skim through and it will contribute to our discussion. In essence, what the website is saying is that"...Mathematics itself is not innate but rather a cultural acquisition. In fact, we might say that mathematics is an emergent property of the human mind and culture combined, in as much as the immense intellectual system that is mathematics today is far greater than the sum of the properties that form it." She denies the notion that mathematical abilities reflect one's level of intellectuality. (Shawn)
 * <span style="font-family: Arial,Helvetica,sans-serif;">Interesting link as well. I was amused to read that only a change in the number of dots on a screen will get the babies' attention, while changing the brightness, width and shape doesn't. This seems to tell us that all humans are born with a natural tendency for math. It was also interesting to read that not only humans but many animals posses the ability to count without actually doing so, a process called subitization. This seems to take place at only one place in our brains, as patients who suffer strokes in that region are unable to do this. Provides good arguments from both sides of the argument of whether or not we are 'born' with math. (Gerard)


 * <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"Why Is Math the Only True Universal Language?."//Math Worksheet Center// n. pag. Web. 8 Feb 2011 <http://www.mathworksheetscenter.com/mathtips/mathlanguage.html>. **
 * ** This website is originally the site for teachers to get math worksheet and use it during the class. However, it explains what math is in one of the pages. It says that math is the only true universal language in the world. Different countries have different languages and different culture. So in order to understand what they are saying, we have to study each others language for a long time and learn about their culture thoroughly in order to understand what they are thinking. According to the passage, we only know Roman and Arabic numerals and those are the only numbers that are used in mathematics. Different countries might have different units but it doesn't matter because it can be converted into different units and be applied in any kinds of situation. This way, math can be used as a language that can be used by any countries in the world in order to communicate with each other. So I think that the primary purpose of math was not to build tall buildings as we use math today. It was used as universal language to communicate with the other countries in order to trade learn about each other. (Chae Young Moon) **

<span style="font-family: Arial,Helvetica,sans-serif;">[|Beauty in Mathematics]"Does Beauty Equal Truth in Physics and Math?."ClockBackward. ClockBackward, 11 March 2000. Web. 9 Feb 2011. <http://www.clockbackward.com/2009/03/11/does-beauty-equal-truth-in-physics-and-math/>.
 * <span style="font-family: Arial,Helvetica,sans-serif;">After our conversation in class, and after having looked at the //Numbers// episode, you ask yourself whether or not elegance in mathematics is of such importance. In the //Numbers// episode, the main character was concerned with an equation that did not predict what it was supposed to; in turn, his 'mentor' told him that his equation was perhaps too 'elegant.' So, is elegance everything in mathematics? The following link includes an article which explores whether or not beauty is a fundamental part of both maths and physics. I think this link would be helpful for those of you who would want to do their TOK essay on mathematics, as it does explore different terms we haven't gone over in class, such as "Occam’s razor" that states "when given many possible explanations for something that are otherwise equally plausible, we should prefer the one that is the simplest." Additionally, there are a multitude of comments at the end of the article that either contest the writer's belief, or add on to his already lengthy analysis of the subject. It might be a long read, but it's interesting to explore how beauty is a "fundamental part of math and science" (Quentin)

<span style="font-family: Arial,Helvetica,sans-serif;">[|Mathematics and Education] Wolfram, Conrad. "Conrad Wolfram: Teaching kids real math with computers." TED. TEDGlobal, November 2010. Web. 9 Feb 2011. <http://www.ted.com/talks/lang/eng/conrad_wolfram_teaching_kids_real_math_with_computers.html>.
 * <span style="font-family: Arial,Helvetica,sans-serif;">Another TED video? I think this video brings to light some issues with mathematics in our education system; in this case, Conrad Wolfram believes the use of computers rather than hand calculation in the process of equation formulation for example, is vital. This is because, he believes that tedious calculation have no application with real mathematics, and the real world. As mathematics surrounds our modern world more and more everyday, from geology to the biology of x-rays; he denies button-pushing by learning on computers, and accentuates the fact that the "outside world" is a place where computers are at the center of things. The "dumbing-down" of computers is a myth, he claims, in a modern world where programming engages students in a unique way. It makes maths "more practical and more conceptual simultaneously." This is an interesting topic, as it opens so many opportunities, as it allows students to "feel" and "interact" with mathematics! He puts mathematics in context with a new economy - an improved outlook - where students have a better time doing mathematics (Quentin).

<span style="font-family: Arial,Helvetica,sans-serif;">[|Wolfram Alpha] Wolfram, Conrad. "Wolfram Alpha." Wolfram Alpha. N.p., n.d. Web. 9 Feb 2011. <http://www.wolframalpha.com/>.
 * <span style="font-family: Arial,Helvetica,sans-serif;">Just a link I think people should check out. A website developed by the man in the above TED video; it is an online service that answers factual queries directly by computing the answer from structured data, rather than providing a list of documents or web pages that might contain the answer as a search engine would. In the TED video, he typed in "Am I drunk?" and a table asking for gender, number of drinks and weight, is available to fill in. Then it calculates your alcohol blood level, and compares it to the legal driving limit, etc. This is, once again, a product of mathematics and quantitative data; from life expectancy to mortgage, it gives you answers in a way no other site I know has before (Quentin).
 * <span style="font-family: Arial,Helvetica,sans-serif;">I really like this site because as you've said, it's not just list of sheer facts. Sometimes I'm surprised by the usefulness of math, especially when it actualizes things that are within our metaphysical world. Often do we imagine living in a world where we just call out something, and it appears, like boom. Many scientists have attempted to organize all sorts of knowledge into a compact form, and one product of such trials is encyclopedia. However, this link goes beyond the level of those thick fat boring books; it's astonishing. (Shawn)

<span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"proof of 1+1=2." //Physicsforum//. Jelsoft Enterprise .Ltd, 05 Jan 2010. Web. 10 Feb 2011. <http://www.physicsforums.com/archive/index.php/t-58069.html>. <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"Math is Everywhere." //Youtube//. Web. 10 Feb 2011. <http://www.youtube.com/watch?v=vFRTgr7MfWw> "Does Mathematics reflect Reality?"[|Does Mathematics Reflect Reality?] <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"Does Mathematics Reflect Reality?." //In Defence of Marxism//. N.p., n.d. Web. 11 Feb 2011. <http://www.marxist.com/>. > This website will provide a motive for you to reconsider the definition and limitations of mathematics as many of us are still blindly advocating math as the "truthful" study. (Shawn)
 * <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">This website makes you think a little bit more about the principles of mathematics rather than just accepting the knowledge we learned during our math classes. How come we know so sure about the answer to 1+1? How are we sure that there are no other answers for this questions? Once you start thinking about these, you will probably find out that it is actually quite hard to prove. This site contains discussions between different internet users with different opinions about the question, "what is 1+1?" (JAngho Seo)
 * <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">This is actually an interesting topic because truly, not a lot of teachers can answer this question with sufficient explanation. Maybe I should ask Mr. Donea one day, but perusing the responses in this link, I can hardly find a definitive answer that can complete prove that 1+1 is really 2. What I liked, though, was that 2 is just a nominal term for a whole number after an isolated term which is 1. The discussion gets even more complex when this guy asked how we can know that, and how we are able to "perceive" it. (Shawn)
 * <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">This video, a scene from the series we watched during class "Numb3rs", shows how deeply we are actually involved in math. Maybe you are wondering why people try so hard to emphasize mathematics in our academics. This video might explain to you why, and how math relates to our life and nature we live in. (Jangho Seo)
 * <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">I must say that the things such as the fibonacci sequence and the golden ratio are amazing things. How these things are found in nature and in the world that we live in.. not only in math textbooks at school, does make me feel that math was not "invented" or "created", but that it was already there for humans to find. (Sean Williams)
 * <span style="color: #333333; font-family: 'times new roman',times,serif; font-size: 12px; letter-spacing: 2px; line-height: normal;">"Math is nature's language. It's a method of communicating directly with us." Despite the fact that it was only a minute and a half, this clip was truly inspiring. Firstly, I didn't know about the Fibonacci sequence, and how it was all around us. It's truly mystifying that this "golden ratio" appears to be foundation of many objects and patterns that not only human beings have created, but of nature as well. Mathematics certainly surrounds us. I still don't quite understand the concept of it being "nature's method of communicating to us," but now I can fathom the importance of learning such a "language" to gain a better understand of our environment. (Harry O'Sullivan)
 * I loved the way how he explains mathematics as the real world, itself. By adducing number of peddles in each row (flower), he gradually developed the idea of 'golden ratio' in Greek to the real world theory. He gives famous examples to make woman understand more clearly like pyramids at giza and the parthenon athens. At the last, he even used the card on the table to explain that it's based number he/she can find flower. The quote "Math is nature's language. It's a method of communicating directly with us" was very inspiring and impressive for me to realize the importance and universality of mathematics in our real world. (Soo Hyung Jung)
 * Marx had his distinct theories about mathematics and epistemology. The website deals with numerous contradictions and complex concepts that are still under discussion of mathematicians, such as Zeno's paradox of infinity, and Galileo's contradiction of axioms. What really drew my attention to the article is this. The attempt to eliminate contradictions from mathematics only led to new and insoluble contradictions. The final blow was struck in 1930, when Kurt Gödel published his famous theorems, which provoked a crisis, even calling into question the fundamental methods of classical mathematics:"As late as 1930 a mathematician might perhaps have been content with accepting one or another of the several foundations of mathematics and declared that his mathematical proofs were at least in accord with the tenets of that school. But disaster struck again in the form of a famous paper by Kurt Gödel in which he proved, among other significant and disturbing results, that the logical principles accepted by the several schools could not prove the consistency of mathematics. This, Gödel showed, cannot be done without involving logical principles so dubious as to question what is accomplished. Gödel’s theorems produced a debacle. Subsequent developments brought further complications. For example, even the axiomatic-deductive method so highly regarded in the past as the approach to exact knowledge was seen to be flawed. The net effect of these newer developments was to add to the variety of possible approaches to mathematics and to divide mathematicians into an even greater number of differing factions." (10)

"Mathematics and the Nature of Language" by <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: small; line-height: normal;">F. David Peat 1990 http://www.fdavidpeat.com/bibliography/essays/maths.htm Feb 13 2011 <span style="font-family: Verdana,Arial,Helvetica,sans-serif; font-size: small; line-height: normal;">* Unlike during our TOK class discussing whether math is real or fake, some people have an unique idea that he thinks math is a language to express the nature surround us. We, human, use language to communicate with other people so that we can express what we think or feel in reality. And to simplify our language, we use mathematics. Although this site is from wikipedia (http://en.wikipedia.org/wiki/Philosophy_of_mathematics) it is talking about how mathematics have so many mechanisms to support the idea of mathematics so it might help us to understand what is mathematics. (Kiyo)

"Mathematics of Billiards ." Web. 13 Feb 2011. <http://www.youtube.com/watch?v=vTCglNcEdOU>. This video is an entertaining way to show how math and some physics is used in billiards. Although the video may be a little cheesy it strongly delivers the point that complex math can be used in a simple game such as billiards. I feel this video is very interesting because I had no idea this much thought could be put into hitting the que ball into another ball. Maybe in the future I can use these skills to beat everyone! (Aaron Olin) "The Hangover." //Math In Movies//. Web. 13 Feb 2011. <http://www.math.harvard.edu/~knill/mathmovies/swf/hangover.html>. This video clip is from a movie that most people know and love, "The Hangover". In this particular scene the character, Alan, uses math to play blackjack to win money. This is also known as counting cards and it is a very difficult skill to acquire. This form of math is used around the world and is featured in many other movies such as 21 and Rain Man. This movie makes it seem easy and quick to pick up but learning how to count cards is very difficult and requires a great amount of memory and concentration. There are many ways to count cards but the most simple is to use the high/low method. Aces, face cards, and 10's are worth -1, cards numbered 2-6 are worth +1, and cards numbered 7-9 are worth 0. You sit at a blackjack table and start to count the deck. The higher the number you have counted the higher chance you have to win. This is because most of the small cards have been used. This method is much more complex than I have described and it is much to difficult for me. I feel that counting cards could be a useful skill to have as shown in the movie "The Hangover.'' (Aaron Olin)
 * I was glad to watch the application of math into biliards, since it included so many concepts of physics. The main things on the video were force, types of energy, elastic collisions and momentum, all explain how closely related mathematics and physics are. This video seems like just playing biliard, but once again, it helps me to notice the fact that mathematics are applied to so many small details of our lives subconsiously. From now on, considering angles, spins, force, and direction would be one of the best ways to become an expert in biliards, I guess. (Soo Hyung Jung)
 * If you think about it, this could be a great way of connecting both movies and mathematics for the extended essay, if someone's interested. Or even, connecting blackjack to mathematics in general for the extended essay. As Aaron explained above, counting cards is more difficult than the movie actually makes you think it is; so perhaps, investigating different methods and maybe coming up with one on your own, or a different version of one, could be something really interesting. Afterwards, you could tie the mathematics of blackjack, with the mathematics of another card-game that involves counting of cards. This subject could be a good TOK essay too, as you could then further explore the societal problems counting cards evokes, and tolerance levels etc (Quentin).

"Math In Daily Life ." //Interactives//. Annenburg Media, 13/03/2003. Web. 15 Feb 2011. <http://www.learner.org/interactives/dailymath/>. [|Math In Daily Life] This website takes us through the usage of mathematics in our daily lives. Although some people think that learning mathematics is redundant, it actually shapes many of our decisions and our opinion. However, upon reading the first web page, I came across the words: "How can math be so universal? First, human beings didn't invent math concepts; we discovered them." I found myself thinking back at our class discussion. This site claims that math was discovered. I believe, personally, this is true. But many question that believe, saying mathematical concepts were invented to allow us to look at the world in a way that we want to look at it; they say that mathematics was invented because at first, humans did not have control over some ideas and thoughts, and so they filled the gaps with mathematical concepts. The site also mentions that mathematics is a language that is understood by everyone. This can also by tied back to our previous class discussion. I found this site to be relevant and interesting. (Harry O'Sullivan)
 * I thought this site was good into putting mathematics under a different light; through the perspective of our daily lives, we come across things that don't immediately strike us as things integrated with mathematics, but however, are revealed to be connected to mathematics in very deep ways. At the bottom of the page there is a link that connects you to a small passage on probability; this is turn, connects you to the probability of "playing" and gambling. There is also a link that brings you to a passage on Savings and Credit, and how to manage your credit cards for example, guiding you through the basics. So, through this site we learn areas of our everyday lives that are connected to mathematics (Quentin).

Moehlis, Jeff. "The Mathematics of the Movie "21" ." //The Mathematics of the Movie "21"//. 15/06/2009, 15/06/2009. Web. 16 Feb 2011. <http://www.me.ucsb.edu/~moehlis/21.html>. [|The mathetics in the movie "21"] I don't know if any of you have seen the film "21," but I definitely thought it was one of the best movies I'd ever seen in my life. Both the plot and the characters were great, but I was most intrigued by the mathematical concepts that were incorporated into the movie. Black Jack is a popular gambling game played by many people around the world. Indeed, many believe that good luck is the key to success in not only this particular gambling game, but many others in way. But a group of kids from Harvard weren't going to rely on luck to prevail against the money making Vagas Casino cooperations; they used teamwork, and a great deal of mathematics to "beat" the game. This page shows us that the maths the characters were rambling on about were legitimate and real, and it helps us understand the intricacies of the mathematics concepts incorporated in this film. Never again will you lose in Black Jack!! (Harry O'Sullivan

Vork, Lauren. "How is mathematics used in music?." "How is mathematics used in music?." 21/02/2008, Web. 17 Feb 2011<[|How is mathematics used in Music?]>

I am very interested in composition and performance through various instruments. I was curious that math would have been applied to music in diverse ways somehow. So, I found the site that tells us about application math into music. This article mainly talks about how we dictate rhythm in terms of symbols that denote, escentially, fractions during composition. Also, it says that musicians are applying math into their performances unconsciously. Mathematics also helps us to assess or analyze the melody and harmony of pieces. I thought that this website would prove that mathematics are subconsciously fused into many different areas of our lives including sports, music, gambling, and even biliards. By reading this article, I realized that mathematics plays an important role in our daily lives; although, most of people treat mathematics as theoretical and impractical subject. (Soo Hyung Jung)

Cecilia, Barnbaum. "The dance of math and physics." //Which comes first, mathematics or physics?//. 10/04/2007, Web. 17 Feb 2011 <[|Which comes first mathematics or physics?]>

Many people assume that mathematics and physics are closely related subject. Yes, it is true that physicists use mathematics as a tool to understand the nature. It is also true that there are many mathematical formulas and equations to help to understand concepts of physics. This kind of too closely linked relationship sometimes makes people misunderstand that there is not much difference in mathematics and physics. I wanted to show that how different they are and how differently they are applied to our daily lives or other fields of subjects through concise examples on this website. However, I am not saying that they are totally different areas of studies; they are in a kind of mutual relationship that helps each other in order for people to learn the concepts more easily. Einstein's general theory of gravity, how imanginary numbers gave us computers, and how Newton invented a new math are the main three examples to explain both physics and mathematics at the same time. (Soo Hyung Jung)
 * This website was interesting to read since it explains whether physics or math came first. It explains math as "abstract" and physics as "concrete." And because we move from "abstract" to "concrete," just like in writing, we now know that math came before physics. On the website, it also says that "some newly discovered abstract formulation in mathematics" eventually leads to the description of physical phenomena which we did not know of until then. (Nari)

[|Math in Architecture]Freiberger, Marianne. "Perfect buildings: the maths of modern architecture." 1 March 2007: n. pag. Web. 17 Feb 2011. <http://plus.maths.org/content/perfect-buildings-maths-modern-architecture>.

This article shows and talks about various modern buildings and how math (geometry) relates to architectural design. For example, the air currents that flow around a building is determined through math. Then, architects may use this information as a reference to design a building which would not have as much air resistance according to the conditions of the site. This is interesting because you usually don't relate designs to math. But, architects and designers use math to create designs as well. By simulating the building's aerodynamic properties before hand using math, you won't be living or working in a building that may collapse due to wind currents. (Nari)

[|Why Golden Ratio?] McVeigh, Karen. "Why golden ratio pleases the eye: US academic says he knows art secret." guardian.co.uk 28 December 2009: n. pag. Web. 17 Feb 2011. <http://www.guardian.co.uk/artanddesign/2009/dec/28/golden-ratio-us-academic>.

Since I take IB Architecture as one of my classes, I encounter the golden ratio a lot. This made me think of some questions; Why are golden ratios pleasing? Whats the difference between a normal design and a design that interprets golden ratios? Well, this article explains everything. Leonardo Da Vinci and Le Corbusier are two from many who interpreted golden ratios in their art design. Finally after many years of research and thoughts, a US academic thinks that he discovered the reason why the golden ratio pleases the eye. Adrian Bejan, a professor of mechanical engineering at Duke University, explains that "the human eye is capable of interpreting an image featuring the golden ratio faster than any other." (Nari)
 * After reading this article, I wondered whether or not if the Golden Ratio was invented intentionally after investigating and experimenting, or if it was stumbled upon by sheer accident and coincidence. I think that this relates to the question "Is math invented?" as although some mathematical formulae are discovered through investigating, others are found through chance, which is why this question is so controversial and thought-provoking. (Edward Cannell)

[|Linking Arts, Math, Perception, and Emotions] Kim, Oliver. "Linking Arts, Math, Perception, and Emotion." //TOKTalk.net//. N.p., 24 Jan 2010. Web. 18 Feb 2011. [].

This link shows two TED videos that link math to other areas and ways of knowing, such as arts, perception, and emotions. The first video is very similar to one we saw in class already - Hans Rosling links math and art (the way he graphs them) to the development of countries, but this time he applies it to the standards of living, such as a nation's economy, life expectancy, income distribution, child survival rate, and so on. The second video involves photographer Chris Jordan linking math to perception and emotion. I found this link very interesting and useful because it links math to other areas/ways of knowledge, which I think is important in TOK. (Edward Cannell)
 * These videos, especially the second video, are interesting because they link math to other areas and ways of knowing, including perception. The way that Hans Rosling and Chris Jordan put their statistics into other forms like art makes us remember and think about these worldwide issues. Something that doesn't enter our mind all the time is represented through real size scales and we can "see" the amount, while comparing it with others. (Nari)

[|Math and Music] Cox, Paul "Math and Music" 1 http://members.cox.net/mathmistakes/music.htm I found this web page on Google. Since I am interested in music, I was curious if music had anything related to math by any chance. Then, it happened to be that there are many strange connections in between these two different subjects. In this web site it shows how some musical theories could be explained by mathematics; for example musical notes are a "sound" which is a "frequency." Therefore musical notes could be displayed as a mathematical equation. The web page also shows how the Pythagorus Theorem relates to harmonics and scales in music. I personally don't feel comfortable about the fact that music could be proved and explained by using math. Music is a way of communicating your internal thoughts with people through the device of instrument. Such like how we have normal conversations with people, we do not calculate about what to talk next and how to talk beforehand. Therefore, I don't like the way how mathematicians try to conclude music as a set of mathematics. (Albert Takagi)

[|The Golden Ratio] Unknown. "The Golden Ratio." //Theory of Knowledge//. N.p., 22 Sep 2008. Web. 18 Feb 2011. [].

This link explains the golden ratio and how they can be pleasing to the eye - in this case, it shows why George Clooney is considered to be so handsome; it is because his face is divided into Golden Ratios. I think that this relates math to perception and fallacies in a way, which we covered in TOK. People may tend to perceive those with "Golden Ratio faces" as being good-looking, and it could also be a hasty generalization. (Edward Cannell)

In response to Eddie's post, I read the article on the golden ratio and being a photography student myself, I can tell you about a very similar concepts that is called the 'Rule of thirds.' The rule basically states that if an image always looks more pleasing to the eye when the subject of the photo is not in the dead center. I personally think that the golden ratio does apply to faces, though it might not be a prime indicator of beauty.

http://video.google.com/videoplay?docid=-5911099858813393554# The beauty of Mathematics by Michael Atiyah February 19th 2011 <span style="font-family: arial,sans-serif; line-height: normal;">Michael Atiyah, one of the world’s foremost mathematicians, talks about beauty in mathematics, which he defines as simplicity, elegance and truth per word, and explains why it is such an important criterion. Although it is a video, the way how he thinks about mathematics is very interesting. He thinks that mathematics is all about "Truth" and in some part i agree with his opinion about mathematics. (Kiyo)

<span style="font-family: arial,sans-serif; line-height: normal;">http://www.intmath.com/numbers/math-of-beauty.php The Math Behind the Beauty By M. Bourne February 19th 2011 This is an another website talks about the beauty of mathematics. It talks about the ratio of our face (the ratio of how attractive you are based on your face). Ratio is something we use alot in our life because it can be found anywhere and anytime. And based on the ratio, we judge what is positive or negative therefore it is something related to perception, which means i believe ratio is related to TOK