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Mathematics TOK Presentation

Mathematics TOK Presentation. Yasser Tsikhlakis and Hanna Bassil. “Mathematics does appear to be an ‘island of stability’ in an ocean of chaos; after all, mathematics appears to be the only area of knowledge that offers ‘definite’ answers to questions”. What is Knowledge in Math?.

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Mathematics TOK Presentation

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  1. MathematicsTOK Presentation Yasser Tsikhlakis and Hanna Bassil

  2. “Mathematics does appear to be an ‘island of stability’ in an ocean of chaos; after all, mathematics appears to be the only area of knowledge that offers ‘definite’ answers to questions”

  3. What is Knowledge in Math? 1) Acknowledging Math around us Mathematics is found in 2 general forms, either as a question or as a real life situation. It is found everywhere in our every day lives even if we don’t recognize it. People such as engineers and architects must acknowledge the mathematical factors in order to be able to create and solve issues around us. In order to identify these issues, one must have a good understanding of core mathematics.

  4. What is Knowledge in Math? Example) Function of a Pringle Each Pringles chip is actually a hyperbolic paraboloid. Its function is y = x2/a2 – z2/b2

  5. What is Knowledge in Math? 2) Understand and Apply Knowledge in Mathematics is very different from other subjects since different people interpret it differently. Knowledge in mathematics is the ability to grasp certain concepts, rules and methods, then being able to convert and apply them to different questions. It is clear that concepts in mathematics cannot be memorized and need to be understood, this is the stage where knowledge in this subject varies amongst people, some minds are more developed when memorizing pieces of information while other minds work at their full potential when they understand why certain rules are used to solve certain questions. Obviously, those who work better when understanding the application of rules will have a more mathematical mind and find it less difficult than others when solving questions based on applying rules to deal with the situation in the question.

  6. What is Knowledge in Math? Example: Find the area of the following shape. 3 3 Person X finds the length of the entire base and then uses the area of the triangle rule (1/2 x b x h) for the triangle as a whole. ½ x 3 x 6 = 9 Person Y realizes that the position of the triangle can be manipulated to form a square. Using the area of a square rule (L x L ). 3 x 3 = 9 3 3

  7. What is Knowledge in Math? Person x applies basic rules that he has memorized across the years without giving the question deep thought. On the other hand, person y is a mathematic thinker and finds a way to manipulate the question in order to make it easier on himself. This proves that there is more than one way in which a person can indulge mathematics.

  8. What is Knowledge in Math? 3) Simple logic behind Mathematics No matter how complicated it can seem, all mathematics is linked to basics and logic. You can take any problem and derive it into its roots thus bringing it back to simple mathematical equations or formulas such as 1+1=2 to solve the question. We know that 1+1=2 from simple reasoning and understanding of the world around because when we take one apple and add it to the other we then have two apples. This will be explained further in the section of axioms which are all so basic, that the only explanation to them are “its obvious”

  9. What is Knowledge in Math? 4) Questioning the issue In order to be able to solve a problem, a good mathematician would question the issue to figure out the steps of work which would lead him to the correct answer.

  10. What is Knowledge in Math? Example: For example, in calculus, if given the following curve, you ask yourself “why is the curve behaving the way it is?” By asking yourself this question, you are able to lead yourself into the path of analyzing the gradient behaviour, trend and nature of the curve to solve any question that is asked about it.

  11. Knowledge Issues in Mathematics

  12. 1- Half correct answers For example: a2 + b2 = c2 . a = 3 and b = 4, find c. a2 + b2 = c2 32 + 42 = c2 9 + 16 = c2 25 = c2 c = 5 This is only half correct, because c2 =25 has 2 answers, since c can equal 5 or -5 ((-5)2=25) and if you put c down as 5 in an exam, you may lose marks for what is technically a half answer.

  13. 2- Axioms An axiom is defined as a statement or proposition that is regarded as being established, accepted, or self-evidently true. The 4 requirements for axioms are: 1) Consistent (The same set of axioms should always produce the same answer) 2) Independent of one another (You should not use one axiom to deduce another axiom) 3) Simple (Should be easily understandable) 4) Diverse (Have various set of applications in terms of theorems produced)

  14. Euclid’s Axioms • It shall be possible to draw a straight line joining any two points. • A finite straight line may be extended without limit in either direction. • It shall be possible to draw a circle with a given centre and through a given point. • All right angles are equal to one another. • There is just one straight line through a given point which is parallel to a given line.

  15. The Knowledge Issue? There is no explanation or justification for axioms, other than the fact that they are obvious. This means that the problem with axioms is that we automatically assume that they are true. A few questions to ask: - Is it possible that the assumptions in axioms are misinterpreted and misused? - To what extent are the assumptions behind axioms vulnerable to error?

  16. 3- Statistics Manipulation Example: Advertising a product, the advertiser claims “90% of people who tested the product decided It was better than any other they have used” People reading this statistic may think that the product must be impressive, but the issue that rises with this is:1) Test Size:9/10 people or 900/1000 people?2) Test Sample: range of age? Both males and females? Rich and middle class? etc.

  17. 4- Misinterpretation of Algebra • Algebra can be used to prove incorrect pieces of information. It is manipulated to prove a certain equation or point which is originally wrong. Nevertheless is it really wrong?

  18. 4- Misinterpretation of Algebra • Algebra will prove the following statement, nevertheless it is unrealistic. 1+1=1.99999999999 • Let x=0.9999999 • So 10x=9.99999999 • 10x-x=9 • Hence 9x=9 and thus x=1 • The same equation was used to prove that x has to values: 0.999999 and 1 • Therefore 1 and 0.99999 are supposedly equal but they are technically not and so 1+1=1.99999 or x+x=1.99999

  19. 4 - Misinterpretation of Algebra Activity: What is wrong with this proof? Given A = B (Multiply both sides by A) A2 = AB Subtract B2 from both sides A2 – B2 =AB - B2 Factorize both sides (A-B)(A+B) = B(A-B) Divide both sides by (A-B) A + B = B Since A=B B + B = B So 2B = B (Divide by B) 2 = 1

  20. 5- Created or Discovered? Since mathematics dates back in History, the question that frequently comes up is as follows: Was Mathematics created by people or was everything already there waiting to be discovered through finding patters and relations between numbers etc. 10 minute in-class debate: Was Mathematics Created or Discovered?

  21. THE GOLDEN RATIO The golden ratio is 1 : 0.618 (or 1.6180339887…) and is said to be “golden” because it is pleasing to the human eye. Symbol for “Phi” which is the number 1.6180339887…

  22. How the Golden Ratio was derived Fibonacci sequence

  23. THE GOLDEN RATIO

  24. Linking Mathematics to AoK and WoK

  25. Linking Math with Reason • Mathematics and its complicated equations are hard to understand and they are not the type of equations that you can memorize and apply. If you don’t know why you use a certain equation to solve a certain problem, then you will find it difficult to be a good mathematician. That is why reason plays a big factor in helping someone understand mathematic concepts. You will begin to be familiar with which equations to apply to familiar problems and get used to answering questions without difficulty. - In mathematics, people grasp concepts differently and at different speeds. Sometimes it is natural, while at other times you need to go over the solution to the question and use reasoning to figure out why each step in the solution was made and where it led you.

  26. Box Activity

  27. Activity- Monty Hall Problem 1 2 3

  28. Reason? 33.3% Chance of winning the goat if you stay with your original choice. 66.6% Chance of winning the car if you switch from your original choice.

  29. Linking Math with the Arts It is unusual to think that these two areas of knowledge can be linked, however, there are a lot of situations where mathematics and mathematical equations are used to create certain effects in Visual and theatrical pieces of art. For example, Leonardo Da Vinci used mathematical equations to create the background effect of his famous Mona Lisa painting. He used calculations based on the position of the Mona Lisa and what is painted onto her left and right to give the illusion of a trailing background. There is even a book called “Math and the Mona Lisa” that describes this further.

  30. Art- The Mona Lisa

  31. Architecture- The Parthenon Phi (derived from the golden ratio) was used by many architects who produced some of the most historically recognized buildings and structures in the world

  32. Modern Architecture- Burj Khalifa

  33. Mathematics behind the perfect face

  34. The mask of the human face is based on phi and the golden ratio discussed earlier. There is a ratio between the proportion of the length of the nose, the position of the eyes and the length of the chin

  35. Perfect examples

  36. Mona Lisa’s Face

  37. Mathematics behind the perfect body

  38. Anthropologist Barnaby Dixson conducted a study to determine what males found attractive in a female across cultures. He claims that there is a ration that men favoured in women almost most of the time. The waist-to-hip ratio should be 0.7 (meaning the waist should measure 70 per cent of the hip circumference).

  39. Reason? Dixson claims that although men may be drawn to a women’s cleavage at first, he then turns his attention to the waist/hip ratio. This is because the 0.7 ratio “sends a biological signal to men that the woman is most fertile and most likely to produce a healthy offspring”

  40. Perfect Examples KimKardashian Rosie Hungtinton

  41. Linking Math with Natural Sciences The easiest two AoK to link are Math and the Natural sciences. All the equations in the Natural Sciences, such as physics and chemistry, rely on mathematically deduced formulas that solve many of the world’s unanswered questions. The use of mathematics is vital in the natural sciences, because without the help of mathematics, the amount of information we know about physics and chemistry would be limited and hard to discover.

  42. MindF#@! http://www.learnenglish.org.uk/games/magic-gopher-central.swf

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