INTRO TO GEOMETRY 1-1 PIB Geometry. Agenda 1. What is a mathematician? 2. What is geometry? 1. Axiomatic system 3. What is a proof? 4. 1-1 Notes 5. 1-1.

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Presentation transcript:

INTRO TO GEOMETRY 1-1 PIB Geometry

Agenda 1. What is a mathematician? 2. What is geometry? 1. Axiomatic system 3. What is a proof? Notes Homework Note: please get a RED PEN for the class.

Mathematicians… i.) Analyze patterns ii.) Make conjectures iii.) Prove things

Let’s try being a mathematician… 1+2=3 2+3=5 3+4=7 … Is the sum of two consecutive integers always odd?  How could we PROVE this?

How geometry works Everything in geometry is proved from previously proven things … …but you have to start somewhere   Postulates Vocabulary:  Definition: Statement of the meaning of a word.  Postulate: Our simplest statements; given without proof.  Conjecture: An unproven statement or generalization.  Theorem: A conjecture that has been proven.

Undefined Terms of Geometry Point: a location; infinitesimally small. Line: Extends forever in two directions; usually defined by 2 points. Plane: A flat surface that extends forever in every direction

What is a mathematical proof? Conjecture: The product of two even numbers is always even. Proof: Conjecture: The product of two odd numbers is always odd. Proof:

What is going on here??? Proof that 2=1:  Let a=b  Then a 2 = abMultiply each side by a  a 2 + a 2 = ab + a 2 Add a 2 to both sides  2 a 2 = ab + a 2 Combine like terms  2 a 2 – 2ab = a 2 + ab - 2abSubtract 2ab from both sides  2 a 2 – 2ab = a 2 – abCombine like terms  2(a 2 – ab) = 1(a 2 – ab)Factor (a 2 – ab) from each side  2 = 1Divide each side by (a 2 – ab)

Let’s play a game p. 1 in your textbooks

Homework p. 3-4: #1-10 Note: homework will always be under “written exercises”, not “classroom exercises”, unless stated otherwise