Suppose you were given a mathematical pattern like h =
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CK12 Conjectures and Counterexamples
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Guidance
A conjecture is an “educated guess” that is based on examples in a pattern. A counterexample is an example that disproves a conjecture.
Example A
Here’s an algebraic equation and a table of values for




1 

0 
2 

0 
3 

0 
After looking at the table, Pablo makes this conjecture:
The value of
Is this a true conjecture?
This is not a valid conjecture. If Pablo were to continue the table to
In this example
Example B
Arthur is making figures for an art project. He drew polygons and some of their diagonals.
From these examples, Arthur made this conjecture:
If a convex polygon has
Is Arthur’s conjecture correct? Or, can you find a counterexample?
The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw
Notice that we have not proved Arthur’s conjecture, but only found several examples that hold true. So, at this point, we say that the conjecture is true.
Example C
Give a counterexample to this statement: Every prime number is an odd number.
The only counterexample is the number 2: an even number (not odd) that is prime.
CK12 Conjectures and Counterexamples
Guided Practice
A car salesman sold 5 used cars to five different couples. He noticed that each couple was under 30 years old. The following day, he sold a new, luxury car to a couple in their 60’s. The salesman determined that only younger couples by used cars.
1. Is the salesman’s conjecture logical? Why or why not?
2. Can you think of a counterexample?
Answers:
1. It is logical based on his experiences, but is not true.
2. A counterexample would be a couple that is 30 years old or older buying a used car.
Practice
Give a counterexample for each of the following statements.
 If
n is a whole number, thenn2>n .  All numbers that end in 1 are prime numbers.
 All positive fractions are between 0 and 1.
 Any three points that are coplanar are also collinear.
 All girls like ice cream.
 All high school students are in choir.
 For any angle there exists a complementary angle.
 All teenagers can drive.
 If
n is an integer, thenn>0 .  All equations have integer solutions.