Download Inductive Reasoning

Section 1-1
Solving Problems by Inductive
Reasoning
Slide 1-1-1
SOLVING PROBLEMS BY INDUCTIVE
REASONING
• Characteristics of Inductive and Deductive
Reasoning
• Pitfalls of Inductive Reasoning
Slide 1-1-2
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.
Slide 1-1-3
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.
Slide 1-1-4
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
Slide 1-1-5
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.
Slide 1-1-6
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.
Slide 1-1-7
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.
Slide 1-1-8
EXAMPLE: PITFALLS OF INDUCTIVE
REASONING
Slide 1-1-9
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
Slide 1-1-10