1/22/2010. Favorite? Topics in geometry. Meeting place. Reconsider the meeting place. Obvious fact? How far are they from each other?
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1 Topics in geometry Dive straight into it! Favorite? Have you every taught geometry? When? How often? Do you enjoy it? What is geometry to you? On a sheet of paper please list your 3 favorite facts from geometry, 3 facts you think are most obvious. Meeting place Three friends are planning to meet before heading to the movie theater. Their goal is to meet so that they each have to walk the same distance to the meeting place. Help them determine where to meet. Reconsider the meeting place Obvious fact? There is one shortest path between any two points! How many shortest paths are there between any two points? How far are they from each other? We know that the shortest distance between points (x 1, y 1 ) and (x, y ) is given by d ( x y 1 x) ( y1 ) How would you find the shortest distance in taxicab geometry? 1
2 Distance formula What is the shortest path between (x 1, y 1 ) and (x, y )? What is geometry? Greeks thought: True and absolute description of physical world We think: Statements are either true or false GIVEN CERTAIN HYPOTHESES We give hypotheses and discover their consequences Questions: Where do hypotheses come from? How do we discover their consequences? HUH? Axiom 1: There are exactly four Fe s. Axiom: Any two distinct Fe s belong to exactly one Fo. Axiom 3: Each Fo contains exactly two Fe s. If two distinct Fo s have a Fe in common, then they have exactly one Fe in common. Go discover things. There are exactly six Fo s. Each Fe belongs to exactly three Fo s.
3 Each Fo has exactly one Fo disjoint to it. UTA problem You are on the UTA s planning commission and it has been agreed that the bus system should satisfy the following conditions: i. One can get from any bus stop to any other without transferring; ii. For any pair of routes there is one and only one bus stop where one can transfer from one route to the other; iii. There are exactly three bus stops on each route. How many bus routes are there in town? Difference? How are Fe/Fo and UTA different? Proving To show that a statement, S, is true: S S 1 S How are Fe/Fo and UTA similar? either: we arrive at a statement that is accepted as true S has been proved we do not arrive at a true statement: Flawed proof Faulty system The sequence of statements is incorrect. We need Agree on language Agree on axioms (statements to be accepted as true without justification) Agree on what constitutes a proof (how do we deduce new statements from old ones?) Two theorems: Contrariwise, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t! No cat has eight tails. Since one cat has one more tail than no cat, it must have nine tails. 3
4 Language: technical terms Exercise: What is a point? What is a line? What is a number? Possible answers What is a point? A sharp or tapered end A decimal point A dimensionless geometric object having no properties except location Euclid: that which has no part. Possible answers What is a line? a geometric figure formed by a point moving along a fixed direction and the reverse direction Euclid: A breadthless length That which lies evenly with the points on itself Undefined terms in our exercise Fe Fo Belongs to Interpretation Give each undefined term a particular meaning interpretation If all axioms are correct statements, the interpretation is called a model Give two interpretation of Fe, Fo: A model A non-model Properties of axiomatic system A set of axioms is said to be Consistent if it is impossible from these axioms to deduce a theorem that would contradict any axiom or previously proved theorem. Independent each of the axioms is independent: can not be deduced from other axioms. 4
5 How students do it (movie #9) Students just can t do it? Easier to follow someone else s argument, than to construct one s own? It s hard to use information that was known before Proof mapping: Forward-backward d technique Ask key question: how can your conclusion come about? It should be abstract. Its answer should be then applied to the specific situation. Decide whether to move forward or backward: Can I now use what I have to advance the argument? Do I need to ask another key question? If in a ABC we have AC BC and CD is the altitude, then ADC BDC Use proof map to prove: Vertical angles are congruent Students questions Consider the questions students ask in geometry classroom given on your handout. With your peers decide how to answer those questions We will give presentations after you had some time to work on them. 5
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