7 Mathematical problem-solving and reasoning
- Syllabus
- 2017
- Section
- 7
- Level
- Higher
Exam analysis
Most tested topics
Problem-solving
- Syllabus
- 2017
- Topic
- —
- Level
- Higher
Translate a problem into mathematics
Problem-solving begins by converting a situation into a mathematical model: identify what is known and required, represent the relationships, carry out a connected process, then interpret the result in the original context.
| Stage | Question to ask |
|---|---|
| understand | What is known, unknown, constrained and being asked? |
| represent | Would a variable, equation, diagram, table, graph or ratio expose the relationships? |
| plan | Which ordered mathematical steps connect the facts to the unknown? |
| execute | Can each step be carried out accurately with units and sufficient precision? |
| interpret | What does the result mean in context? |
| validate | Is it feasible, correctly rounded and consistent with the original conditions? |
Translate phrases into relationships before calculating. For example, 'three more than x' is x+3, 'shared in the ratio 2:5' means seven equal parts, and a fixed total creates an equation whose parts sum to that total.
A series of processes may cross topics: form an equation from a diagram, solve it, substitute the value into a formula, convert units, then make a decision. Record the chain so every result has a clear source.
Use bounds and common sense: lengths and counts cannot be negative, a probability must lie from 0 to 1, dimensions must use compatible units, and a rounded answer must satisfy the requested accuracy.
Do not start with an operation just because a number appears in the question. A correct calculation on the wrong model does not solve the problem; the final answer must address the stated unknown and context.
Mathematical reasoning
- Syllabus
- 2017
- Topic
- —
- Level
- Higher
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.