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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 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 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 9 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 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. Conjecture: The value of x 2 is always greater than the value of x. Lesson 1-1 Point, Line, Plane 12 2.2 Analyzing Conditional Statements Conditional Statements: are statements that can be put in “In-Then” form “If-Then” Form: If A (Hypothesis), then B (Conclusion) Ex. Conditional Statement: “All birds have feathers.” Hypothesis (A) Conclusion (B) Conditional Statement (In “If-Then” form): “If it is a bird, then it has feathers.” Hypothesis (A) Conclusion (B) Converse:If B, then A. (Reverse Order) Conditional Statement “If it is a bird, then it has feathers.” Conditional Statement “If it has a feathers, then it has bird.” On your own… Rewrite the conditional statements in if-then form: “All vertebrates have a backbone” “All triangles have 3 sides” 2 “When x=2, x =4 Biconditional Statement - - a statement that contain the phrase “if and only if” Used when both the conditional statement and it’s converse are true Ex: Definition: If two lines intersect to form a right angle, then they are perpendicular Converse: If two lines are perpendicular, then they intersect to form a right angle Biconditional: Two lines are perpendicular if and only if they intersect to form a right angle