Mathematical reasoning
- Syllabus
- 2017
- Topic
- —
- Level
- Foundation
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 justified deductions and conclusions
A deduction is a statement that must follow from given mathematical information and accepted facts. A conclusion is justified only when the information is sufficient and the logical link is stated.
| Step | Action |
|---|---|
| identify | separate given facts from what must be shown |
| connect | choose a definition, theorem, property or calculation that links them |
| infer | state the new fact that necessarily follows |
| conclude | state the required result and cite the decisive reason |
If two circle centres are 17 units apart and their radii total 13+6=19, then the distance between centres is less than the sum of the radii, so the circles intersect.
A pattern, diagram appearance or plausible claim is not a deduction. The conclusion must follow from the stated facts; if another case is possible, more information is needed.
Construct a coherent chain of reasoning
A chain of reasoning is an ordered sequence in which every statement follows from earlier information and moves toward the required result.
| Link | What to write |
|---|---|
| fact | the given value, condition or earlier result |
| reason | the rule, definition, theorem or valid operation used |
| consequence | the new statement produced |
| next link | use that consequence as information for the following step |
| conclusion | connect the final statement directly to what was required |
Use connective language such as 'because', 'therefore', 'so' and 'hence' to expose the dependency between steps. Define symbols before using them and keep equations equivalent when rearranging.
Several correct statements do not form a chain if their order or connection is missing. Do not hide a necessary assumption or jump from evidence to a result without the intermediate reason.
Present mathematical arguments and proofs
A mathematical argument supports a claim with valid reasoning. A proof establishes that a statement is true for every case covered by its conditions, not just for selected examples.
| Purpose | Suitable structure |
|---|---|
| prove a universal algebraic claim | represent a general value, transform logically, reach the claim |
| prove finitely many cases | exhaust all cases without omission or repetition |
| prove an implication | assume the conditions and derive the conclusion |
| disprove a universal claim | give one valid counterexample |
| prove impossibility | assume the contrary and derive a contradiction |
State what is assumed, justify each transformation, and end by naming the claim established. A diagram or numerical check may guide a proof but cannot replace general reasoning.
Many confirming examples do not prove a universal statement. Conversely, one valid counterexample is enough to disprove a claim that says 'all' or 'always'.
Interpret and communicate mathematics accurately
Accurate mathematical communication preserves the meaning of information when moving among words, symbols, tables, graphs and diagrams, and states results with the context and precision needed.
| Feature | Accuracy check |
|---|---|
| notation | symbols, inequalities and equality signs express the intended relationship |
| labels | variables, axes, sets, angles and points are defined |
| units | quantities use compatible units and answers include required units |
| precision | exact values are retained until the requested rounding stage |
| interpretation | the final sentence answers the contextual question |
| qualification | assumptions, estimates and limitations are stated where relevant |
Read scales, legends, intervals and wording before extracting data. Distinguish < from ≤, an estimate from an exact value, and correlation from a claim of causation.
A bare number or unexplained diagram may be mathematically correct yet communicate inadequately. Do not report more precision or certainty than the evidence supports.