Warm Up

1. Warm Up Describe the pattern and sketch the next figure. 1.) Adds one more side to the polygon. 2.) Face rotates 90 degrees clockwise.

2. Geometry Sections 2.1, 2.2 Use Inductive Reasoning and Analyze Conditional Statements

3. Describe a Number Pattern Describe the pattern in the numbers and find the next number in the sequence. 1.) 3, 6, 12, 24, 48 Multiply by 2 2.) 0, 1, 4, 9, 16, 25, 36 Perfect squares

4. Vocabulary • Conjecture: An unproven statement that is based on observations. • You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case.

5. Conjecture/ Counterexamples • Use these sums of odd integers: 3+7=10, 1+7=8, 17+21=38 Conjecture: The sum of any two odd integers is__ even _______.

6. Vocabulary • Counterexample:An example that proves a conjecture is false. *Example: • Conjecture: The sum of two numbers is always greater than the larger number. Counterexample: -3 + -2 = -5

7. Conjecture/ Counterexamples • Show the conjecture is false by finding a counterexample. • If the product of two numbers is positive, then the two numbers must both be positive. -8 * -2 = 16

8. Vocabulary Conditional statement: a logical statement that has two parts, a hypothesis and a conclusion. When a conditional statement is written in if-then form the “if” part contains the hypothesis and the “then” part contains the conclusion. If hypothesis, then conclusion.

9. Conditional Statements Rewrite statement in if-then form: Guitar players are musicians If you are a guitar player then you are a musician.

10. Conditional Statement Rewrite the statement in if-then form: An even number is divisible by two. If a number is even, then it is divisible by two.

11. Converse The converse of a conditional statement is formed by switching the conclusion and the hypothesis. Conditional Statement: If you are a guitar player then you are a musician. Converse: If you are a musician then you are a guitar player.

12. Converse Example Write the converse of: If a number is even, then it is divisible by two. Converse: If a number is divisible by two, then it is an even number.

13. Inverse The inverse of a conditional statement is when you negate the hypothesis and the conclusion. (Turn them into nots) Conditional Statement: If you are a guitar player then you are a musician. Inverse: If you are not a guitar player than you are not a musician.

14. Inverse Example Write the Inverse of: If a number is even, then it is divisible by two. Inverse: If a number is not even, then it is not divisible by two.

15. Contrapositive The contrapositive of a conditional statement is when you negate the hypothesis and conclusion of the converse statement. Converse Statement: If you are a musician, then you are a guitar player. Contrapositive: If you are not a musician, then you are not a guitar player

16. Contrapositive Example Write the contrapositive of: If a number is divisible by two, then it is an even number. Contrapositive: If a number is not divisible by two, then it is not an even number.

17. Related Conditionals • Conditional statement: If hypothesis, then conclusion. • Inverse: If not hypothesis, then not conclusion. • Converse: If conclusion, then hypothesis. • Contrapositive: If not conclusion, then not hypothesis.

18. Write the inverse, converse, & contrapositive. • Conditional: If water is frozen, then its temp is below zero. • Inverse: If water is not frozen, then its temp is not below zero. • Converse: If the temp is below zero, then water is frozen. • Contrapositive: If the temp is not below zero, then the water is not frozen.

19. Homework: • Page 67- 68 • #6-10(evens),16, 17 • Page 74 • #3-6