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

Find the next item in the sequence:

1. December, November, October, ...

2. 3, 6, 9, 12, ...

3. , , , ...

4. 1, 1, 2, 3, 5, 8, ...

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.

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 ______ .

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

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.

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.

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.

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

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

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

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

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

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).

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).

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.

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