 2012 Pearson Education, Inc. Slide Chapter 1 The Art of Problem Solving

 2012 Pearson Education, Inc. Slide Chapter 1: The Art of Problem Solving 1.1 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

 2012 Pearson Education, Inc. Slide Section 1-1 Solving Problems by Inductive Reasoning

 2012 Pearson Education, Inc. Slide Solving Problems by Inductive Reasoning Characteristics of Inductive and Deductive Reasoning Pitfalls of Inductive Reasoning

 2012 Pearson Education, Inc. Slide Characteristics of Inductive and Deductive Reasoning Inductive Reasoning Draw a general conclusion (a conjecture) from repeated observations of specific examples. There is no assurance that the observed conjecture is always true. Deductive Reasoning Apply general principles to specific examples.

 2012 Pearson Education, Inc. Slide Example: determine the type of reasoning Determine 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. Solution Because the reasoning goes from general to specific, deductive reasoning was used.

 2012 Pearson Education, Inc. Slide Example: predict the product of two numbers Use the list of equations and inductive reasoning to predict the next multiplication fact in the list: 37 × 3 = × 6 = × 9 = × 12 = 444 Solution 37 × 15 = 555

 2012 Pearson Education, Inc. Slide Example: predicting the next number in a sequence Use inductive reasoning to determine the probable next number in the list below. 2, 9, 16, 23, 30 Solution Each number in the list is obtained by adding 7 to the previous number. The probable next number is = 37.

 2012 Pearson Education, Inc. Slide Pitfalls of Inductive Reasoning One can not be sure about a conjecture until a general relationship has been proven. One counterexample is sufficient to make the conjecture false.

 2012 Pearson Education, Inc. Slide Example: pitfalls of inductive reasoning We 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.

 2012 Pearson Education, Inc. Slide Example: pitfalls of inductive reasoning

 2012 Pearson Education, Inc. Slide Example: use deductive reasoning Find 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. Solution c 2 = c 2 = = 25 c = 5