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Ma T h OK

Ma T h OK. Yosef Karasik. Bertrand Russell. "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music .”. Reasoning as a WOK. Find a function for: X=1 Y=1 X=2 Y=2 X=3 Y=3 X=4 Y=4 X=5 Y=125.

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Ma T h OK

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  1. MaThOK Yosef Karasik

  2. Bertrand Russell • "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music.”

  3. Reasoning as a WOK Find a function for: • X=1 Y=1 • X=2 Y=2 • X=3 Y=3 • X=4 Y=4 • X=5 Y=125

  4. Inductive reasoning • Is reasoning that takes specific information and makes a broader generalization that is considered probable, allowing for the fact that the conclusion may not be accurate. • Jesse and the divas

  5. Get to work! • Draw 3 different triangles on paper and measure the sum of their angles (as accurate as possible) using your protractors. • What results did you get?

  6. Questions • Can we come up with a conjecture for the sum of angles in a triangle based on your measurements? What will that conjecture be? • (KQ) Under what conditions can we ignore results that come as contradictions to the hypothesis?

  7. Deductive reasoning • Is a type of reasoning which goes from general to specific. Deductive reasoning is based on premises and if the premises are true, then the reasoning will be valid. • Come up with a different proof for the sum of angles in a triangle.

  8. Inductive or deductive? • Jennifer leaves for school at 7:00 a.m. and is on time. Jennifer assumes, then, that she will always be on time if she leaves at 7:00 a.m. • Snakes are reptiles and reptiles are cold-blooded; therefore, snakes are cold-blooded. • All observed women in one area wear high heels, so all women must wear high heels. • All dogs are mammals. Spot is a dog. Spot is a mammal. • Robert is a teacher. All teachers are nice. Therefore, it can be assumed that Robert is nice.

  9. Questions • Were there any assumptions (premises) in your proof? • Can you prove them as well? • Axioms – a self-evident truth that requires no proof. A universally accepted principle or rule.

  10. The postulates of geometry • You can only draw one line between two points. • There's 360° around a circle • Two lines can intersect at ONLY one point • A line segment has ONLY one midpoint • An angle can only have one bisector • Any geometric shape can be moved without changing its shape • And more…

  11. Proof • What methods of proving are there? • Direct proof • Proof by induction • Proof by contraposition • Proof by contradiction • Proof by construction • Proof by exhaustion • Others…

  12. Further explore • What makes a proving method valid? • What makes a sound proof? • Can one proof method be better then the other?

  13. Balloons! • Draw triangles on balloons and measure the angles. • Calculate the sum of the three angles • Measure side lengths of any right angle triangle.

  14. WHAT THE…!?!?!? • Why not as expected? • Does size of balloon make a difference? • What if you were inside the balloon? • How many right angles can the triangle contain? • How can you decide which one is the hypotenuse?

  15. The postulates stop working! • They work for planes, but not curved surfaces • How about earth then? Is it flat?

  16. Spherical Geometry • Redefine “straight line”, “circle”, “Plane” • Lobachevsky’s Geometry

  17. Further explore • What role perception plays in mathematical understanding? • Does context affect mathematical understanding?

  18. Is Zero Nothing? • What do you think of when you see O? 0? • Do they bring the same or different things to your mind? • History of the symbol (quill pen)

  19. A few questions • What remains when 7 is taken away from 7? • A place value symbol as in 402.6 , giving value to other digits around it but having no value itself. • The number that causes great difficulties if you try to divide by it 12/0=? • Undefined? • Infinity? • What is ?

  20. Does 2=1?

  21. More of nothing • What happened to the certainty of Math? • Nothing, emptiness, lack of anything, absence, void, black, darkness, oblivion… and therefore the embodiment of evil!

  22. A bit of history • Zero as absence in ancient times. • Ancient Greeks and Romans • Babylonians • Chinese traders • Zero as a number, entity, presence • A shift from NAME to SYMBOL

  23. Magic number • Zero counts the totality of what isn’t there • It is the gateway between the positive and negative numbers • A number without qualities

  24. Zero or nothing? Presence or absence? • {} null set – called ϕ • {0} the set containing 0 • So how about {ϕ}?

  25. In art • Nothing is? Black(absence of color) or White(presence of every color)?

  26. In Music • John Cage “The material of music is sound and silence. Integrating these is composing. I have nothing to say and I’m saying it” He sat there turning pages from time to time, then stood up and bowed

  27. The simpsons • YouTube S09E17 14:30 • http://www.animefo.com/watchcartoon-the-simpsons-season-9-episode-17-lisa-the-simpson?ver=1

  28. In Literature • George Gershwin – in Porgy and Bess “I got plenty of nothin’, nothin’s plenty for me” • Lewis Carroll – Alice in conversation with the king • A: I see nobody on the road • K: I only wish I had such eyes, to be able to see nobody, and at such a distance too

  29. The church • God exists! • Demonstrating that something can come from nothing if nothing is acted on by an infinite being – hence the act of creation can take place.

  30. Zero and infinity – a love story • A young child when asked about what is infinity: “The biggest number ever was thousand thousand thousands” • What about thousand thousand thousands plus 1? • “Well”, she said, “I was very close wasn’t I?” • Geometric series – adding infinite numbers and end up with finite sum.

  31. Nonsense Rhyme • As I was going up the stair, I met a man who wasn’t there. He wasn’t there again today. I wish, I wish he’d stay away.

  32. Knowledge questions for you (you’re welcome!) • Can ethics (and others areas of knowledge) be reduced to equations? (George Price) • Does mathematics provide us with more practicable knowledge than less quantitative data? • Are those who understand mathematics better equipped to deal with the modern world? (algorithms) • What role does mathematics play in our knowledge acquisition? Do those who understand math have a clearer understanding of the world?

  33. More KQs • To what extent does our perception of ordinary life depend on statistical knowledge? What are the dangers associated with this in terms of providing us with a realistic view of the world? • To what extent is our perception of the world shaped by the way a ‘base 10 approach’ is instilled in us from an early age? • Correlation VS causation – To what extent interpretation of statistics affects our understanding. • Why does it have very little effect to tell people that they can’t statistically beat the odds in a casino?

  34. Thanks for nothing

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