Use examples to support a conjecture

Detailed Explanation: Use examples to support a conjecture

A conjecture is a mathematical idea that seems true because examples show a pattern. To use examples well:

1. Notice a pattern.
2. State the pattern as a conjecture.
3. Test the conjecture with several new examples.
4. Decide whether the examples support it or whether a counterexample shows it is false.

Worked example

Problem: Look at the sums below. Make a conjecture about the sum of two consecutive whole numbers.

Step 1: Notice the pattern

The numbers in each pair are consecutive: they are next to each other when counting.

The sums are $3$, $7$, and $11$. Each sum is odd.

Step 2: State a conjecture

A possible conjecture is:

The sum of any two consecutive whole numbers is odd.

Step 3: Test the conjecture with new examples

Try pairs that were not in the original list:

Both $21$ and $49$ are odd, so these examples support the conjecture.

Step 4: Explain what the examples show

The examples give evidence that the conjecture may be true. However, even many examples do not prove that it works for every pair of consecutive whole numbers. To disprove the conjecture, you would only need to find one counterexample, such as a pair of consecutive numbers whose sum is not odd.