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MaThOKYosef Karasik
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0011 0010 1010 1101 0001 0100 1011Bertrand Russell
• "Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, without the gorgeous trappings of painting or music.”
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0011 0010 1010 1101 0001 0100 1011Reasoning 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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0011 0010 1010 1101 0001 0100 1011Inductive 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
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0011 0010 1010 1101 0001 0100 1011Get 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?
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0011 0010 1010 1101 0001 0100 1011Questions
• 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?
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0011 0010 1010 1101 0001 0100 1011Deductive 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.
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0011 0010 1010 1101 0001 0100 1011Inductive 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.
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0011 0010 1010 1101 0001 0100 1011Questions
• 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.
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0011 0010 1010 1101 0001 0100 1011The 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…
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0011 0010 1010 1101 0001 0100 1011Proof
• What methods of proving are there?Direct proofProof by inductionProof by contrapositionProof by contradictionProof by constructionProof by exhaustionOthers…
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0011 0010 1010 1101 0001 0100 1011Further explore
• What makes a proving method valid?
• What makes a sound proof?
• Can one proof method be better then the other?
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0011 0010 1010 1101 0001 0100 1011Balloons!
• Draw triangles on balloons and measure the angles.– Calculate the sum of the three angles
• Measure side lengths of any right angle triangle.
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0011 0010 1010 1101 0001 0100 1011WHAT 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?
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0011 0010 1010 1101 0001 0100 1011The postulates stop working!
• They work for planes, but not curved surfaces
• How about earth then? Is it flat?
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0011 0010 1010 1101 0001 0100 1011Spherical Geometry
• Redefine “straight line”, “circle”, “Plane”
• Lobachevsky’s Geometry
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0011 0010 1010 1101 0001 0100 1011Further explore
• What role perception plays in mathematical understanding?
• Does context affect mathematical understanding?
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0011 0010 1010 1101 0001 0100 1011Is 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)
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0011 0010 1010 1101 0001 0100 1011A 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 ?
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0011 0010 1010 1101 0001 0100 1011Does 2=1?
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0011 0010 1010 1101 0001 0100 1011More 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!
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0011 0010 1010 1101 0001 0100 1011A 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
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0011 0010 1010 1101 0001 0100 1011Magic number
• Zero counts the totality of what isn’t there
• It is the gateway between the positive and negative numbers
• A number without qualities
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0011 0010 1010 1101 0001 0100 1011Zero or nothing? Presence or absence?
• {} null set – called ϕ
• {0} the set containing 0
• So how about {ϕ}?
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0011 0010 1010 1101 0001 0100 1011In art
• Nothing is? Black(absence of color) or White(presence of every color)?
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0011 0010 1010 1101 0001 0100 1011In 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
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0011 0010 1010 1101 0001 0100 1011The simpsons
• YouTube S09E17 14:30
• http://www.animefo.com/watchcartoon-the-simpsons-season-9-episode-17-lisa-the-simpson?ver=1
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0011 0010 1010 1101 0001 0100 1011In 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
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0011 0010 1010 1101 0001 0100 1011The 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.
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0011 0010 1010 1101 0001 0100 1011Zero 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.
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0011 0010 1010 1101 0001 0100 1011Nonsense 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.
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0011 0010 1010 1101 0001 0100 1011
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?
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0011 0010 1010 1101 0001 0100 1011More 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?
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0011 0010 1010 1101 0001 0100 1011
Thanks for nothing