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Exploring Inductive Reasoning: Patterns, Conjectures & Counterexamples

Understand inductive reasoning by recognizing patterns, making predictions, and testing conjectures with relevant examples and counterexamples. Explore the process of making educated guesses based on observations.

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Exploring Inductive Reasoning: Patterns, Conjectures & Counterexamples

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  1. 2.1 Inductive Reasoning

  2. Inductive Reasoning • Process of considering several facts and then making an educated guess based on these facts. • Recognizing patterns and making predictions.

  3. Describe the pattern. Find the next number. 625, multiply the previous number by 5 • 1, 5, 25, 125, … • 1, 2, 4, 7, 11, … • 1, 4, 9, 16, 25, …. 16, add the next consecutive integer to the previous # 36, square of consecutive whole numbers

  4. Based on the chart…make a function rule relating x and y. y = 4x y = 2x + 1

  5. Conjectures • A conjecture is an unproven statement that is based on observations. • Not necessarily a true statement.

  6. Complete the conjecture based on the pattern. • Conjecture: The product of any two even numbers is ________________. 4 x 2 = 8 6 x 8 = 48 -2 x -8 = 16 10 x 4 = 40 -4 x 6 = -24 -6 x 12 = -72 EVEN • Conjecture: The sum of any two consecutive whole numbers is ________________. 4 + 5 = 9 6 + 7 = 13 11 + 12 = 23 7 + 8 = 15 12 + 13 = 25 20 + 21 = 41 ODD

  7. Is the conjecture true or false? • Any four-sided polygon is a square. FALSE! Four-Sided Polygons: rectangle, rhombus, trapezoid, parallelogram. • For a conjecture to be true it must be true for ALL cases. • An example that proves a conjecture false is called a counterexample.

  8. Determine if the conjecture is true or false. Give a counterexample if false. Jose is a child living in Argentina, where spring begins in September and ends in December. Because he sees the days getting longer in these months, he makes a conjecture that the days are getting longer all over the world. Is his conjecture true or false? FALSE! Counterexample: New Jersey – September – December the days are getting shorter!

  9. D E F Determine if the conjecture is true or false. Give a counterexample if false. 1.) Given: Collinear points D, E, and F. Conjecture: DE + EF = DF False. Counterexample: EF + FD = ED 2.)Conjecture: The difference of two positive numbers is positive. False. Counterexample: 20 – 25 = -5

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