Slide 1-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.

1 Copyright © 2005 Pearson Education, Inc.

SEVENTH EDITION and EXPANDED SEVENTH EDITION

Copyright © 2005 Pearson Education, Inc.

Chapter 1

Critical Thinking Skills


1.1

Inductive Reasoning


Natural Numbers

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

The three dots, called an ellipsis, mean that 4 is not the last number but that the numbers continue in the same pattern.

{1,2,3,4,...}


Divisibility

If 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,…

a b


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.


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.


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.


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.


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.


1.2

Estimation


Estimation

The process of arriving at an approximate answer to a question is called estimation.

Estimates are not meant to give exact values for answers but are a means of determining whether your answer is reasonable.

We often round numbers to estimate, or approximate, an answer.

The symbol means is approximately equal to.


Example: Estimation

For a staff meeting, Judith Spangler wants to buy her staff bagels and cream cheese. Estimate her cost to purchase 5 dozen bagels and 3 pounds of cream cheese if bagels are $6.59 a dozen and cream cheese is $3.29 per pound.

Solution: Round the amounts to obtain an estimate. Bagels: Cream cheese: Estimated total cost is

5 $7 $35 3 $3 $9$44.


Example: Using Estimates in Calculations

After one year Kara has 16,248.3 miles on her vehicle that she uses only for her job.

Estimate how many miles she drives monthly. If her company pays her $0.29 per mile,

estimate the amount that she is compensated monthly.


Example: Using Estimates in Calculations continued

Solution: Round the numbers to obtain an estimate.

Therefore the Kara drives approximately 1700 miles per month.

Rounding the compensation to $0.30 per mile, the monthly compensation is 1600 x $0.30, or $480.

16,248.3 16,000

160012 10


Estimates on a Map

Sometimes when working with measurements on a map, it may be difficult to get an accurate estimate because of the curves on the map.

To get a more accurate estimate, you may want to use a piece of string and tape or pins to mark the ends.


Estimates on a Photo

In order to estimate large areas, a photograph can be helpful. We can divide the photo into rectangles with equal areas, then pick one area that looks representative of all the areas.

Estimate (count) the number of items in this single area. Then multiply by the number of equal areas.


1.3

Problem Solving


Polya’s Procedure

George Polya (1887-1985) developed a general procedure for solving problems.


Guidelines for Problem Solving

Understand the Problem. Devise a Plan. Carry Out the Plan. Check the Results.


1. Understand the Problem.

Read the problem carefully, at least twice. Try to make a sketch of the problem. Label the

given 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.


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. Use a table to list information to help solve. Can you make an educated guess at the solution?


3. Carrying Out the Plan.

Use the plan you devised in step 2 to solve the problem.


4. Check the Results.

Ask yourself, “Does the answer make sense?” and “Is it reasonable?” If the answer is not reasonable, recheck your method and

calculations. Check the solution using the original statement, if

possible. Is there a different method to arrive at the same

conclusion? Can the results of this problem be used to solve other

problems?


Example: Selling a House

The Sharlow’s are planning to sell their home. They want to be left with $139,500 after paying commission to the realtor.

If a realtor receives 7% of the selling price, how much must they sell the house for?

If a realtor receives a flat $5000 and then 3% of the selling price, how much must they sell their house for?


Solution

Translate the problem in algebraic terms as follows:

selling price less commission is amount left.

x - 0.07x = $139,500

Thus, the Sharlow’s need to sell their home for $150,000.


Solution continued

If the realtor receives a flat fee of $5000, then 3% commission, first add 3% to the amount they wish to be left with.

$139,500 + 3% $143,814.

Then add $5000 to this total.

$143,814 + $5000 = $148,814.


Example: Taxi Rates

In Mexico, a taxi ride costs $4.80 plus $1.68 for each mile traveled. Diego and Juanita budgeted $25 for a taxi ride (excluding tip).

How far can they travel on their $25 budget? If they include a $2 tip, then how far can they

travel?


Solution

We know that the initial charge plus the mileage charge can equal $25. So, if we let x equal the distance, in miles, driven by the taxi for $25, then

If they wish to give a $2 tip, we solve the same way only allowing only $23 to be the budget.

$4.80 ($1.68) $25

12 miles.

x

x

$4.80 ($1.68) $23

10.8 miles.

x

x

Slide 1-1 Copyright © 2005 Pearson Education, Inc. SEVENTH EDITION and EXPANDED SEVENTH EDITION.

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