2.1 Use Inductive Reasoning
Objectives
1. To form conjectures through inductive
reasoning
2. To disprove a conjecture with a
counterexample
3. To avoid fallacies of inductive reasoning
Activity: Murder Mystery
Once upon a time long, long ago in a far, far
away land known as Geometrica there
occurred an unspeakable crime. On a dark
and dreary night as the Circular family lay
sleeping in their soft, round beds and
dreaming of their favorite dessert, pi, a violent
criminal murdered them. Their neighbor, Mrs.
Equi Angular said that she and her husband,
Mr. Tri Angular, heard the awful blood
curdling screams.
Activity: Murder Mystery
So, they sprang from their bed to see what
was the matter, and what to their
wandering eyes did appear (not eight tiny
reindeer) but a strange four-sided figure
leaping from the Circular’s upstairs
window. Well, the Angulars gave a
description of the terrible beast and so did
many other Geometrica residents.
However, to this day, the mystery remains.
Activity: Murder Mystery
Therefore, Detective Pentagonal Walsh of
Geometrica’s Most Wanted has asked for
your assistance in solving this crime.
Below you will find descriptions that
tipsters have given the authorities. Your
job is to list the suspects from your line-up
of twelve that meet each set of criteria.
Activity: Murder Mystery
In this activity, you used your observational
skills to draw a conclusion about who each
tip was. That’s called inductive
reasoning. You then used your conjecture
to help you solve the who committed the
horrible, disastrous murder.
Example 1
You’re at school eating lunch. You ingest some air
while eating, which causes you to belch.
Afterward, you notice a number of students
staring at you with disgust. You burp again, and
looks of distaste greet your natural bodily
function. You have similar experiences over the
course of the next couple of days. Finally, you
conclude that belching in public is social
unacceptable. The process that lead you to this
conclusion is called inductive reasoning.
Inductive Reasoning
Inductive reasoning
is the process of
observing data,
recognizing
patterns, and
making
generalizations
based on your
observations.
Generalization
Generalization:
statement that
applies to every
member of a group
• Science =
hypothesis
• Math = conjecture
Conjecture
A conjecture is a
general, unproven
statement believed
to be true based on
investigation or
observation
Inductive Reasoning
Inductive reasoning can
be used to make
predictions about the
future based on the
past or to make
conjectures about the
past based on the
present.
Example 2
A scientist takes a piece of salt, turns it over
a Bunsen burner, and observes that it
burns with a yellow flame. She does this
with many other pieces of salt, finding they
all burn with a yellow flame. She therefore
makes the conjecture: “All salt burns with
a yellow flame.”
Inductive Reasoning
Inductive Reasoning
Example 3
Find the 12th number in the following
sequence:
1, 1, 2, 3, 5, 8, …
This is called the Fibonacci Sequence!
Example 4
Numbers such as 3, 4, and 5 are
consecutive numbers. Make and test a
conjecture about the sum of any three
consecutive numbers.
Example 5
Use the map of
Texas provided
to formulate a
conjecture about
the numbering
system of
Interstate
Highways.
Example 5
Use the map of
the US provided
to formulate a
conjecture about
the numbering
system of
Interstate
Highways.
Example 6
(An allegory) Student A neglected to do his/her
homework on numerous occasions. When Student
A's mean teacher popped a quiz on the class,
Student A failed. After the quiz, Student A had
several other HW assignments that he/she also
neglected to complete. When test time rolled
around, Student A failed the exam . Students B-F
behaved in a similar, academically deplorable
manner. Use inductive reasoning to make a
conjecture about the relationship between homework
and test/quiz performance.
Example 7
Inductive reasoning does not
always lead to the truth.
What are some famous
examples of conjectures
that were later discovered to
be false?
To Prove or To Disprove
In science, experiments are used to prove or
disprove an hypothesis.
In math, deductive reasoning is used to
prove conjectures and counterexamples
are used to disprove them.
Counterexample
A counterexample is a single case in which
a conjecture is not true.
Example 8
On her first road trip, Little Window Watcher Wilma
observes a number of vehicles. Each one she
observes has four wheels. She conjectures “All
vehicles have four wheels.” What is wrong with
her conjecture? What counterexample will
disprove it?
Conjecture: All
vehicles have 4
wheels
Example 9
Prove or disprove the following conjecture:
For every integer x, x2 + x + 41 is prime.
For more information on prime numbers, visit
http://www.utm.edu/research/primes/.
Example 10
Kenny makes the following conjecture about
the sum of two numbers. Find a
counterexample to disprove Kenny’s
conjecture.
Conjecture: The sum of two numbers is
always greater than the larger number.
Example 11
Joe has a friend who just happens to be a
Native American named Victor. One day
Victor gave Joe a CD. The next day Victor
decided that he wanted the CD back, and
so he confronted Joe. After reluctantly
giving the CD back to his friend, Joe made
the conjecture: “Victor, like all Native
Americans, is an Indian Giver.” What is
wrong with his conjecture? What does this
example illustrate?
Inductive Fallacies
The previous example illustrated an inductive
fallacy, where a reliable conjecture cannot be
justifiably made. Joe was guilty of a Hasty
Generalization, basing a conclusion on too little
information. Here are some others:
• Unrepresentative Sample
• False Analogy
• Slothful Induction
• Fallacy of Exclusion
Inductive Fallacies
As a group, match each inductive fallacy
definition with the corresponding example.
Be sure to take some notes, as this
priceless information is not in your
textbook.
Assignment
• Inductive Fallacy
Examples
• Challenge
Problems