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Chapter 1Chapter 1The Art of The Art of Problem SolvingProblem Solving

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Chapter 1: Chapter 1: The Art of Problem SolvingThe 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

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Section 1-1Section 1-1

Solving Problems by Inductive Reasoning

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Solving Problems by Inductive Solving Problems by Inductive ReasoningReasoning

• Characteristics of Inductive and Deductive Reasoning

• Pitfalls of Inductive Reasoning

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Characteristics of Inductive and Characteristics of Inductive and Deductive ReasoningDeductive 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.

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Example: determine the type of Example: determine the type of reasoningreasoning

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.

SolutionBecause the reasoning goes from general to specific, deductive reasoning was used.

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Example: predict the product of two Example: predict the product of two numbersnumbers

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

Solution37 × 15 = 555

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Example: predicting the next number Example: predicting the next number in a sequencein a sequence

Use inductive reasoning to determine the probable next number in the list below.

2, 9, 16, 23, 30

SolutionEach number in the list is obtained by adding 7 to the previous number.

The probable next number is 30 + 7 = 37.

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Pitfalls of Inductive ReasoningPitfalls 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.

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Example: pitfalls of inductive reasoningExample: 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.

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Example: pitfalls of inductive reasoningExample: pitfalls of inductive reasoning

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Example: use deductive reasoningExample: 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.

Solutionc 2 = 3 2 + 4 2

c 2 = 9 + 16 = 25

c = 5