Download 2-1 indcutive reasoning

Chapter 2
2-1 Using inductive
reasoning to make
conjectures
Objectives
Use inductive reasoning to identify
patterns and make conjectures.
Find counterexamples to disprove
conjectures.
Identifying Patterns
 Find
the next 2 items in the following
pattern.
 January, March, May, ...
 The next month is July. The next month is
september
 Alternating months of the year make up
the pattern
Identifying patterns
 Find
the next 2 items in the following
pattern.
 1,8,27,64,…………….
 1,1,2,3,5,8,…………………
Inductive reasoning
 When
several examples form a pattern
and you assume the pattern will continue,
you are applying inductive reasoning
What is inductive reasoning ?


Inductive reasoning is the process of reasoning
that a rule or statement is true because specific
cases are true. You may use inductive reasoning
to draw a conclusion from a pattern.
Inductive reasoning is the process of observing,
recognizing patterns and making conjectures
about the observed patterns. Inductive reasoning
is used commonly outside of the Geometry
classroom; for example, if you touch a hot pan
and burn yourself, you realize that touching
another hot pan would produce a similar
(undesired) effect.
What is conjecture?
A
statement you believe to be true based
on inductive reasoning is called a
conjecture.
Making conjectures Ex#1
 Complete
 The



the conjecture.
sum of two positive numbers is ? .
List some examples and look for a pattern.
1+1=2
3.14 + 0.01 = 3.15
3,900 + 1,000,017 = 1,003,917
 The
sum of two positive numbers is
positive
Example #2
 Complete
the conjecture.
 The number of lines formed by 4 points, no
three of which are collinear, is ? .
 Draw four points. Make sure no three
points are collinear. Count the number of
lines formed:
 The number of lines formed by four points,
no three of which are collinear, is 6.
Example #3
 The
sum of two odd numbers is __________
Example #4
 Make
a conjecture about the lengths of
male and female whales based on the
data.
 Average length female
49
Average length male
47
51
45
50
44
48
46
51
48
47
48
Example #4 continue
 In
5 of the 6 pairs of numbers above the
female is longer.
 Conjecture:
 Female whales are longer than male whales.
Counter Example
 To
show that a conjecture is always true,
you must prove it.
 To show that a conjecture is false, you
have to find only one example in which
the conjecture is not true. This case is
called a counterexample.
 A counterexample can be a drawing, a
statement, or a number.
Counter example ex.#1
 Show
that the conjecture is false by
finding a counterexample.
 For every integer n, n3 is positive.
 Pick integers and substitute them into the
expression to see if the conjecture holds.
Counterexample ex.#2
 Show
that the conjecture is false by
finding a counterexample.
 Two complementary angles are not
congruent.
 If the two congruent angles both measure
45°, the conjecture is false.
Counterexample ex.#3
 Show
that the conjecture is false by
finding a counterexample.
 For any real number x, x2 ≥ x.
Counterexample ex.#4
 Sow
that each conjecture is false by
finding a counterexample
 For all positive numbers n,1/n≤n
How do inductive reasoning
works
 Inductive
Reasoning
 1. Look for a pattern.
 2. Make a conjecture.
 3. Prove the conjecture or find a
counterexample.
Student guided practice
 Lets
77
do problems 2-10 on the book page
Homework !!!
 Do
problems 11-23 in the book page 77
Closure
 Today
we saw about inductive reasoning
and how to make conjectures and
counterexamples.
 Next class we are going to continue with
conditional statements.