Mathematical conjectures are tentative statements about patterns, quantities, or relationships formed from examples, diagrams, tables, or calculations and expressed clearly enough to test. At this level, reasoning includes checking a conjecture across varied cases, identifying a counterexample that shows it is not always true, and distinguishing supporting evidence from a convincing explanation; formal proof and highly general abstract conjectures are beyond this scope.
A conjecture is a statement about a pattern that you think is true. You make one by looking at examples, then test it with more examples.
Example: Look at the sums of consecutive whole numbers.
The answers are , , and , which are all prime numbers.
A possible conjecture is:
The sum of two consecutive whole numbers is always prime.
Try the next pair:
But is not prime because .
So is a counterexample. One counterexample is enough to show that the conjecture is not always true.
Although the sums are not always prime, they are all odd:
Test some more varied examples:
Each answer is odd. This supports the new conjecture:
The sum of two consecutive whole numbers is always odd.
The reason this pattern makes sense is that two consecutive whole numbers always include one even number and one odd number. An even number plus an odd number is odd.
Remember: several examples can support a conjecture, but a counterexample can show that a conjecture is false.
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