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# Tok- Maths Presentation

## by Iain Williamson, Teacher/Head of Film/Media at South Island School on Aug 25, 2010

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A useful presentation for TOK (AOK- Maths)

A useful presentation for TOK (AOK- Maths)

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• Three man in an air balloon are lost in the Grand Canyon. In a desperate attempt, they scream simultaneously: “Where are we”? After 15 minutes, an answer comes back: “You are lost!” “This must have been a Mathematician”, says one of them. “How do you know?” asks another. “ The first one replies promptly: “Because the answer came very late, was true, but completely useless’”
• My words are: Beautiful, creative,inspirational, logical, intuitive Students’ words will be probably different and this activity could generate in small groups interesting discussion. This activity will be repeated at the end of the unit. Has student’s view changed?
• Answer to question 1 can be found in Marcus DuSautoy’sChristmas lectures – “Ch1 – The Story of Never Ending Numbers”, starts at 13:25, finishes 18:20.Answer to question 2 and a good introduction to fractals can be found in the same series, “Ch2 – The story of elusive shapes”, starts at 12:49 and ends at 18:06
• The answer depends on what we mean by “average”. If we meant “add the numbers and divide by 11”, the answer is 96. If we meant “the most common number in the list”, the answer is 1. If we meant “the number in the middle”, the answer is 4. Which one is then true? Mathematicians name each of these averages differently. They are called mean, mode and median respectively. Some definitions will be then useful in order to make mathematical progress.Definitions are called axioms. Starting from axioms and performing a chain of logical steps Mathematicians build theorem. They represent the new knowledge.
• Notice that all this had started as a intellectual game.Where the Euclidian geometry is wrong? The problem that someone find is that is does not correctly represent the three-dimensional universe that we live in.
• At early age mathematical knowledge is formed more using intuition. With time students get more familiar with the concept of mathematical proof.
• In the previous slide we can comment on the fact that at early stage the area of a triangle was introduced intuitively. Can I count on my intuition?Answers: 1, 2, 4, 8, 16, 31The students are surprised by the result. Aren’t all mathematical examples straightforward and predictable? This example makes us think. If a seemingly obvious conjecture cannot be make after 5 steps, how many of them should I take in order to make sure that it is true?