A diagram or table can suggest a result, but only a general reason establishes it. The safe habit is to say what the picture suggests and then say what has actually been justified.
This lesson is part of proving or disproving an everyday mathematical claim. It pairs well with finding one counterexample.
How can a pattern mislead you?
A short run of matching cases feels like evidence, and it is. It is not proof, because the next case might break the pattern.
Take points on a circle. Join every pair of points with a straight line, and count the regions inside the circle.
| Number of points | Regions |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 3 | 4 |
| 4 | 8 |
| 5 | 16 |
The regions are 1, 2, 4, 8, 16, which double each time. The diagram and the table both suggest that 6 points give 32 regions.
Worked example: what happens at six points
Draw six points on a circle and join every pair. Count the regions carefully, including those made by lines crossing inside.
The count is 31, not 32. The doubling pattern holds for five cases and breaks on the sixth.
What went wrong? The pattern was a coincidence of small numbers, and the diagram for 6 points is crowded enough that the count is easy to misjudge by eye. The safe statement would have been: “The first five cases suggest doubling, but this has not been proved.”
A single drawing can also mislead in geometry. A triangle may look right-angled in a sketch marked “not drawn to scale”, and measuring it with a protractor gives 88°. You must use the given values and stated relationships, not the picture.
The mistake that costs marks
The common slip is writing “from the diagram” as the reason in a question that asks you to justify. It describes the picture, and it does not explain the result.
| Step | Wrong | Right |
|---|---|---|
| Observation | regions double: 1, 2, 4, 8, 16 | regions double: 1, 2, 4, 8, 16 |
| Claim | so 6 points give 32 | the pattern suggests 32, but is unchecked |
| Check | not done | count for 6 points gives 31 |
| Conclusion | doubling holds | doubling fails at 6 points |
The right column asks one extra question: has the next case been tested?
Check yourself
The sums of the first odd numbers are 1 = 1, 1 + 3 = 4, 1 + 3 + 5 = 9, and 1 + 3 + 5 + 7 = 16. Write one sentence that says what the pattern suggests, and one sentence that says what has not been established.
Answer
The pattern suggests that the sum of the first n odd numbers is n², since 1, 4, 9 and 16 are the squares of 1, 2, 3 and 4.
It has not been established for every n, because four cases do not cover all whole numbers. A general argument, such as grouping the numbers in pairs, would be needed to prove it.
What to study next
Test your wording in the integrated practice set. If you keep writing descriptions when a reason is needed, log it in the mistake log.
For a teacher to help you turn pictures into justified answers, see online one-to-one Mathematics tuition.