Transcript Chapter 1 Lecture Notes

Chapter 1
The Art of
Problem
Solving
© 2007 Pearson Addison-Wesley.
All rights reserved
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
2
Chapter 1
Section 1-1
Solving Problems by Inductive
Reasoning
3
Solving Problems by Inductive or
Deductive Reasoning
• Characteristics of Inductive and Deductive
Reasoning
• Pitfalls of Inductive Reasoning
• Examples of Inductive and Deductive
Reasoning
4
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.
Prof Darini is a math teacher. Therefore, Prof
Darini must have a great sense of humor.
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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 = 111
37 × 6 = 222
37 × 9 = 333
37 × 12 = 444
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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
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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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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.
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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 1-2
An Application of Inductive
Reasoning: Number Patterns
13
An Application of Inductive
Reasoning: Number Patterns
•
•
•
•
Number Sequences
Successive Differences
Number Patterns and Sum Formulas
Figurate Numbers
14
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
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
8
32
10
2
44
12
2
58
14
2
Find differences
Find differences
Build up to next term: 58
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Number Patterns and Sum
Formulas
Sum of the First n Odd Counting Numbers
If n is any counting number, then
1 3  5 
 (2n  1)  n 2 .
Special Sum Formulas
For any counting number n,
(1  2  3 
 n) 2  13  23   n3
n(n  1)
and 1  2  3   n 
.
2
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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
For any natural number n,
n(n  1)
the nth triangular number is given by Tn 
,
2
the nth square number is given by Sn  n , and
2
n(3n  1)
the nth pentagonal number is given by Pn 
.
2
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Example: Figurate Numbers
Use a formula to find the sixth pentagonal
number
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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
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Section 1.2: An Application of Inductive
Reasoning: Number Patterns
When applying the sum formula
2
1  3  5   (2n  1)  n , to
1  3  5   51, what is the value of n?
a) 25
b) 26
c) 51
d) 52
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Chapter 1
Section 1-3
Strategies 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 Sense
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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
1st
1
Number
Produced
0
Number of
Pairs at the End
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
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.
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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
x
4 x  x
8
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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 the
result by 2, and then add 3 to get 15. What is
the 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 1-4
Calculating, Estimating, and
Reading Graphs
43
Calculating, Estimating, and Reading
Graphs
• Calculation
• Estimation
• Interpretation of Graphs
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Calculation
There are many types of calculators such as fourfunction, 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) 2601
4
c) 1.5
Solution
a) 3.14159265 (approximately)
b) 51
c) 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.
48
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)
Use the circle graph below to determine how
many of the 140 students made an A or a B.
Letter Grades in College Algebra
D
10%
F
10%
C
40%
A
15%
B
25%
50
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)
10
8
6
4
2
0
2001 2002 2003 2004 2005
a) Estimate the number of cups in 2004
b) What year shows the greatest decrease in cups?
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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
34
30
26
22
18
14
’01 ’02
’03
’04
’05
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?
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Section 1.4: Calculating, Estimating, and
Reading Graphs
Compute 3 16.387064.
a) 2.54
b) 8.191532
c) 4.095766
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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 to
complete the trip.
a) 3 hours
b) 6 hours
c) 30 hours
d) 300 hours
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