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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%
So
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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
So
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he W
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Alm
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Fac
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003
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