2.12.1Using Inductive Using Inductive

Reasoning to Make Reasoning to Make ConjecturesConjectures

Find the patternFind the pattern If everyone at a party of people shakes everyone If everyone at a party of people shakes everyone

else’s hand one time, how many handshakes will else’s hand one time, how many handshakes will there be?there be?

January, March, May ____________January, March, May ____________ 7,14,21,28,_________7,14,21,28,_________ 1,2,4,__________1,2,4,__________ ________________________

peoplepeople 11 22 33 44 55 66 77 …….. 2020Hand-Hand-shakesshakes

Inductive ReasoningInductive Reasoning When several examples form a pattern When several examples form a pattern

and you assume or draw the conclusion and you assume or draw the conclusion that the pattern will continue (like on the that the pattern will continue (like on the previous slide) you are applying inductive previous slide) you are applying inductive reasoning.reasoning.

Inductive reasoning is the process of Inductive reasoning is the process of reasoning that a rule or statement is true reasoning that a rule or statement is true because specific cases are true.because specific cases are true.

ConjectureConjecture A statement you believe to be true based A statement you believe to be true based

on inductive reasoning.on inductive reasoning. Examples:Examples:

The product of an even number and an odd The product of an even number and an odd number is ________.number is ________.

The sum of two odd numbers is ________.The sum of two odd numbers is ________.

How do you show that a How do you show that a conjecture is always true?conjecture is always true?

PROVE IT!!!PROVE IT!!!

How do you show that a How do you show that a conjecture is false?conjecture is false?

Find a counterexample.Find a counterexample.

Counterexamples can be…..Counterexamples can be….. A drawingA drawing A statement, orA statement, or A number.A number.

True or False?True or False?

If x = 3, then xIf x = 3, then x² = 9.² = 9. If If xx² = 9, then x = 3.² = 9, then x = 3. For every integer, n³ is positive.For every integer, n³ is positive. If the grass is wet, then it rained.If the grass is wet, then it rained. If B is the midpoint of AC, then AB = BC.If B is the midpoint of AC, then AB = BC. If AB = BC, then B is the midpoint of AC.If AB = BC, then B is the midpoint of AC. For any real number x, For any real number x, xx² > x.² > x. The winner of the door decoration contest The winner of the door decoration contest

will get a pizza party!will get a pizza party!

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Inductive Reasoning

Look for a pattern

Make a conjecture

Is it true?If yes, PROVE IT! If no, provide a Counterexample.