Download Chapter 1 - Basics of Geometry Section 1.1

Chapter 1 - Basics of Geometry
Section 1.1 - Patterns and Inductive Reasoning
*Watch this video to give you a better grasp of the concepts listed below.*
3 Stages of Reasoning in Geometry (Inductive Reasoning)
1. Look for a pattern.
look at several examples
use pictures/tables
2. Make a conjecture. (conjecture: an unproven statement that is based on observations.)
use examples to make conjecture
3. Verify the conjecture.
test it (verify that it works for all cases)
Counterexample: an example that shows a conjecture is false.
Patterns.
Example 1: Sketch the next figure in the pattern:
Example 2: Describe the pattern in the sequence of numbers. Predict the next number.
a)
b)2, 5, 10, 17, …
Example 3: Complete the conjecture.
a) The sum of the first n odd positive integers is_______.
List some specific examples first and look for a pattern.
1 odd: 1=
2 odd: 1+3=
3 odd: 1+3+5=
4 odd: 1+3+5+7=
5 odd: 1+3+5+7+9=
b) The product of any two consecutive positive integers is_________.
How do you know if a conjecture is true?
How do you know if it is false?
Example 4: Find a counterexample.
a) Conjecture: The difference of two positive numbers is always positive.
b) Conjecture: For all real numbers x, the expression x² is greater than or equal to x.
*Watch this video to see the examples worked through*