Mathematical reasoning
- Syllabus
- 2016
- Topic
- —
- Level
- —
Learning objectives
—Make deductions and draw conclusionsMake deductions and draw conclusions from mathematical information.—Construct chains of reasoningConstruct coherent chains of mathematical reasoning.—Present arguments and proofsPresent mathematical arguments and proofs.—Interpret and communicate accuratelyInterpret and communicate mathematical information accurately.
Make deductions from mathematical information
A deduction is a statement that must follow from the given facts and valid mathematical rules. A conclusion answers what those deductions establish.
| Move | Question to ask |
|---|---|
| identify facts | What is explicitly given or already proved? |
| infer | Which definition, theorem, operation or pattern justifies the next statement? |
| test | Must it be true in every allowed case? |
| conclude | Does the final statement answer the claim with its conditions? |
To reject a universal claim, one valid counterexample is enough. To establish it, checking examples is not enough; use a general argument.
Do not treat a diagram's appearance, a few numerical cases or the desired answer as evidence. A deduction must follow from stated information, not an unstated assumption.
Build a coherent chain of reasoning
A coherent chain links each statement to the previous facts by an explicit reason, so another reader can verify every step.
| Stage | Purpose |
|---|---|
| start | state givens, definitions and target |
| transform | perform one justified algebraic, numerical, geometric or statistical step |
| connect | explain why the new fact advances the target |
| finish | state the required result and its conditions |
Work forwards from known facts or backwards from the target to find a bridge, then present the final solution in forward logical order. Keep symbols defined and equalities genuinely equivalent.
A list of correct facts is not a chain unless the links are shown. Avoid circular reasoning, unexplained jumps and changing notation or assumptions midway.
Present an argument or proof
| Purpose | Suitable form |
|---|---|
| prove a universal algebraic claim | represent a general case and simplify logically |
| prove a geometric result | cite angle, congruence, similarity or vector facts |
| disprove a universal claim | give and verify one counterexample |
| establish equivalence | prove both directions or use reversible steps |
State the claim and assumptions, define variables, give each necessary step with its reason, and end with the exact conclusion. Use exact values where approximation could weaken the claim.
Examples may suggest a conjecture and help test it, but even many examples do not prove a statement about all cases.
Do not assume what you are trying to prove, rely on a not-to-scale diagram, or cite a theorem whose conditions have not been established.
Interpret and communicate mathematics accurately
| Element | Accurate communication |
|---|---|
| symbols | define variables and use equality/inequality signs correctly |
| quantities | include units, scale, direction and appropriate precision |
| data/graphs | name axes, intervals and what heights, areas or trends mean |
| conclusions | answer in context and state limitations or conditions |
Translate tables, graphs, diagrams and prose into precise relationships before calculating. Afterward translate the result back into a complete sentence that matches the requested quantity.
Check notation, units, rounding, domain restrictions and whether the conclusion is supported rather than merely plausible. Distinguish exact values from estimates.
Correct arithmetic can still be communicated inaccurately through missing units, ambiguous notation, over-rounded values, causal claims from association, or conclusions beyond the available information.