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2.1 Inductive Reasoning Inductive Reasoning: is the process of forming a conjecture based on a set of observations Conjecture is a statement that is believed to be true but not yet proved. Ex. 1 Use inductive reasoning to form a conjecture 1, 2, 3, 4, 5, 6, 7, …. *Purple is your given *Green is your guess of what comes next Conjecture The sequence increases by one each time Lesson 1-1 Point, Line, Plane 2 Ex. 2 Inductive Reasoning 5, 10, 15, 20, 25, 30, 35, 40, 45, …. Conjecture 1: The sequence goes up by five each time. Conjecture 2: ends in a 5, then a 0, then a 5, and so on…and all the numbers in the tens place appear twice in the sequence. Lesson 1-1 Point, Line, Plane 3 Ex. 3 Inductive Reasoning Conjecture 1: every term in the sequence is rotated counterclockwise 90 degrees. Conjecture 2: continuously repeats the four positions of right, bottom, left, top, right, bottom, left, top, … Lesson 1-1 Point, Line, Plane 4 Ex 4 Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure. Conjecture: ?? Lesson 1-1 Point, Line, Plane 5 EXAMPLE I Describe the pattern in the numbers –7, –21, –63, –189,… and write the next three numbers in the pattern. Lesson 1-1 Point, Line, Plane 6 EXAMPLE II Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers using inductive reasoning. Hint: We must first gather data before we make predictions Lesson 1-1 Point, Line, Plane 7 EXAMPLE IV Make and test a conjecture about the sign of the product of any three negative integers. Hint: We must first gather data before we make predictions Lesson 1-1 Point, Line, Plane 8 Counterexamples in Real Life All birds can fly. A basketball player must be tall in order to be good at dunking baskets. Students with low grade-point averages in high school do not contribute to the academic community. Lesson 1-1 Point, Line, Plane 9 Counterexample in Math A counter example in math is an example for which the conjecture is false. * It is one number or one picture or one set of numbers….it is NOT a written reason!!! Lesson 1-1 Point, Line, Plane 10 EXAMPLE A – Counter Examples Conjecture: The sum of two numbers is always greater than the larger number. Are they any counterexamples that exist to disprove this conjecture? Lesson 1-1 Point, Line, Plane 11 EXAMPLE V – Counter Examples Find a counterexample to show that the following conjecture is false. 2 Conjecture: The value of x is always greater than the value of x. Lesson 1-1 Point, Line, Plane 12 2.3 Apply Deductive Reasoning Deductive Reasoning uses facts, definitions, and the laws of logic to form a logical argument. Laws of Logic Law of Detachment If the hypothesis of a conditional statement is true, then the conclusion is also true. Law of Syllogism If A, then B. If B, then C. If A, then C Ex Ex. 2. 1 Ex. Mary 1 If two segments have the Ex.2 goes to the movies every If Ifx2Rick >25,takes thenChemistry x 2 >20 this year, same length, then they are Friday and Saturday night. then Jesse will be Rick’s partner. If congruent. If x>5, then x 2 >25. Jess is Rick’s lab partner, then You know that BC =XY Rick will get an A in chemistry. Today is Friday Lesson 1-1 Point, Line, Plane 14 Logic Puzzle (Nicknames) Four friends: Dave, Mike, John, and Terry, are nicknamed Stick, Batman, Atomic Head, and Feaser, but not in that order. Which friend has which nickname? A. John is faster than Batman but not as strong as Atomic Head. B. Batman is stronger than Terry but slower than Feaser. C. Dave is faster than both Stick and John, but not as strong as Batman Strength and speed are independent qualities. Dave Stick Mike John Terry x x x 0 Batma n x 0 x x Atomic Head 0 x x x x x 0 x Feaser Lesson 1-1 Point, Line, Plane 15