Mathematical Reasoning and Communication
What the domain tests
This domain is assessed inside every other task: can you show a complete chain of reasoning, justify a claim rather than just assert it, use notation and units correctly, and present the solution so a reader can follow it? Exam tasks include justify, explain why, show that, decide whether the statement is true, and give a counterexample.
What a full solution contains
- A starting statement of what is given and what is to be found.
- Each step connected logically to the previous one, with the operation or property named.
- Correct symbols: = only between quantities that are genuinely equal, ≈ for rounded values, and the inequality signs pointing the right way.
- Units on the answer, plus a short concluding sentence.
- Rounding applied once, at the end, with the rounding stated.
Justifying in different areas
- Number and algebra: prove a statement about all numbers by writing it with letters. Three consecutive integers are n, n + 1, n + 2. Two consecutive odd numbers are 2n + 1 and 2n + 3.
- Geometry: each line needs a reason - angles on a straight line sum to 180°, base angles of an isosceles triangle are equal, alternate angles between parallel lines are equal, vertically opposite angles are equal.
- Disproving: one counterexample is enough to show a claim is false; a hundred examples never prove it true.
Worked example
Show that the sum of three consecutive integers is always divisible by 3. Let the integers be n, n + 1 and n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1). Since n + 1 is an integer, the sum is 3 multiplied by an integer, so it is divisible by 3 for every whole number n. Note that testing 1 + 2 + 3 = 6 and 4 + 5 + 6 = 15 shows the pattern but does not prove it; the algebraic step is what makes the claim general.
Traps that cost marks
- Giving an example as if it were a proof.
- Writing a string of expressions joined by equals signs when the quantities are not equal, or omitting the equals sign altogether.
- Leaving a geometric statement with no reason attached (for example claiming two angles are equal because they look equal).
- Dropping units halfway through, then quoting a bare number.
- Contradicting your own earlier rounding, or ending without stating the final answer clearly.