1. 1.1 – PATTERNS AND INDUCTIVE REASONING Chapter 1: Basics of Geometry

2. Where did Geometry come from anyhow? ‘geometry’ = ‘geo’, meaning earth, and ‘metria’, meaning measure. Euclid = “Father of Geometry” 300 BC Greeks used Geometry for building Modern Geometry enables our computers to work so fast.

3. Notice the Pattern Much of Geometry came from people recognizing and describing patterns.

4. Ex 1: Sketch the next figure in the pattern: 1 432 Visual Patterns

5. Number Patterns Ex 2: Describe the pattern in this sequence. Predict the next number. 1, 4, 16, 64, 256

6. Number Patterns Ex 3: Describe the pattern in this sequence. Predict the next number. -5, -2, 4, 13, 25

7. Number Patterns Ex 4: Describe the pattern in this sequence. Predict the next number. 3, 7, 15, 31, 63

8. Using Inductive Reasoning Inductive Reasoning is the process of arriving at a general conclusion based on observations of specific examples. SpecificGeneral Specific General Ex 5: You purchased notebooks for 4 classes. Each notebook costs more than $5.00 Conclusion: All notebooks cost more than $5.00

9. The Three Stages to Reason 1) Look for a pattern 2) Make a Conjecture 3) Verify the Conjecture: make sure its ALWAYS true What even is a Conjecture? Its an unproven statement that’s based on observations. You can discuss it and modify it if necessary. It ain’t a rule yet!

10. Making a Conjecture Ask a Question: What is the sum of the 1 st n odd positive integers? 1) List some examples and look for a pattern. 1) First odd positive integer1 = 1 2 2) Sum of first 2 odd integers1 + 3 = 4 = 2 2 3) Sum of first 3 odd integers1 + 3 + 5 = 9 = 3 2 4) Sum of first 4 odd integers1 + 3 + 5 + 7 = 16 = 4 2 2) Conjecture: the sum of the 1 st n odd positive integers is n 2

11. Proving a Conjecture is TRUE To prove true, you must prove it is true for EVERY case. (Every example must fit the conjecture) To prove a conjecture false, you only need to provide one counter example Ex 6: Everyone in our class has blonde hair. Counter Example: Mrs. Pfeiffer, Mr. Nguyen…

12. Find the Counter Example Ex 7: Show the conjecture is false by finding a counterexample: Conjecture: The difference of two positive numbers is always positive. Counter Example: 2 - 8 = -6

13. Find the Counter Example Ex 8: Find the Counter Example: Conjecture: the square of any positive number is always greater than the number itself. Counter Examples: 1) 1 2 = 1 2) (0.5) 2 = 0.25 which is NOT greater than 0.5

14. Unproven Conjecture Some Conjectures have been around for hundreds of years and are still unknown to be true or false. GGoldbach’s Conjecture: all even numbers greater than 2 can be written as the sum of 2 primes. 44 = 2 + 2 66 = 3 + 3 88 = 3 + 5, etc. IIt is unknown whether this is true for ALL cases, but it has not yet been disproven. A $1,000,000 prize is offered to the person that can crack Goldbach’s Conjecture!

15. Pattern Puzzle To get the next number in a sequence, you multiply the previous number by 2 and subtract 1. If the fourth number is 17, what is the first number in the sequence?

16. Find the Pentagon that has no twin… the one that is different from all the others.

17. Homework Page 6: # 5 – 39 Odd, 53-71 Odd