This is a brief, I mean brief, introduction to mathematics that I used this year. I also introduced the different types of Geometry, and steps to solving a geometry problem.

1. History of Mathematics Prentice Hall Geometry Geometry August 2009

2. Mayans MEXICO Mayans are from ____________. Base 20 system One of the 1st cultures to invent _________. Their calendar had 18 months a year; 20 days a month ZERO

3. According to the Mayans: The WORLD will END on… The WORLD will END on… 21 DECEMBER 2012

4. Romans Roman Numeral System I, II, III, IV, V,…,X, L, C, D, M,… The system is based on Subtractive Pairs The Line above Roman Numerals means multiply by a thousand

5. Egyptians 1st to have fractions! Geometry invented by Egyptians Geo means earth; meter comes from measures Used geometry to measure land to assess taxes

6. Egyptians Came closest to developing pi pi is the ratio of diameter of a circle to the circumference of the circle

7. Famous Mathematicians Rene Descartes Pierre de Fermat Blaise Pascal Robert Hooke Isaac Newton James Bernoulli GirolamoSaccheri John Bernoulli Christian Goldbach Leonhard Euler Joseph Louise Lagrange Carl Freidrich Gauss Bernhard Riemann David Hilbert John von Neumann Thales of Miletus Pythagoras of Samos Hippocrates of Chios Euclid Archimedes Galileo Galilei

8. QUIZ Quiz When is the world going to end according to the Mayans? Where are the Mayans from? What is the system of Roman Numerals based on? Egyptians invented _______ for what two reasons? What base system do we use today?

9. Geometry Three Types of Geometry: Euclidean (what we will study) Non- Euclidean Elliptic Geometry (Spherical Geometry) Hyperbolic Geometry

10. EuclideanGeometry

11. Undefined Terms can be described but cannot be given precise definitions using simpler known terms. 3 Main Undefined Terms: Undefined Terms Point is thought to be a circular dot that is shrunk until it has no size. Line is thought to be a wire stretched as tightly as possible of infinite length having no thickness. Plane is thought to be a sheet of paper that has no thickness, stretched tightly , and extending infinitely in all directions.

12. Using the undefined terms (point, line, plane) allows us to define other terms in geometry, e.g. space: a set of all points Definitions

13. Postulates and axioms are one in the same. They are accepted as statements of fact. Postulates and Axioms

14. Theorems Theorems are results that are deduced from undefined terms, definitions, postulates, and/ or results that follow from them.

15. EuclideanGeometry

16. how to solve geometry? Scientific Method? Ask a Question Do Background Research Construct a Hypothesis Test Your Hypothesis by Doing an Experiment Analyze Your Data and Draw a Conclusion Communicate Your Results

17. how to solve geometry? Four Steps: Understand the Problem Devise a Plan Carry Out the Plan Look Back

18. 1. Understand the Problem Is it clear to you what is to be found? Do you understand the terminology? Is there enough information? Is there irrelevant information? Are there any restrictions or special conditions to be considered?

19. 2. Devise a Plan. How should the problem be approached? Does the problem appear similar to any others you have solved? What strategy might you use to solve the problem?

20. 3. Carry Out the Plan Apply the strategy or course of action chosen in Step 2 until a solution is found or you decide to try another strategy

21. 4. Look Back Is your solution correct? Do you see another way to solve the problem? Can your results be extended to a more general case?

22. Some Strategies… Draw a picture! Guess and check. Use a variable. Look for a pattern. Make a table. Solve a simpler problem.

23. What to do when you start working a geometry problem? Understand the problem. Devise a plan. Carry out the plan. Look Back.

24. POINT LINE B A A l AB Line l

25. plane B A C Plane ABC or Plane P P