Inductive Reasoning Inductive Reasoning: The process of using examples, patterns or specific observations to form a conclusion, conjecture, or generalization.

Inductive Reasoning 1. The sun rose this morning. 2. The sun rose yesterday morning. 3. The sun rose two days ago. 4. The sun rose three days ago. What conclusion can we make?

More Examples You go to the cafeteria every Friday for the first 12 weeks of school. Every Friday they have burgers on the grill. What conjecture can you make about what you will see when you go in there on Friday of the 13 th week? Is this necessarily true?

More Examples The Bad News Bears have lost every game this season going into the last week. What would you expect to happen their last game of the season?

More Examples Can you think of your own example?

Deductive Reasoning The process of using accepted definitions or theories to form more specific conclusions.

Deductive Reasoning Examples: 1.All students eat pizza. Rachel is a student at Hiram. Therefore, Rachel eats pizza. 2. All athletes work out in the gym. Matt Ryan is an athlete. Therefore, Matt Ryan works out in the gym.

EOCT Practice a)Every time John goes to a Hillgrove basketball game they win. He concludes that they are going to win the next game he goes to. b)If the last digit of a number is divisible by 5, then the number itself is divisible by 5. c)McDonald’s has 49 cent burgers every Tuesday for the first 9 weeks of the year. Next time I go there on a Tuesday, I expect to be able to buy some for 49 cents. d)It snows every Christmas in Boise, Idaho where Tina’s grandparents live. She expects to see snow again this year when she goes to see them for Christmas. b All of the following are examples of inductive reasoning except:

EOCT Practice a) b) c) d) d Goldbach’s conjecture states: every even number greater than 2 can be written as the sum of two primes. Which sum for 30 supports his conjecture?

EOCT Practice a)Inductive reasoning b)Deductive reasoning c)Indirect proof d)Conjecture a For the past 6 weeks, your aunt has asked you to watch your cousin on Wednesday night. You conclude that you are will be asked to watch your cousin next Wednesday. Which of the following did you use to reach this conclusion?

Conditional Statements Conditional statement: A statement usually written in “If-Then” form that has two parts: Hypothesis: The condition Conclusion: What follows when the condition is met Example 1: If it is noon in Florida, then it is 9:00 AM in California. Hypothesis: “it is noon in Florida” Conclusion: “it is 9:00 AM in CA”

Practice 1: Identify Hypothesis / Conclusion If the weather is warm, then we should go swimming. Hypothesis: ______________________________ Conclusion: ______________________________

If the sun is shining, then we should go to the beach. Hypothesis: ___________________________ Conclusion: __________________________

Rewrite Into If-Then Form Example : “Today is Monday if yesterday was Sunday.” If-Then form: If yesterday was Sunday, then today is Monday Practice : “An angle is acute if it measures less than 90°.” If-Then form: ______________________ ___________________________________

Counterexamples Conditional statements can be TRUE or FALSE To show a statement is TRUE, you must prove it for ALL cases To show a statement is FALSE, you must show only ONE counterexample

Provide a Counterexample Decide if the statement is true or false If false, provide a counterexample (a) Statement: If x² = 16, then x = 4. True / False? ____________ Counterexample: _____________________________

Provide a Counterexample (b) Statement: If it is February 14, then it is Valentine’s Day. True / False? ____________ Counterexample: _____________________________

Provide a Counterexample (a) Statement: If you visited New York, then you visited the Statue of Liberty. True / False? ________________ Counterexample: ___________________________________

Provide a Counterexample (b) Statement: If a number is odd, then it is divisible by 3. True / False? _____________ Counterexample: _________________________________

Other Forms of If-Then Statements Statement: If you like volleyball, then you like to be at the beach. Converse: (Think of “FLIP FLOPS”) SWITCH hypothesis & conclusion If you like to be at the beach, then you like volleyball.

Other Forms of If-Then Statements Statement: If you like volleyball, then you like to be at the beach. Inverse: NEGATE hypothesis & conclusion If you do not like volleyball, then you do not like to be at the beach.

Other Forms of If-Then Statements Statement: If you like volleyball, then you like to be at the beach. Contrapositive: SWITCH and NEGATE!!! If you do not like to be at the beach, then you do not like volleyball.

Write the converse, inverse, and contrapositive If x is odd, then 2x is even. Converse: _________________________________ Inverse: _________________________________ Contrapositive: _________________________________

If there is snow, then flowers are not in bloom. Converse: _________________________________ Inverse: _________________________________ Contrapositive: _________________________________

If an angle measures 90°, then it is a right angle. Converse: _________________________________ Inverse: _________________________________ Contrapositive: _________________________________

Equivalent Statements Two conditional statements are equivalent if they are BOTH true or BOTH false A conditional statement and its contrapositive are equivalent statements The inverse and converse are equivalent The statements themselves are not the same, but their logic is the same

Statement: If you like volleyball, then you like to be at the beach. Converse: If you like to be at the beach, then you like volleyball. Inverse: If you do not like volleyball, then you do not like to be at the beach. Contrapositive: If you do not like to be at the beach, then you do not like volleyball.

Ticket out the Door! Closure If you do homework, then you will succeed. Hypothesis: _____________________________ Conclusion: _____________________________ Converse: ______________________________ Inverse: _________________________________ Contrapositive: ________________________________________

Homework Geometry Worksheet