Chapter 1 Lesson 2

Inductive Reasoning

Inductive Reasoning

Reasoning based on patterns that you observe

Finding the next term in a sequence is a form of inductive reasoning

Conjecture

A conclusion that you reach based on inductive reasoning

Example of a conjectureMake a chart with the following information:

Count the number of ways 2 people shake hands

Count the number of ways 3 people shake hands

Count the number of ways 4 people shake hands

Count the number of ways 5 people shake hands

Make a conjecture about the number of ways 6 people shake hands

People

Handshakes

2 3 4 5 6

1 3 6 10 ?

Make a conjecture

Finish the statement: The sum of any two odd numbers is ____________.

Begin by writing several examples:

What do you notice about each sum? Answer: The sum of any 2 odd

numbers is:

1+1=2 1+3=4 3+5 = 8

5+7=14 7+9= 16 11+13 = 24

even

Make a conjecture

Complete the conjecture: The sum of the first 30 odd numbers is ____________________.

1 = 1 1+3 = 4 1+3+5 = 9 1+3+5+7 = 16 1+3+5+7+9 = 25 1+3+5+7+9+11 = 36

What do you notice about the pattern?

Conjecture: The sum of the first 30 odd numbers is 302.

Counterexamples

Just because a statement is true for several examples does not mean that it is true for all cases

If a conjecture is not always true, then it is considered false

To prove that a conjecture is false, you need ONE counterexample

Counterexample: an example that shows a conjecture is false.

Examples:

You can connect any three points to form a triangle.

Counterexample: three points on the same line

Any number and its absolute value are opposites

55

Show that the conjecture is false by providing a counterexample:

1. If the product of two numbers is even, then the numbers must be even.

2. If it is Monday, then there is school.

Assignment:

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