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Inductive Reasoning and Counterexamples in Problem Solving

Learn how to use inductive reasoning to make conjectures and find counterexamples. Explore the concept of consecutive integers and test conjectures about their sums. Discover the importance of counterexamples in disproving conjectures.

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Inductive Reasoning and Counterexamples in Problem Solving

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  1. Unit 01 – Lesson 08 – Inductive Reasoning • Essential Question • How can you use reasoning to solve problems? • Scholars will • Make conjectures based on inductive reasoning • Find counterexamples

  2. What is inductive reasoning? A conjectureis an unproven statement that is based on observations. You see inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general cause.

  3. How would you describe the visual pattern?

  4. How can you make and test a conjecture? Numbers such as 3, 4, and 5 are called consecutive integers. Make and test a conjecture about the sum of any three consecutive integers.

  5. What is a counterexample? To show that a conjecture is true, you must show that it is true for all cases. You can show that a conjecture is false, however, by finding just one counterexample. A counterexampleis a specific case for which the conjecture is false.

  6. How do you find a counterexample? A student makes the following conjecture about the sum of two numbers. Find a counter example to disprove the student’s conjecture. CONJECTURE – The sum of two numbers is always more than the greater number. To find a counterexample, you need to find a sum that is less that the greater number. -2 + (-3) = -5 -5 ≯ -2 Because a counterexample exists, the conjecture is false.

  7. Practice • Find a counterexample to show that the conjecture is false. • The value of x2 is always greater than the value of x. • The sum of two numbers is always greater than their difference.

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