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

1.1 – PATTERNS AND INDUCTIVE REASONING

Chapter 1: Basics of Geometry

Where did Geometry come from anyhow?

‘geometry’ = ‘geo’, meaning earth, and ‘metria’, meaning measure.

Euclid = “Father of Geometry” 300 BCGreeks used Geometry for building Modern Geometry enables our computers to

work so fast.

Notice the Pattern

Much of Geometry came from people recognizing and describing

patterns.

Ex 1: Sketch the next figure in the pattern:

1 432

Visual Patterns

Number Patterns

Ex 2: Describe the pattern in this sequence. Predict the next number.

1, 4, 16, 64, 256

Number Patterns

Ex 3: Describe the pattern in this sequence. Predict the next number.

-5, -2, 4, 13, 25

Number Patterns

Ex 4: Describe the pattern in this sequence. Predict the next number.

3, 7, 15, 31,63

Using Inductive Reasoning

Inductive Reasoning is the process of arriving at a general conclusion based on observations of specific examples.

Specific General

Ex 5: You purchased notebooks for 4 classes. Each notebook costs more than $5.00Conclusion: All notebooks cost more than $5.00

The Three Stages to Reason

1) Look for a pattern2) Make a Conjecture3) 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!

Making a Conjecture

Ask a Question: What is the sum of the 1st n odd positive integers?

1) List some examples and look for a pattern.1) First odd positive integer 1 = 12

2) Sum of first 2 odd integers 1 + 3 = 4 = 22

3) Sum of first 3 odd integers 1 + 3 + 5 = 9 = 32

4) Sum of first 4 odd integers 1 + 3 + 5 + 7 = 16 = 42

2) Conjecture: the sum of the 1st n odd positive integers is n2

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…

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

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) 12 = 12) (0.5)2 = 0.25 which is NOT greater than

0.5

Unproven Conjecture

Some Conjectures have been around for hundreds of years and are still unknown to be true or false. Goldbach’s Conjecture: all even numbers greater than

2 can be written as the sum of 2 primes. 4 = 2 + 2 6 = 3 + 3 8 = 3 + 5, etc.

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!

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?

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

Homework

Page 6: # 5 – 39 Odd, 53-71 Odd