Patterns and
Inductive Reasoning
Geometry
Objectives/Assignment:
• Find and describe patterns.
• Use inductive reasoning to make real-life
conjectures.
Finding & Describing Patterns
Geometry, like much of mathematics and
science, developed when people began
recognizing and describing patterns. In
this course, you will study many amazing
patterns that were discovered by people
throughout history and all around the
world. You will also learn how to recognize
and describe patterns of your own.
Sometimes, patterns allow you to make
accurate predictions. This is called
inductive reasoning.
Ex. 1: Describing a Visual
Pattern
• Describe the next figure in the pattern.
1
2
3
4
5
Ex. 1: Describing a Visual
Pattern - Solution
• The sixth figure in the pattern has 6 squares in
the bottom row.
5
6
Ex. 1: Describing a Number
Pattern
Describe a pattern in the sequence of
numbers. Predict the next number.
a. 1, 4, 16, 64
Many times in a number pattern, it is easiest
listing the numbers vertically rather than
horizontally.
•
Ex. 1: Describing a Number
Pattern
 Describe a
pattern in the
sequence of
numbers.
Predict the
next number.
a. 1
4
16
64
How do you get to the
next number?
That’s right. Each
number is 4 times the
previous number. So,
the next number is
256, right!!!
Ex. 1: Describing a Number
Pattern
 Describe a
pattern in the
sequence of
numbers.
Predict the
next number.
b. -5
-2
4
13
How do you get to the
next number?
That’s right. You add
3 to get to the next
number, then 6, then
9. To find the fifth
number, you add
another multiple of 3
which is +12 or
25, That’s right!!!
Ex. 1: Describing a Number
Pattern
 Describe a pattern in the sequence
of numbers. Predict the next
number.
 5,10,20,40…
80
Ex. 1: Describing a Number
Pattern
 Describe a pattern in the sequence
of numbers. Predict the next
number.
 1, -1, 2, -2, 3…
-3
Ex. 1: Describing a Number
Pattern
 Describe a pattern in the sequence
of numbers. Predict the next
number.
 15,12,9,6,…
3
Ex. 1: Describing a Number
Pattern
 Describe a pattern in the sequence
of numbers. Predict the next
number.
 1,2,6,24,120…
720
Ex. 1: Describing a Number
Pattern
 Describe a pattern in the sequence.
Predict the next letter.
 O,T,T,F,F,S,S,E,…

1,2,3,4,5,6,7,8,
N

9
Goal 2: Using Inductive
Reasoning
Much of the reasoning you need in geometry consists of
3 stages:
Look for a Pattern: Look at several examples. Use
diagrams and tables to help discover a pattern.
•
1.
2.
Make a Conjecture. Use the example to make a
general conjecture. Okay, what is that?
•
3.
A conjecture is an unproven statement that is based on
observations. Discuss the conjecture with others. Modify the
conjecture, if necessary.
Verify the conjecture. Use logical reasoning to verify
the conjecture is true IN ALL CASES. (You will do this in
Chapter 2 and throughout the book).
Ex. 2: Making a Conjecture
• Complete the conjecture.
Conjecture: The sum of the first n odd positive
integers is ________?_________.
How to proceed:
List some specific examples and look for a
pattern.
Ex. 2: Making a Conjecture
First odd
positive integer:
=12
1 = 1 =22
1 + 3 = 4 =32
1 + 3 + 5 = 9 =42
1 + 3 + 5 + 7 = 16
The sum of the first n odd positive integers is
n2.
Ex. 2: Making a Conjecture
First even positive integer:
2 = 2 = 1x2
2 + 4 = 6 = 2x3
2 + 4 + 6= 12 = 3x4
= 4x5
2 + 4 + 6 + 8 = 20
= 5x6
2 + 4 + 6 + 8 + 10 = 30
The sum of the first 6 even positive integers is
what?

6x7 = 42
Note:
• To prove that a conjecture is true, you need to
prove it is true in all cases. To prove that a
conjecture is false, you need to provide a
single counter example. A counterexample is
an example that shows a conjecture is false.
Ex. 3: Finding a
counterexample
• Show the conjecture is false by finding a
counterexample.
Conjecture: For all real numbers x, the
expressions x2 is greater than or equal to x.
Ex. 3: Finding a
counterexample- Solution
Conjecture: For all real numbers x, the
expressions x2 is greater than or equal
to x.
The conjecture is false. Here is a
counterexample: (0.5)2 = 0.25, and 0.25
is NOT greater than or equal to 0.5. In
fact, any number between 0 and 1 is a
counterexample.
Ex. 3: Finding a counterexample
Conjecture: The sum of two numbers is
greater than either number.

The conjecture is false. Here is a
counterexample: -2 + -5 = -7

-7 is NOT greater than -2 or -5. In
fact, the sum of any two negative
numbers is a counterexample.
Ex. 3: Finding a counterexample
Conjecture: The difference of two
integers is less than either integer.

The conjecture is false. Here is a
counterexample: -2 - -5 = 3

3 is NOT less than -2 or -5. In fact,
the difference of any two negative
numbers is a counterexample.
Ex. 4: Using Inductive
Reasoning in Real-Life
• Moon cycles. A full moon occurs when the
moon is on the opposite side of Earth from the
sun. During a full moon, the moon appears as
a complete circle.
Ex. 4: Using Inductive
Reasoning in Real-Life
• Use inductive reasoning and the information
below to make a conjecture about how often a
full moon occurs.
• Specific cases: In 2005, the first six full moons
occur on January 25, February 24, March 25,
April 24, May 23 and June 22.
Ex. 4: Using Inductive
Reasoning in Real-Life - Solution
• A full moon occurs every 29 or 30 days.
• This conjecture is true. The moon
revolves around the Earth approximately
every 29.5 days.
Ex. 4: Using Inductive
Reasoning in Real-Life - NOTE
• Inductive reasoning is very important to the
study of mathematics. You look for a pattern
in specific cases and then you write a
conjecture that you think describes the
general case. Remember, though, that just
because something is true for several specific
cases does not prove that it is true in general.
Ex. 4: Using Inductive
Reasoning in Real-Life
• Use inductive reasoning and the information
below to make a conjecture about the
temperature.
• The speed with which a cricket chirps is
affected by the temperature. If a cricket
chirps 20 times in 14 seconds, what is the
temperature?

5 chirps
45 degrees
10 chirps
55 degrees
15 chirps
65 degrees
The temperature would be 75 degrees.
Find a pattern for each sequence.
Use the pattern to show the next
two terms or figures.
Use the table and inductive reasoning.
1. 3, –6, 18, –72, 360
–2160; 15,120
2.
3. Find the sum of the first 10 counting numbers.
55
4. Find the sum of the first 1000
counting numbers.
500,500
Show that the conjecture is false by finding one
counterexample.
5. The sum of two prime numbers is an
even number.
Sample: 2 + 3 = 5, and 5 is not even
-1
Conditional Statements
If…then…
Conditional Statements
*A conditional statement is also known as an ifthen statement.
If you are not completely satisfied then your money will be refunded.
Every conditional has two parts, the part following
the if, called the hypothesis, and the part
following then, which is the conclusion.
If Texas won the 2006 Rose Bowl game, then Texas was college
football’s 2005 national champion.
Hypothesis – Texas won the 2006 Rose Bowl game.
Conclusion – Texas was college football’s 2005 national champion.
Conditional Statements
This also works with mathematical statements.
If T – 38 = 3, then T = 41
Hypothesis: T – 38 = 3
Conclusion: T = 41
Conditional Statements
Sometimes the “then” is not written, but it is still
understood:
If someone throws a brick at me, I can catch it and throw it back.
Hypothesis: Someone throws a brick at me
Conclusion: I can catch it and throw it back.
Name the hypothesis:
• If you want to be fit, then get plenty of exercise.
A.
B.
C.
D.
If
You want to be fit
Then
Get plenty of exercise
• If you can see the magic in a fairy tale, you can
face the future.
A.
B.
C.
D.
If
You can see the magic in a fairy tale
Then
You can face the future
Name the conclusion:
• “If you can accept a deal and open your pay
envelope without feeling guilty, you’re stealing.” –
George Allen, NFL Coach
A. If
B. You can accept a deal and open your pay envelope
without feeling guilty
C. Then
D. You’re stealing
Name the conclusion:
• “If my fans think I can do everything I say I can do,
then they are crazier than I am.” – Muhammed Ali
A.
B.
C.
D.
If
My fans think I can do everything I say I can
Then
They are crazier than I am
Writing a Conditional
You can write many sentences as conditionals.
A rectangle has four sides.
If a figure is a rectangle, then it has four sides.
A tiger is an animal.
If something is a tiger, then it is an animal.
An integer that ends with 0 is divisible by 5.
A square has four congruent sides.
Truth Value
• A conditional has a truth value of either true
or false. To show a conditional is true, show
that every time the hypothesis is true, then
the conclusion is true. If you can find one
counterexample where the hypothesis is true,
but the conclusion is false, then the
conditional is considered false.
Counterexamples
 To show that a conditional is false, find one counterexample.
If it is February, then there are only 28 days in the month.
This is typically true, however, because during leap years February
has 29 days, this makes the conditional false.
If a state begins with New, then the state borders an ocean.
This is true of New York, and New Hampshire, but New Mexico has
no borders on the water, therefore this is a false conditional.
Using a Venn Diagram
If you live in Chicago, then you live in Illinois.
Residents
of Chicago
Residents of Illinois
Converse
The converse of a conditional switches the
hypothesis and the conclusion.
Conditional: If two lines intersect to form right angles, then
they are perpendicular.
Converse: If two lines are perpendicular, then they intersect to
form right angles
If two lines are not parallel and do not intersect, then they are
skew.
Finding the Truth of a Converse
It is possible for a conditional and its converse to have
two different truth values.
If an animal is a chicken, then it is covered in feathers.
If an animal is covered in feathers, then it is a chicken.
Although the first conditional is true, as all chickens
have feathers, the second is false, as chickens are not
the only animal with feathers.
Finding the Truth of a Converse
Finding the Truth of a Converse