While investigating the properties of parallelograms using a dynamic
geometry software program, a student notices that the diagonals of a
parallelogram seem to bisect each other. The student conjectures
that this statement is true for all parallelograms. Which of the
following would be the most reasonable next step for the student to
take to evaluate the validity of this conjecture? My chooses are as followed.
A. Try to find more examples supporting the conjecture.
B. Try to find a counterexample to the conjecture.
C. Try to prove the conjecture using mathematical induction.
D. Try to generalize the conjecture to apply to other
quadrilaterals.
To evaluate the validity of a conjecture would require one of the following:
1. Mathematically prove that the conjecture is correct,
2. provide a counter-example to prove that the conjecture is incorrect.
The first case applies to believers, and the second to non-believers.
More examples do not provide mathematical proof, for example:
Conjecture:
"(2^p)-1 is prime if p is prime".
Examples:
2^2-1=3 (prime)
2^3-1=7 (prime)
More examples:
2^5-1=31 (prime)
2^7-1=127 (prime)
BUT:
2^11-1=2047 = 23*89
Even though more examples do not prove the conjecture, there are merits of taking this option. In the search for more examples, perhaps the student will come across counter-examples, in which case the conjecture will be proven invalid.
(read more about Mersene primes here:
(Broken Link Removed)
Mathematical induction applies very well in number theory where we can establish a proof based on known examples that are discrete, or in steps such as integers. To me, this particular problem requires a geometric proof.
Generalizing the conjecture would require that the conjecture is established (or proved) for the case of parallelograms.
The most reasonable next step for the student to take to evaluate the validity of the conjecture is B. Try to find a counterexample to the conjecture.
To find a counterexample, the student would need to search for a parallelogram where the diagonals do not bisect each other. If such an example is found, it would disprove the conjecture and show that it is not true for all parallelograms.
By attempting to find a counterexample, the student can test the validity of the conjecture directly rather than relying on more examples to support it (option A), proving it using mathematical induction (option C), or generalizing it to other quadrilaterals (option D).