2.1 Inductive Reasoning and 2.1 Inductive Reasoning and ConjectureConjecture

ObjectivesObjectives

Make conjectures based on inductive Make conjectures based on inductive reasoningreasoning

Find counterexamplesFind counterexamples

Making ConjecturesMaking Conjectures

ConjectureConjecture – an educated guess – an educated guess based on known informationbased on known information

Inductive ReasoningInductive Reasoning – using a – using a number of specific examples to arrive number of specific examples to arrive at a plausible generalization or at a plausible generalization or predictionprediction

We use inductive reasoning to create We use inductive reasoning to create a conjecture. a conjecture.

Make a conjecture about the next number based on the pattern.

2, 4, 12, 48, 240

Answer: 1440

Find a pattern:

2 4 12 48 240

×2

The numbers are multiplied by 2, 3, 4, and 5.

Conjecture: The next number will be multiplied by 6. So, it will be or 1440.

×3 ×4 ×5

Example 1:Example 1:

Make a conjecture about the next number based on the pattern.

Answer: The next number will be

Your Turn:Your Turn:

Given: points L, M, and N; Examine the measures of the segments. Since the points can be collinear with point N between points L and M.

Answer:

Conjecture: L, M, and N are collinear.

For points L, M, and N, and , make a conjecture and draw a figure to illustrate your conjecture.

Example 2:Example 2:

ACE is a right triangle with Make a conjecture and draw a figure to illustrate your conjecture.

Answer:

Conjecture: In ACE, C is a right angle and is the hypotenuse.

Your Turn:Your Turn:

CounterexamplesCounterexamples

Recall, conjectures are based on Recall, conjectures are based on multiple observations. Whenever we multiple observations. Whenever we are able to find an instance in which are able to find an instance in which the conjecture is false, the entire the conjecture is false, the entire conjecture is untrue. This false conjecture is untrue. This false example is referred to as a 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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Example 3:Example 3:

Examine the data in the table. Find two cities such that the population of the first is greater than the population of the second while the unemployment rate of the first is less than the unemployment rate of the second. Shawnee has a greater population than Osage while Shawnee has a lower unemployment rate than Osage.

Answer: Osage has only 10,182 people on its civilian labor force, and it has a higher rate of unemployment than Shawnee, which has 90,254 people on its civilian labor force.

Example 3:Example 3:

DRIVING The table on the next screen shows selected states, the 2000 population of each state, and the number of people per 1000 residents who are licensed drivers in each state. Based on the table, find a counterexample for the following statement.

The greater the population of a state, the lower the number of drivers per 1000 residents.

Your Turn:Your Turn:

StateState PopulationPopulation Licensed Drivers Licensed Drivers per 1000per 1000

AlabamaAlabama 4,447,1004,447,100 792792

CaliforniaCalifornia 33,871,64833,871,648 627627

TexasTexas 20,851,82020,851,820 646646

VermontVermont 608,827608,827 831831

West VirginiaWest Virginia 1,808,3441,808,344 745745

WisconsinWisconsin 5,363,6755,363,675 703703

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Answer: Alabama has a greater population than West Virginia, and it has more drivers per 1000 than West Virginia.

Your Turn:Your Turn:

AssignmentAssignment

Geometry:Geometry:

Pg. 64 – 65Pg. 64 – 65

#11 – 36 #11 – 36

Pre-AP Geometry:Pre-AP Geometry:

Pg. 64 – 65 Pg. 64 – 65 #11 – 36, 38 - 40#11 – 36, 38 - 40