Thinking
Mathematically
Problem Solving and Critical
Thinking
1.1 Inductive and Deductive Reasoning
Inductive Reasoning
“Inductive reasoning” is the process of
arriving at a general conclusion based
on observations of specific examples.
Patterns
Conjecture / hypothesis
Counter-example
Inductive Reasoning
There is no guarantee that the conclusions
reached by “inductive reasoning” are correct
with no exceptions.
A strong inductive argument does not guarantee
the truth of the conclusion, but rather provides
strong support for the conclusion.
Inductive Reasoning
In mathematics “inductive reasoning” is often
used to find patterns.
Exercise Set 1.1 #15, #23, #27
Identify a pattern, then use the pattern to find
the next number:
1, 2, 4, 8, 16, __
11 1
1
, , , ,___
3 9 27
3, 6, 11, 18, 27, 38, __
Deductive Reasoning
“Deductive reasoning” is the process of
proving a specific conclusion from one or
more general statements. A conclusion that is
proved true by deductive reasoning is called
a theorem.
Applying the rules of logic (chapter 3)
Inductive/Deductive Examples
Inductive or Deductive?
• We examine the fingerprints of 1000 people. No
two individuals in this group of people have
identical fingerprints. We conclude that for all
people, no two people have identical fingerprints.
• All mammals are warm-blooded animals. No
snakes are warm-blooded. I have a pet snake. We
conclude that my pet snake cannot be a mammal.
Counterexample
If there is one case where for which a
conjecture/hypothesis does not work, the
conjecture is false. That one case is called a
counterexample.
Example: Exercise Set 1.1 #3
If a number is multiplied by itself, the result is even.
Disprove by counter-example.
Thinking
Mathematically
Problem Solving and Critical
Thinking
1.2 Estimation and Graphs
“Rounding” Numbers
To “round” to a particular place:
1. Look at the digit to the right of the digit
where rounding is to occur.
2. a. If the digit to the right is 5 or greater,
add 1 to the digit to be rounded. Replace
all digits to the right with zeros.
b. If the digit to the right is less than 5, do
not change the digit to be rounded. Replace
all digits to the right with zeros.
Estimation in Calculations
Exercise Set 1.2 #3, 11, 19, 21, 25
• Estimate e = 2.718281828459045 to the nearest
thousandth.
• Estimate 8.93 + 1.04 + 19.26
• Estimate 47.83 / 2.9
• Estimate 32% of 187,253
• A full-time employee who works 40 hours per
week earns $19.50 per hour. Estimate annual
income.
Thinking
Mathematically
Chapter 1: Problem Solving and
Critical Thinking