1. Chapter 1 The Art of Problem Solving © 2007 Pearson Addison-Wesley.
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2. 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

3. Solving Problems by Inductive Reasoning
Chapter 1
Section 1-1
Solving Problems by Inductive Reasoning

4. Solving Problems by Inductive or Deductive Reasoning
Characteristics of Inductive and Deductive Reasoning
Pitfalls of Inductive Reasoning
Examples of Inductive and Deductive Reasoning

5. 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.

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

7. Example: predict the product of two numbers
Use the list of equations and inductive reasoning to predict the next multiplication fact in the list:
37 × 3 = × 6 = 222
37 × 9 = × 12 = 444

8. 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

9. 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.

10. Example: 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.

11. 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 11

12. 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 12

13. An Application of Inductive Reasoning: Number Patterns
Chapter 1
Section 1-2
An Application of Inductive Reasoning: Number Patterns 13 13

14. An Application of Inductive Reasoning: Number Patterns
Number Sequences
Successive Differences
Number Patterns and Sum Formulas
Figurate Numbers 14

15. Number Sequences Number Sequence
A list of numbers having a first number, a second number, and so on, called the terms of the sequence.
Arithmetic Sequence
A sequence that has a common difference between successive terms.
Geometric Sequence
A sequence that has a common ratio between successive terms. 15

16. Successive Differences
Process to determine the next term of a sequence using subtraction to find a common difference. 16

17. Example: Successive Differences
Use the method of successive differences to find the next number in the sequence.
14, 22, 32, 44,...
Find differences
Find differences 58 14 2
Build up to next term: 58 17

18. Number Patterns and Sum Formulas
Sum of the First n Odd Counting Numbers
If n is any counting number, then
Special Sum Formulas
For any counting number n, 18

19. Example: Sum Formula
Use a sum formula to find the sum 19 19

20. Figurate Numbers 20

21. Formulas for Triangular, Square, and Pentagonal Numbers
For any natural number n, 21

22. Example: Figurate Numbers
Use a formula to find the sixth pentagonal number 22 22

23. Section 1.2: An Application of Inductive Reasoning: Number Patterns
Find the probable next number in the sequence
1, 5, 13, 25, 41,…
a) 51
b) 58
c) 61 23

24. 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 24

25. Strategies for Problem Solving
Chapter 1
Section 1-3
Strategies for Problem Solving 25

26. 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 Sense 26

27. 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. 27

28. 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? 28

29. 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 29

30. 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
2nd 2
3rd 3
4th 5
5th 8 30

31. 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. 31

32. 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? 32

33. 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 Divide by 4 to undo quadruple = 15.
Add 47 to get 62, then divide by 2 = Add 5 to get 36 and divide by 3 = 12. 33

34. Example: Solution Solution The original unknown number was 12.
Step 4 Look back and check. We can take 12 and run through the computations to get 60. 34

35. 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? 35

36. Example: Guessing and Checking
Find a positive natural number that satisfies the equation below. 36

37. Example: Considering a Simpler Problem
What is the ones (or units) digit in 3200? 37

38. 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. 38

39. Example: Solution
Through trial and error with different attempts such as 39

40. Example: Using Common Sense
Two currently minted United States coins together have a total value of $ One is not a quarter. What are the two coins? 40

41. Section 1.3: Strategies for Problem Solving
Given a number, you subtract 6, divide the
result by 2, and then add 3 to get 15. What is
the original number?
a) 3
b) 24
c) 30 41

42. 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 42

43. Calculating, Estimating, and Reading Graphs
Chapter 1
Section 1-4
Calculating, Estimating, and Reading Graphs 43 43

44. Calculating, Estimating, and Reading Graphs
Calculation
Estimation
Interpretation of Graphs 44

45. 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. 45

46. Example: Calculation Solution
Use your calculator to find the following: a) b) c)
Solution
a) (approximately)
b) 51 c) 46

47. Estimation
There are many times when we only need an estimate to a problem and a calculator is not necessary. 47

48. Example: Estimation
A 20-ounce box of cereal sells for $ Approximate the cost per ounce. 48

49. 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. 49

50. Example: Circle Graph (Pie Chart)
Use the circle graph below to determine how many of the 140 students made an A or a B.
Letter Grades in College Algebra F
10% D
10% A
15% B
25% C
40% 50

51. Example: Bar Graph
The bar graph shows the number of cups of coffee, in hundreds of cups, that a professor had in a given year.
Cups (in hundreds)
a) Estimate the number of cups in 2004
What year shows the greatest decrease in cups? 51

52. Example: Line Graph
The line graph shows the average class size of a first grade class at a grade school for years 2001 though 2005.
Students per class
’01 ’ ’ ’ ’05
In which years did the average class size increase
from the previous year?
How much did the average size increase from 2001
to 2003? 52

53. Section 1.4: Calculating, Estimating, and Reading Graphs
Compute
a) 2.54 b) c) 53

54. 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 to
complete the trip.
a) 3 hours
b) 6 hours
c) 30 hours
d) 300 hours 54