MAT-105

Applied Liberal Arts

Mathematics

Module 1

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Chapter 1

Critical Thinking Skills

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WHAT YOU WILL LEARN

• Inductive and deductive reasoning processes • Estimation

• Problem-solving techniques

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1.1

Inductive Reasoning

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Natural Numbers

The set of natural numbers is also called the set of counting numbers.

•

{1,2,3,4,5,6,7,8,...}

The three dots, called an ellipsis, mean that 8 is not the last number but that the numbers continue in the same manner.

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Divisibility

If a b has a remainder of zero, then a is divisible by b.

The even counting numbers are divisible by 2.

They are 2, 4, 6, 8,… .

The odd counting numbers are not divisible by

2. They are 1, 3, 5, 7,… .

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Inductive Reasoning

The process of reasoning to a general conclusion through observations of specific cases. Also called induction.

Often used by mathematicians and scientists to predict answers to complicated problems.

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Scientific Method

Inductive reasoning is a part of the scientific method. When we make a prediction based on specific observations, it is called a hypothesis or conjecture. Copyright © 2009 Pearson Education, Inc.

Counterexample

In testing a hypothesis, if a special case is found that satisfies the conditions of the conjecture but produces a different result, that case is called a counterexample.

Only one exception is necessary to prove a hypothesis false.

If a counterexample cannot be found, the conjecture is neither proven nor disproven.

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Deductive Reasoning

A second type of reasoning process.

Also called deduction.

Deductive reasoning is the process of reasoning to a specific conclusion from a general statement. Copyright © 2009 Pearson Education, Inc.

Example: Inductive Reasoning

Use inductive reasoning to predict the next three numbers in the pattern (or sequence).

7, 11, 15, 19, 23, 27, 31,…

Solution:

We can see that four is added to each term to get the following term.

31 + 4 = 35,

35 + 4 = 39,

39 + 4 = 43

Therefore, the next three numbers in the sequence are 35, 39, and 43.

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1.3

Problem Solving

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Polya’s Procedure

George Polya (1887-1985) a mathematician who was educated in Europe and taught at

Stanford developed a general procedure for solving problems.

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Guidelines for Problem Solving

Understand the Problem.

Devise a Plan to Solve the Problem.

Carry Out the Plan.

Check the Results.

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1. Understand the Problem.

Read the problem carefully at least twice.

Try to make a sketch of the problem. Label the information given.

Make a list of the given facts that are pertinent to the problem.

Decide if you have enough information to solve the problem.

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2. Devise a Plan to Solve the Problem.

Can you relate this problem to a previous problem that you’ve worked before?

Can you express the problem in terms of an algebraic equation?

Look for patterns or relationships.

Simplify the problem, if possible.

Can you substitute smaller or simpler numbers to make the problem more understandable?

Use a table to list information to help solve.

Can you make an educated guess at the solution? Copyright © 2009 Pearson Education, Inc.

3. Carrying Out the Plan.

Use the plan you devised in step 2 to solve the problem. Copyright © 2009 Pearson Education, Inc.

4. Check the Results.

Ask yourself, “Does the answer make sense?” and “Is it reasonable?”

If the answer is not reasonable, recheck your method for