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Learn how to use inductive reasoning, identify patterns, make conjectures, and disprove them with counterexamples. Practice steps of inductive reasoning with examples and counterexamples. Improve your problem-solving skills!
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Lesson Objectives • Use inductive reasoning to identify patterns and make conjectures • Find counterexamples to disprove conjectures
Vocabulary • inductive reasoning: using specific cases to prove that a rule or statement is true • conjecture: a statement based on inductive reasoning that is believed to be true • counterexample: an example that shows a conjecture is NOT true
Steps of Inductive Reasoning FIRST Identify a pattern SECOND Make a conjecture LASTLY Prove the conjecture as true or find a counterexample
Example: Identifying a Pattern Find the next item in each pattern. • January, March, May, … Pattern: every other month (odd months) Next item: July B. 7, 14, 21, 28, … Pattern: multiples of 7 Next item: 35
Example: Making a Conjecture Complete each conjecture. • The sum of two positive numbers is ____. List some examples and look for a pattern. The sum of two positive numbers is positive. • The area of a square with side length greater than 4 is _____ (greater/less) than its perimeter. Example: s = 5, A = 52 = 25, P = 4(5) = 20 The area of such a square is greater than its perimeter.
Example: Making a Conjecture The heights of eight students in a class are recorded below. Make a conjecture based on the data.
Example: Finding a Counterexample Show that each conjecture is false by finding a counterexample. • For every integer n, n3is positive. n = -3 (-3)3 = (-3)(-3)(-3) = -27 • Two complimentary angles are not congruent. 45 + 45 = 90 Two 45˚ angles are complimentary and congruent
Example: Finding a Counterexample • Based on the data of students’ heights, every boy is at least 3 inches taller than the tallest girl.
