Transcript Inductive reasoning

Using Inductive Reasoning to
2-1 Make Conjectures
Objectives
Use inductive reasoning to identify
patterns and make conjectures.
Find counterexamples to disprove
conjectures.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Vocabulary
inductive reasoning
conjecture
counterexample
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Warm Up
Find the next item in the pattern.
1. January, March, May, ...
Alternating months of the year make up the pattern.
The next month is July.
2. 7, 14, 21, 28, …
Multiples of 7 make up the pattern.
The next multiple is 35.
3.
In this pattern, the figure
rotates 90° counter-clockwise
each time.
The next figure is
Holt McDougal Geometry
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Using Inductive Reasoning to
2-1 Make Conjectures
When several examples form a pattern and you
assume the pattern will continue, you are
applying 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.
A statement you believe to be true based on
inductive reasoning is called a conjecture.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1A: Making a Conjecture
Complete the conjecture.
The 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.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1B: Making a Conjecture
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:
AB
AC AD BC BD CD
The number of lines formed by four points, no
three of which are collinear, is 6.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 1
Complete the conjecture.
The product of two odd numbers is
? .
List some examples and look for a pattern.
11=1
33=9
5  7 = 35
The product of two odd numbers is odd.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
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.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Steps of Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 2A: Finding a Counterexample
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.
Let n = 1. Since n3 = 1 and 1 > 0, the conjecture holds.
Let n = –3. Since n3 = –27 and –27  0, the
conjecture is false.
n = –3 is a counterexample.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 2B: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
Two complementary angles are not congruent.
45° + 45° = 90°
If the two congruent angles both measure 45°, the
conjecture is false.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Check It Out! Example 2
Show that the conjecture is false by finding a
counterexample.
For any real number x, x2 ≥ x.
1
Let x = 2 .
1
Since 2
2
1 1
1
= 4, 4 ≥ 2 .
The conjecture is false.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Directions
Test your skills at proving conjectures TRUE or FALSE with a
counterexample.
With your partner - Go around the room and read the
various conjectures:
1. If you think its FALSE, write an counterexample to prove it
false on the paper below and put names by counterexample.
2. If you think its TRUE, write TRUE on paper below and put
names by it.
One point will be awarded to each unique counterexample
or correct TRUE statement.
One point will be deducted for each incorrect statement.
Holt McDougal Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Determine if each conjecture is true. If false,
give a counterexample.
A. The quotient of two negative numbers is a positive
number. true
B. Every prime number is odd. false; 2
false; 90° and 90°
C. Two supplementary angles are not congruent.
D. The square of an odd integer is odd. true
E. Three points on a plane always form a triangle
F. If y-1>0, then 0<y<1
Holt McDougal Geometry