Chapter 1 The Art of Problem Solving © 2007 Pearson Addison-Wesley. All rights reserved

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 © 2008 Pearson Addison-Wesley. All rights reserved

Section 1-1 Chapter 1 Solving Problems by Inductive Reasoning © 2008 Pearson Addison-Wesley. All rights reserved

Solving Problems by Inductive Reasoning Characteristics of Inductive and Deductive Reasoning Pitfalls of Inductive Reasoning © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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 = 111 37 × 6 = 222 37 × 9 = 333 37 × 12 = 444 Solution 37 × 15 = 555 © 2008 Pearson Addison-Wesley. All rights reserved

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 30 + 7 = 37. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

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. © 2008 Pearson Addison-Wesley. All rights reserved

Example: pitfalls of inductive reasoning © 2008 Pearson Addison-Wesley. All rights reserved

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 = 3 2 + 4 2 c 2 = 9 + 16 = 25 c = 5 © 2008 Pearson Addison-Wesley. All rights reserved