Transcript Geo 2.1 Using Inductive Reasoning to Make Conjectures

Using
Inductive
Reasoning
to
Using
Inductive
Reasoning
to
2-1
2-1 Make
Conjectures
Make
Conjectures
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Are you ready?
Complete each sentence.
1.
2.
?
?
points are points that lie on the same line.
points are points that lie in the same plane.
3. The sum of the measures of two
Holt Geometry
?
angles is 90°.
Using Inductive Reasoning to
2-1 Make Conjectures
Objectives
TSW use inductive reasoning to identify
patterns and make conjectures.
TSW find counterexamples to disprove
conjectures.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Biologists use inductive
reasoning to develop theories
about migration
patterns.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Vocabulary
inductive reasoning
conjecture
counterexample
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 1: Identifying a Pattern
Find the next item in the pattern.
January, March, May, ...
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 2: Identifying a Pattern
Find the next item in the pattern.
7, 14, 21, 28, …
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 3: Identifying a Pattern
Find the next item in the pattern.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 4
Find the next item in the pattern 0.4, 0.04, 0.004, …
Holt Geometry
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 Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 5: Making a Conjecture
Complete the conjecture.
The sum of two positive numbers is
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? .
Using Inductive Reasoning to
2-1 Make Conjectures
Example 6: Making a Conjecture
Complete the conjecture.
The number of lines formed by 4 points, no
three of which are collinear, is ? .
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 7
Complete the conjecture.
The product of two odd numbers is
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? .
Using Inductive Reasoning to
2-1 Make Conjectures
Example 8: Biology Application
The cloud of water leaving a whale’s blowhole
when it exhales is called its blow. A biologist
observed blue-whale blows of 25 ft, 29 ft, 27 ft,
and 24 ft. Another biologist recorded humpbackwhale blows of 8 ft, 7 ft, 8 ft, and 9 ft. Make a
conjecture based on the data.
Heights of Whale Blows
Height of Blue-whale Blows 25
Height of Humpback-whale
Blows
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8
29
27
24
7
8
9
Using Inductive Reasoning to
2-1 Make Conjectures
Example 9
Make a conjecture about the lengths of male and
female whales based on the data.
Average Whale Lengths
Length of Female (ft)
49
51
50
48
51
47
Length of Male (ft)
47
45
44
46
48
48
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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.
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Using Inductive Reasoning to
2-1 Make Conjectures
Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a
counterexample.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 10: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
For every integer n, n3 is positive.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 11: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
Two complementary angles are not congruent.
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Example 12: Finding a Counterexample
Show that the conjecture is false by finding a
counterexample.
The monthly high temperature in Abilene is
never below 90°F for two months in a row.
Monthly High Temperatures (ºF) in Abilene, Texas
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug Sep
88
89
97
99
107
109
110
107 106 103
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Oct Nov Dec
92
89
Using Inductive Reasoning to
2-1 Make Conjectures
Example 13
Show that the conjecture is false by finding a
counterexample.
For any real number x, x2 ≥ x.
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Using Inductive Reasoning to
2-1 Make Conjectures
Example 14
Show that the conjecture is false by finding a
counterexample.
Supplementary angles are adjacent.
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Using Inductive Reasoning to
2-1 Make Conjectures
Example 15
Show that the conjecture is false by finding a
counterexample.
The radius of every planet in the solar system is
less than 50,000 km.
Planets’ Diameters (km)
Mercury Venus Earth Mars
4880
12,100
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12,800
6790
Jupiter
Saturn
Uranus
Neptune
Pluto
143,000
121,000
51,100
49,500
2300
Using Inductive Reasoning to
2-1 Make Conjectures
Holt Geometry
Using Inductive Reasoning to
2-1 Make Conjectures
Lesson Quiz
Find the next item in each pattern.
1. 0.7, 0.07, 0.007, … 2.
0.0007
Determine if each conjecture is true. If false,
give a counterexample.
3. The quotient of two negative numbers is a positive
number. true
4. Every prime number is odd. false; 2
false; 90° and 90°
5. Two supplementary angles are not congruent.
6. The square of an odd integer is odd. true
Holt Geometry