Chapter 1 Basics of Geometry.

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Presentation transcript:

Chapter 1 Basics of Geometry

Section 1 Patterns and Inductive Reasoning

GOAL 1: Finding and Describing Patterns Study patterns discovered throughout history Learn to recognize and describe patterns of your own Make accurate predictions based on patterns

Example 1: Describing a Visual Pattern Sketch the next figure in the pattern.

Example 2: Describing a Number Pattern Describe a pattern in the sequence of numbers. Predict the next number. a. 1, 4, 16, 64, … b. -5, -2, 4, 13, …

GOAL 2: Using Inductive Reasoning Three stages: Look for a pattern Make a conjecture Verify the conjecture

Example 3: Making a Conjecture Complete the conjecture Example 3: Making a Conjecture Complete the conjecture. Conjecture: The sum of the first n odd positive integers is ____.

What is a counterexample?

Example 4: Finding a Counterexample Show the conjecture is false by finding a counterexample. Conjecture: For all real numbers x, the expression x2 is greater than or equal to x.

Example 5: Examining an Unproven Conjecture In the early 1700s a Prussian mathematician named Goldbach noticed many even numbers greater than 2 can be written as the sum of two primes. Specific Cases: 4 = 2 + 2; 6 = 3 + 3; 8 = 3 + 5; 10 = 3 + 7; 12 = 5 + 7; 14 = 3 + 11; 16 = 3 + 13 Conjecture: Every even number greater than 2 can be written as the sum of two primes. This is called Goldbach’s Conjecture. No one has ever proved that this is true or found a counterexample to show it is false.

Example 6: Using Inductive Reasoning in Real Life A full moon occurs when the moon is on the opposite side of Earth from the sun. During a full moon, the moon appears as a complete circle. Use inductive reasoning and the information below to make a conjecture about how often a full moon occurs. Specific Cases: In 2005, the first six full moons occurred on January 25, February 24, March 25, Aril 24, May 23, and June 22. Conjecture: