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Inductive and Deductive Reasoning
Objectives:
The student is able to (I can):
Use inductive reasoning to identify patterns and make
conjectures
Find counterexamples to disprove conjectures
Understand the differences between inductive and deductive
reasoning
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Find the next item in the sequence:
1. December, November, October, ...
2. 3, 6, 9, 12, ...
3. , , , ...
4. 1, 1, 2, 3, 5, 8, ...
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Find the next item in the sequence:
1. December, November, October, ...
SeptemberSeptemberSeptemberSeptember
2. 3, 6, 9, 12, ...
15151515
3. , , , ...
4. 1, 1, 2, 3, 5, 8, ...
13 13 13 13 This is called the Fibonacci This is called the
Fibonacci This is called the Fibonacci This is called the Fibonacci
sequence.sequence.sequence.sequence.
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inductive reasoning
conjecture
Reasoning that a rule or statement is true because specific
cases are true.
A statement believed true based on inductive reasoning.
Complete the conjecture:
The product of an odd and an even number is ______ .
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inductive reasoning
conjecture
Reasoning that a rule or statement is true because specific
cases are true.
A statement believed true based on inductive reasoning.
Complete the conjecture:
The product of an odd and an even number is ______ .
To do this, we consider some examples:
(2)(3) = 6 (4)(7) = 28 (2)(5) = 10
eveneveneveneven
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counterexample
If a conjecture is true, it must be true for every case. Just
one exampleJust one exampleJust one exampleJust one example for
which the conjecture is false will disprove it.
A case that proves a conjecture false.
To be a counterexample, the first part must be truetruetruetrue,
and the second part must be falsefalsefalsefalse.
Example: Find a counterexample to the conjecture that all
students who take Geometry are 10th graders.
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counterexample
If a conjecture is true, it must be true for every case. Just
one exampleJust one exampleJust one exampleJust one example for
which the conjecture is false will disprove it.
A case that proves a conjecture false.
To be a counterexample, the first part must be truetruetruetrue,
and the second part must be falsefalsefalsefalse.
Example: Find a counterexample to the conjecture that all
students who take Geometry are 10th graders.
Most of the students in our class are taking Geometry, but are
not 10th graders.
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Examples
To Use Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a counterexample to disprove
it.
Show that each conjecture is false by giving a
counterexample.
1. The product of any two numbers is greater than the numbers
themselves.
2. Two complementary angles are not congruent.
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Examples
To Use Inductive Reasoning
1. Look for a pattern.
2. Make a conjecture.
3. Prove the conjecture or find a counterexample to disprove
it.
Show that each conjecture is false by giving a
counterexample.
1. The product of any two numbers is greater than the numbers
themselves.
((((----1)(5) = 1)(5) = 1)(5) = 1)(5) = ----5555
2. Two complementary angles are not congruent.
45 and 4545 and 4545 and 4545 and 45
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Sometimes we can use inductive reasoning to solve a problem that
does not appear to have a pattern.
Example: Find the sum of the first 20 odd numbers.
Sum of first 20 odd numbers?
1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1
4
9
16
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Sometimes we can use inductive reasoning to solve a problem that
does not appear to have a pattern.
Example: Find the sum of the first 20 odd numbers.
Sum of first 20 odd numbers?
1
1 + 3
1 + 3 + 5
1 + 3 + 5 + 7
1
4
9
16
12
22
32
42
202 = 400
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These patterns can be expanded to find the nth term using
algebra. When you complete these sequences by applying a rule, it
is called a functionfunctionfunctionfunction.
Examples: Find the missing terms and the rule.
To find the pattern when the difference between each term is the
same, the coefficient of n is the difference between each term, and
the value at 0 is what is added or subtracted.
1 2 3 4 5 8 20 n
-3 -2 -1 0 1
1 2 3 4 5 8 20 n
32 39 46 53 60
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These patterns can be expanded to find the nth term using
algebra. When you complete these sequences by applying a rule, it
is called a functionfunctionfunctionfunction.
Examples: Find the missing terms and the rule.
To find the pattern when the difference between each term is the
same, the coefficient of n is the difference between each term, and
the value at 0 is what is added or subtracted.
1 2 3 4 5 8 20 n
-3 -2 -1 0 1 4 16 n 4
1 2 3 4 5 8 20 n
32 39 46 53 60 81 165 7n+25
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Geometry is based on a deductive structurea system of thought in
which conclusions are justified by means of previously assumed or
proved statements. Every deductive structure contains the following
four elements:
Undefined terms (points, lines, planes)
Assumptions known as postulates
Definitions
Theorems and other conclusions
A deductive system is very much like a gameto play, you have to
learn the terms being used (definitions) and the rules
(postulates).
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deductive reasoning
The process of using logic to draw conclusions from given facts,
definitions, and properties.
Inductive reasoning uses specific cases and observations to form
conclusions about general ones (circumstantial evidence).
Deductive reasoning uses facts about general cases to form
conclusions about specific cases (direct evidence).
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Example Decide whether each conclusion uses inductive or
deductive reasoning.
1. Police arrest a person for robbery when they find him in
possession of stolen merchandise.
2. Gunpowder residue tests show that a suspect had fired a gun
recently.
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Example Decide whether each conclusion uses inductive or
deductive reasoning.
1. Police arrest a person for robbery when they find him in
possession of stolen merchandise.
Inductive reasoningInductive reasoningInductive
reasoningInductive reasoning
2. Gunpowder residue tests show that a suspect had fired a gun
recently.
Deductive reasoningDeductive reasoningDeductive
reasoningDeductive reasoning