Deductive Reasoning BIG IDEA: REASONING AND PROOF ESSENTIAL UNDERSTANDING: Given true statements, deductive reasoning can be used to make a valid or true conclusion. Deductive reasoning often involves the Laws of Syllogism and Detachment MATHEMATICAL PRACTICE: Reason abstractly and quantitatively

You want to use the coupon to buy three different pairs of jeans You want to use the coupon to buy three different pairs of jeans. You have narrowed your choices to four pairs. The costs of the different pairs are $24.99, $39.99, $40.99 and $50.00. If you spend as little as possible, what is the average amount per pair of jeans that you will pay? Explain. BUY TWO PAIRS OF JEANS GET A THIRD FREE* Free jeans must be of equal or lesser value GETTING READY

Symbolic notation Conditional Statement: If p, then q. Converse Statement: If q, then p. Biconditional Statement: p if and only if q Inverse Statement: If not p, then not q. Contrapositive Statement: If not q, then not p. Symbolic notation

Deductive Reasoning: a process of reasoning _____________ from given ____________ to a conclusion. Law of Detachment: If the __________________ of a true conditional is __________, then the _____________ is true. Law of Syllogism: If and are __________ conditional statements, then is true. Deductive Reasoning

a) If it is Friday, then the Smith’s family has pizza for dinner a) If it is Friday, then the Smith’s family has pizza for dinner. Today is Friday, therefore, the Smith’s family will have pizza for dinner. b) Josh knows that Brand X computers cost less than Brand Y computers. All other brands that Josh knows of cost less than Brand X. Josh reasons that Brand Y costs more than all other brands. EX 1: State whether the following argument used Inductive or Deductive Reasoning.

To use the Law of Detachment, identify the hypothesis of the given true conditional. If the second given statement matches the hypothesis of the conditional or the given information is “detached” from it, then you can make or draw a valid conclusion. To use the Law of Syllogism, the conclusion of one statement is the hypothesis of the other statement which allows you to state a conclusion from two true conditional statements. Laws of Logic

Ex 2: What can you conclude from the given true statements? a) GIVEN: If a student gets an A on a final exam, then the student will pass the course. Felicia got an A on her History final exam. b) GIVEN: If two angles are adjacent, then they share a common vertex. c) GIVEN: If there is lightning, then it is not safe to be out in the open. Marla sees lightning from the soccer field Ex 2: What can you conclude from the given true statements?

Ex 3: What can you conclude from the given true statements? a) GIVEN: If a figure is a square, then the figure is a rectangle. If a figure is a rectangle, then the figure has four sides. b) GIVEN: If a whole number ends in 0, then it is divisible by 10. If a whole number is divisible by 10, then it is divisible by 5. c) GIVEN: If you do gymnastics, then you are flexible. If you do ballet, then you are flexible. Ex 3: What can you conclude from the given true statements?

EX 4: State whether the argument is valid. a) Michael knows that all sophomores take driver’s education in his school. Joe takes driver education, so Joe is a sophomore. b) If , then is an acute angle. The so it is an acute angle. EX 4: State whether the argument is valid.

Ex 5: What can you conclude from the given true statements? 1. If a fish swims at 68 mph, then it swims at 110 kmph. 2. If a fish can swim at 110 kmph, then it is a sailfish. 3. If a fish is the largest species of fish, then it is a great white shark. 4. If a fish weighs over 2000 pounds, then it is the largest species of fish. 5. If a fish is the fastest species of fish, then it can reach speeds of 68 mph. Ex 5: What can you conclude from the given true statements?

2.3 p. 91 8 – 13 all, 14 – 20 evens, 21, 22, 24, 25, 30, 32, 33 – 42 all, 46, 48, 50 29 questions