Lesson 2.1 Use Inductive ReasoningLesson 2.1 Use Inductive Reasoning

Describe how to sketch the fourth figure in the pattern. Then sketch the fourth figure.

What is your reasoning behind the fourth figure?

Describe the pattern in the numbers -7, -21, -63, -189, . . . And write the next three numbers in the pattern.

Inductive ReasoningInductive ReasoningMaking a unproven statement about

something by observation is called a conjecture.

Conjectures are made using inductive reasoning. You recognize a pattern based on specific cases.

It may not be true for all cases.

Test a ConjectureTest a ConjectureNumbers such as 3, 4, and 5 are called

consecutive integers. Make and test a conjecture about the sum of any three consecutive integers.

Step 1: Try a few sums.

Conjecture:

Step 2: Test the conjecture with other numbers.

Recall, conjectures are based on multiple Recall, conjectures are based on multiple observations. Whenever we are able to observations. Whenever we are able to find an instance in which the conjecture is find an instance in which the conjecture is false, the entire conjecture is untrue. This false, the entire conjecture is untrue. This false example is referred to as a false example is referred to as a counterexamplecounterexample. .

UNEMPLOYMENT Based on the table showing unemployment rates for various cities in Kansas, find a counterexample for the following statement. The unemployment rate is highest in the cities with the most people.

CountyCounty Civilian Labor ForceCivilian Labor Force RateRate

ShawneeShawnee 90,25490,254 3.1%3.1%

JeffersonJefferson 9,9379,937 3.0%3.0%

JacksonJackson 8,915 8,915 2.8%2.8%

DouglasDouglas 55,73055,730 3.2%3.2%

OsageOsage 10,18210,182 4.0%4.0%

WabaunseeWabaunsee 3,5753,575 3.0%3.0%

PottawatomiePottawatomie 11,02511,025 2.1%2.1%

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Counterexample: A specific case Counterexample: A specific case for which a conjecture is false.for which a conjecture is false.

Example: The sum of two numbers is always greater than the larger number.

ExampleExample

More ExamplesMore Examples

Find a counterexample: The value of x² is always greater than the value of x.

Supplementary angels are always adjacent.

How do you use inductive reasoning in mathematics?