1. Chapter 1Tools of Geometry Section 1.1 Patterns and Inductive Reasoning

2. Inductive Reasoning: reasoning based on patterns you observe • Finding and Using a Pattern • Examples: • Find the next two terms of the sequence • 3, 6, 12, 24,… • Each term is being multiplied by 2 • Next two terms: 48, 96 • 1, 2, 4, 7, 11, 16, 22, … • Adding another 1 to the previous • Next two terms: 29, 37

3. Examples (cont.) • Monday, Tuesday, Wednesday… • Thursday, Friday

4. Conjecture: a conclusion you reach using inductive reasoning • Make a conjecture about the sum of the first 30 odd numbers • 1 = 1 • 1 + 3 = 4 • 1 + 3 + 5 = 9 • 1 + 3 + 5 + 7 = 16 • 1 + 3 + 5 + 7 + 9 = 25 • 1 + 3 + 5 + 7 + 9 + 11 = 36 • Notice the pattern created • (number of odd numbers, squared) • The sum of the first 30 odd numbers would be 302 = 12 .= 22 .= 32 = 42 .= 52 .= 62

5. Examples (cont.) • Make a conjecture about the sum of the first 35 odd numbers • Since the sum of the first 30 odd numbers is 302, then the sum of the first 35 odd numbers would be: 352 352 = 35 * 35, so:

6. True or False? • If a statement is true, we can prove it is true for all cases • If a statement is false, we need to provide ONE counterexample. • A counterexample is a specific example for which the conjecture is false

7. Examples: Find a counterexample for each conjecture • The square of any number is greater than the original number. • Counterexample: Is • You can connect any three points to form a triangle • Counterexample: a straight line

8. Examples (cont) • ANY number and its absolute value are opposites. • Counterexample: A positive number and its absolute value • ie: and • Alana makes a conjecture about slicing pizza. She says that if you use only straight cuts, the number of pieces will be twice the number of cuts. Draw an example that shows you can make 7 pieces using 3 cuts.

9. Classwork: pg. 6 # 1 – 6, 8, 11, 12, 19 – 22, 25 - 29 Homework: WS 1-1 #1-6, 13, 14