Consider this scenario:

Suppose you just got your driver’s license. You are begging your parents to buy you a car, but they refuse. An argument you might make is that ALL the kids at school are getting a new car when they get their license.

Does that argument work?

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All the kids? Really?!Close Pop Up

Is the statement, “All my friends are getting a car when they get their license” even true?

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Not unless every single student at your school gets a car when they get their license.Close Pop Up

Your parents could easily disprove this statement. How?

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Find ONE student at your school who did not get a car.Close Pop Up

It only takes one counterexample to show that your statement is false.  

What's a counter example that your parents could use to disprove the claim about your friends getting a car?

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Name any student without a car that has their license. That student is a counter example. Close Pop Up

In geometry, for a statement to have a truth value of TRUE, it must always be true. Sometimes, it just takes some clarification to make the statement true. For example, if we said, "For three collinear points P, Q, and R, then PQ + QR = PR.

Is this always true, sometimes true, or never true?

TRUE here: line with points P, Q, R

NOT TRUE here: line with points Q, P, R

How can you change the statement to make it always true?

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For three collinear points P, Q, and R where Q is between P and R, then PQ + QR = PR. Close Pop Up

A statement or conjecture can be written based on seeing several examples. This is called using inductive reasoning. Sometimes a conjecture made this way seems true, but one counter example proves it to be false.

Example:

hexagon with 60 degree exterior angle, pentagon with 72 degree exterior angle, octagon with a 45 degree exterior angle

Conjecture: The exterior angles of a regular polygon are acute.

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Counter example:

triangle with a 120 degree exterior angle Close Pop Up