Chapter 1

The Art of Problem Solving

© 2007 Pearson Addison-Wesley.All rights reserved

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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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Chapter 1

Section 1-1Solving Problems by Inductive

Reasoning

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Solving Problems by Inductive or Deductive Reasoning

• Characteristics of Inductive and Deductive Reasoning

• Pitfalls of Inductive Reasoning

• Examples of Inductive and Deductive Reasoning

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Characteristics of Inductive and Deductive Reasoning

Inductive ReasoningDraw a general conclusion (a conjecture) from repeated observations of specific examples. There is no assurance that the observed conjecture is always true.

Deductive ReasoningApply general principles to specific examples.

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Determine whether the reasoning is an example of deductive or inductive reasoning.

All math teachers have a great sense of humor. Prof Darini is a math teacher. Therefore, Prof Darini must have a great sense of humor.

Example: Determine the type of reasoning

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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

Example: predict the product of two numbers

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

2, 9, 16, 23, 30

Example: predicting the next number in a sequence

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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.

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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.

Example: Use deductive reasoning

Section 1.1: Solving Problems by Inductive Reasoning

Is the reasoning an example of inductive or deductive reasoning?

If it rains, then Jess will stay home. It is raining.Therefore, Jess is at home.

a) Deductive

b) Inductive

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Section 1.1: Solving Problems by Inductive Reasoning

Is the reasoning an example of inductive or deductive reasoning?

It was sunny yesterday, and it is sunny today.Therefore it will be sunny tomorrow.

a) Deductive

b) Inductive

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Chapter 1

Section 1-2An Application of Inductive Reasoning: Number Patterns

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An Application of Inductive Reasoning: Number Patterns

• Number Sequences

• Successive Differences

• Number Patterns and Sum Formulas

• Figurate Numbers

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Number Sequences

Number SequenceA list of numbers having a first number, a second number, and so on, called the terms of the sequence.

Arithmetic SequenceA sequence that has a common difference between successive terms.

Geometric SequenceA sequence that has a common ratio between successive terms.

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Successive Differences

Process to determine the next term of a sequence using subtraction to find a common difference.

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Example: Successive Differences

Use the method of successive differences to find the next number in the sequence.

14, 22, 32, 44,...

14 22 32 44

8 10 12 Find differences

2 2 Find differences

Build up to next term: 58

2

14

58

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Number Patterns and Sum Formulas

21 3 5 (2 1) .n n

Sum of the First n Odd Counting NumbersIf n is any counting number, then

Special Sum FormulasFor any counting number n,

2 3 3 3(1 2 3 ) 1 2n n ( 1)

and 1 2 3 .2

n nn

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Example: Sum Formula

Use a sum formula to find the sum

1 2 3 48.

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Figurate Numbers

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Formulas for Triangular, Square, and Pentagonal Numbers

2the th square number is given by , andnn S n

For any natural number n,

( 1)the th triangular number is given by ,

2nn n

n T

(3 1)the th pentagonal number is given by .

2nn n

n P

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Example: Figurate Numbers

Use a formula to find the sixth pentagonal number

Section 1.2: An Application of Inductive Reasoning: Number Patterns

Find the probable next number in the sequence1, 5, 13, 25, 41,…

a) 51

b) 58

c) 61

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Section 1.2: An Application of Inductive Reasoning: Number Patterns

When applying the sum formula to what is the value of n?

a) 25

b) 26

c) 51

d) 52

1 3 5 51,

21 3 5 (2 1) ,n n

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Chapter 1

Section 1-3Strategies for Problem Solving

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Strategies for Problem Solving

• A General Problem-Solving Method

• Using a Table or Chart

• Working Backward

• Using Trial and Error

• Guessing and Checking

• Considering a Similar Simpler Problem

• Drawing a Sketch

• Using Common Sense26

A General Problem-Solving Method

Polya’s Four-Step Method

Step 1 Understand the problem. Read and analyze carefully. What are you to find?

Step 2 Devise a plan.

Step 3 Carry out the plan. Be persistent.

Step 4 Look back and check. Make sure that your answer is reasonable and that you’ve answered the question.

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Example: Using a Table or Chart

A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring but each month thereafter produced one new pair of rabbits. If each new pair produced reproduces in the same manner, how many pairs of rabbits will there be at the end of the 5th month?

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Example: Solution

Step 1 Understand the problem. How many pairs of rabbits will there be at the end of five months? The first month, each pair produces no new rabbits, but each month thereafter each pair produces a new pair.

Step 2 Devise a plan. Construct a table to help with the pattern.

Month Number of Pairs at Start

Number Produced

Number of Pairs at the End

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Example (solution continued)

Step 3 Carry out the plan.

Month Number of Pairs at Start

Number Produced

Number of Pairs at the End

1st 1 0 1

2nd 1 1 2

3rd 2 1 3

4th 3 2 5

5th 5 3 8

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Example (solution continued)

Solution: There will be 8 pairs of rabbits.

Step 4 Look back and check. This can be checked by going back and making sure that it has been interpreted correctly. Double-check the arithmetic.

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Example: Working Backward

Start with an unknown number. Triple it and then subtract 5. Now, take the new number and double it but then subtract 47. If you take this latest total and quadruple it you have 60. What was the original unknown number?

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Example: Solution

Step 1 Understand the problem. We are looking for a number that goes through a series of changes to turn into 60.

Step 2 Devise a plan. Work backwards to undo the changes.

Step 3 Carry out the plan. The final amount was 60. Divide by 4 to undo quadruple = 15. Add 47 to get 62, then divide by 2 = 31. Add 5 to get 36 and divide by 3 = 12.

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Example: Solution

SolutionThe original unknown number was 12.

Step 4 Look back and check. We can take 12 and run through the computations to get 60.

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Example: Using Trial and Error

The mathematician Augustus De Morgan lived in the nineteenth century. He made the following statement: “I was x years old in the year x

2.” In what year was he born?

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Example: Guessing and Checking

Find a positive natural number that satisfies the equation below.

2

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xx x

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Example: Considering a Simpler Problem

What is the ones (or units) digit in 3200?

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Example: Drawing a Sketch

An array of nine dots is arranged in a 3 x 3 square as shown below. Join the dots with exactly four straight lines segments. You are not allowed to pick up your pencil from the paper and may not trace over a segment that has already been drawn.

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Example: Solution

Through trial and error with different attempts such as

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Example: Using Common Sense

Two currently minted United States coins together have a total value of $0.30. One is not a quarter. What are the two coins?

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Section 1.3: Strategies for Problem Solving

Given a number, you subtract 6, divide theresult by 2, and then add 3 to get 15. What isthe original number?

a) 3

b) 24

c) 30

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Section 1.3: Strategies for Problem Solving

How many ways can you make change for fifty cents using only nickels and pennies?

a) 9

b) 10

c) 11

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Chapter 1

Section 1-4Calculating, Estimating, and

Reading Graphs

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Calculating, Estimating, and Reading Graphs

• Calculation

• Estimation

• Interpretation of Graphs

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Calculation

There are many types of calculators such as four-function, scientific, and graphing.

There are also many different models available and you may need to refer to your owner’s manual for assistance. Other resources for help are instructors and students that have experience with that model.

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Example: Calculation

Use your calculator to find the following:a) b)c)

2601

41.5

Solutiona) 3.14159265 (approximately)b) 51c) 5.0625

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Estimation

There are many times when we only need an estimate to a problem and a calculator is not necessary.

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Example: Estimation

A 20-ounce box of cereal sells for $3.12. Approximate the cost per ounce.

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Interpretation of Graphs

Using graphs is an efficient way to transmit information. Some of the common types of graphs are circle graphs (pie charts), bar graphs, and line graphs.

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Example: Circle Graph (Pie Chart)

A 15%

D 10%

C 40%

B 25%

F 10%

Use the circle graph below to determine how many of the 140 students made an A or a B.

Letter Grades in College Algebra

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Example: Bar Graph

0

2

4

6

8

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The bar graph shows the number of cups of coffee, in hundreds of cups, that a professor had in a given year.

2001 2002 2003 2004 2005

Cup

s

(in

hun

dred

s)

a) Estimate the number of cups in 2004b) What year shows the greatest decrease in cups?

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Example: Line Graph

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18

22

26

30

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The line graph shows the average class size of a first grade class at a grade school for years 2001 though 2005.

a) In which years did the average class size increase from the previous year?b) How much did the average size increase from 2001 to 2003?

Stu

dent

s pe

r cl

ass

’01 ’02 ’03 ’04 ’05

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Section 1.4: Calculating, Estimating, and Reading Graphs

Compute a) 2.54

b) 8.191532

c) 4.095766

3 16.387064.

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Section 1.4: Calculating, Estimating, and Reading Graphs

If you drive 1823 miles at an average speed of 62

miles per hour, estimate the time it would take tocomplete the trip.

a) 3 hours

b) 6 hours

c) 30 hours

d) 300 hours

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