Math Homework

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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

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Section 1-1

Solving Problems by Inductive

Reasoning

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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.

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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.

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Example: predict the product of two numbers.

Use Inductive reasoning: Draw a general conclusion (a conjecture) from repeated observations

of specific examples

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

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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.

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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.

(Note: if you get a different answer and explain it well after a

test, I have to accept your answers as correct.)

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Example: pitfalls of inductive reasoning

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Example: use deductive reasoning Apply general principles to specific examples

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