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2-1 Using Inductive Reasoning to Make Conjectures

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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2-1 Using Inductive Reasoning to Make Conjectures

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  1. 2-1 Using Inductive Reasoningto Make Conjectures

  2. Lesson Objectives • Use inductive reasoning to identify patterns and make conjectures • Find counterexamples to disprove conjectures

  3. 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

  4. Steps of Inductive Reasoning FIRST Identify a pattern SECOND Make a conjecture LASTLY Prove the conjecture as true or find a counterexample

  5. 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

  6. 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.

  7. Example: Making a Conjecture The heights of eight students in a class are recorded below. Make a conjecture based on the data.

  8. 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

  9. Example: Finding a Counterexample • Based on the data of students’ heights, every boy is at least 3 inches taller than the tallest girl.

  10. HW #10: p.77 #11-33 odd

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