Download Solving Problems by Inductive Reasoning

Chapter 1
The Art of
Problem Solving
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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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Section 1-1
Solving Problems by Inductive
Reasoning
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Solving Problems by Inductive
Reasoning
•  Characteristics of Inductive and Deductive
Reasoning
•  Pitfalls of Inductive Reasoning
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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.
Patrick is a math teacher. Therefore, Patrick must
have a great sense of humor.
Solution
Because the reasoning goes from general to specific,
deductive reasoning was used.
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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
Solution
37 × 15 = 555
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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
Solution
Each number in the list is obtained by adding 7 to
the previous number.
The probable next number is 30 + 7 = 37.
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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: pitfalls of inductive reasoning
We concluded that the probable next number in the
list 2, 9, 16, 23, 30 is 37.
If the list 2, 9, 16, 23, 30 actually represents the
dates of Mondays in June, then the date of the
Monday after June 30 is July 7 (see the figure on the
next slide). The next number on the list would then
be 7, not 37.
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Example: pitfalls of inductive reasoning
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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.
Solution
c 2 = 32 + 4 2
c 2 = 9 + 16 = 25
c=5
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