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2.2 :Deductive Reasoning Objectives: 1. To use deductive reasoning to prove a conjecture is true Assignment: • P. 22: 15-19 • P. 24: 11-19 Objective 1 You will be able to use deductive reasoning to prove a conjecture is true Exercise 1 Numbers such as 3, 4, and 5 are consecutive integers. Make and test a conjecture about the sum of any three consecutive integers. History When the architects designed this school building, they were approached by an ancient secret society whose members make up numerous Texas dignitaries. They convinced the architects to add several secret passages and hidden conference rooms to their design plans. History Sometimes when I stay after school late into the evening grading papers, planning lessons, and contacting parents, I hear strange and inauspicious sounds emanating from behind one of my walls. Conspiracy? Thus, it is my conjecture that one of the secret society's hidden passages lies between the walls of Room D202 and D204. This is a bold and perhaps conspiratorial conjecture, but I am confident that it is true. (You should hear the sounds--Oh, my!) Stop Making Fun of Me! I have told few people of my theory, and they unanimously dismiss my conviction with ridicule. (Then they ask me if I frequently watch re-runs of the XFiles with the notion that the story lines are largely nonfiction!) Redemption To convince the skeptics and to redeem my reputation, I need absolute and conclusive proof that there exists a hidden passage between these classrooms. Proof The Principle of Laplace: The weight of evidence for an extraordinary claim must be proportional to its strangeness. In Other Words… “Extraordinary claims require extraordinary evidence.” -Carl Sagan Exercise 2 In your group, come up with a nondestructive method for proving or disproving the extraordinary claim that there’s a secret tunnel between D202 and D204. Exercise 3 In the Sudoku puzzle shown, what number must be written in the blue box? Why? ? Deductive Reasoning The process of demonstrating that if certain statements are accepted as true, then other statements can be shown to follow from them. Deductive Reasoning The “accepted” statements are sometimes premises or assumptions, and all deductive arguments must have them. Deductive reasoning uses logical inference to build on these assumptions. Unlike inductive reasoning, deductive reasoning will always lead to the truth as long as the assumptions are true. Exercise 4 All humans have skeletons is a reasonable assumption. So, since Mr. Noel is a human, what must be true about him? Inductive Reasoning Deductive Reasoning Inductive vs. Deductive 1. We use inductive reasoning to investigate and discover things about our world. 2. Since the conjectures we make using our inductive reasoning is based on our fallible observation skills, we can be wrong. 3. We can search for a counterexample to disprove our conjectures. 4. In mathematics, we use our deductive reasoning to prove our conjectures beyond all uncertainty. 2.2 :Deductive Reasoning Objectives: 1. To use deductive reasoning to prove a conjecture is true Assignment: • P. 22: 15-19 • P. 24: 11-19