Slide 1Chapter 1: The Art of Problem SolvingSlide 3Solving Problems by Inductive ReasoningCharacteristics of Inductive and Deductive ReasoningExample: determine the type of reasoningExample: predict the product of two numbersExample: predicting the next number in a sequencePitfalls of Inductive ReasoningExample: pitfalls of inductive reasoningSlide 11Example: use deductive reasoningChapter 1The Art of Problem Solving© 2007 Pearson Addison-Wesley.All rights reserved© 2008 Pearson Addison-Wesley. All rights reserved1-2Chapter 1: The Art of Problem Solving1.1 Solving Problems by Inductive Reasoning 1.2 An Application of Inductive Reasoning: Number Patterns 1.3 Strategies for Problem Solving 1.4 Calculating, Estimating, and Reading Graphs© 2008 Pearson Addison-Wesley. All rights reserved1-3Chapter 1Section 1-1Solving Problems by Inductive Reasoning© 2008 Pearson Addison-Wesley. All rights reserved1-4Solving Problems by Inductive Reasoning•Characteristics of Inductive and Deductive Reasoning•Pitfalls of Inductive Reasoning© 2008 Pearson Addison-Wesley. All rights reserved1-5Characteristics of Inductive and Deductive ReasoningInductive ReasoningDraw a general conclusion (a conjecture) from repeated observations of specific examples. There is no assurance that the observed conjecture is always true. Deductive ReasoningApply general principles to specific examples.© 2008 Pearson Addison-Wesley. All rights reserved1-6Determine whether the reasoning is an example of deductive or inductive reasoning.All math teachers have a great sense of humor. Patrick is a math teacher. Therefore, Patrick must have a great sense of humor.SolutionBecause the reasoning goes from general to specific, deductive reasoning was used.Example: determine the type of reasoning© 2008 Pearson Addison-Wesley. All rights reserved1-7Use the list of equations and inductive reasoning to predict the next multiplication fact in the list:37 × 3 = 111 37 × 6 = 222 37 × 9 = 333 37 × 12 = 444Solution37 × 15 = 555Example: predict the product of two numbers© 2008 Pearson Addison-Wesley. All rights reserved1-8Use inductive reasoning to determine the probable next number in the list below.2, 9, 16, 23, 30SolutionEach number in the list is obtained by adding 7 to the previous number. The probable next number is 30 + 7 = 37.Example: predicting the next number in a sequence© 2008 Pearson Addison-Wesley. All rights reserved1-9Pitfalls of Inductive ReasoningOne can not be sure about a conjecture until ageneral relationship has been proven.One counterexample is sufficient tomake the conjecture false.© 2008 Pearson Addison-Wesley. All rights reserved1-10We concluded that the probable next number in the list 2, 9, 16, 23, 30 is 37.If the list 2, 9, 16, 23, 30 actually represents the dates of Mondays in June, then the date of the Monday after June 30 is July 7 (see the figure on the next slide). The next number on the list would then be 7, not 37.Example: pitfalls of inductive reasoning© 2008 Pearson Addison-Wesley. All rights reserved1-11Example: pitfalls of inductive reasoning© 2008 Pearson Addison-Wesley. All rights reserved1-12Find the length of the hypotenuse in a right triangle with legs 3 and 4. Use the Pythagorean Theorem: c 2 = a 2 + b 2, where c is the hypotenuse and a and b are legs.Solutionc 2 = 3 2 + 4 2 c 2 = 9 + 16 = 25c = 5Example: use deductive