In a PSMT — the problem-solving and modelling task set as IA1 in General Mathematics, Mathematical Methods and Specialist Mathematics — an observation is a relevant fact or pattern you identify in the task, context, diagram or data, and an assumption is a condition you accept as true so the mathematical model can work. The difference is where the statement comes from: observations are noticed in the available evidence; assumptions are introduced or accepted to simplify a situation that cannot be modelled in every real-world detail.
Both live in the Formulate criterion, which is worth 4 of the PSMT's 20 marks. Four marks sounds small until you notice that two of Formulate's three top-band requirements are about exactly these two statement types — and that the Evaluate criterion, worth 5 more marks, sends you back to them. Getting the distinction right is worth more of the task than its own section suggests. If you want the whole write-up mapped out first, start with the PSMT meaning and criterion-by-criterion guide.
Observations vs assumptions: the difference
Both belong in the Formulate section because both influence the mathematical choices that follow. They are not interchangeable labels for a list of introductory facts. Each statement should help explain why you selected particular variables, representations, procedures or technology.
| Question | Observation | Assumption |
|---|---|---|
| Where does it come from? | The task, supplied information, a diagram, research or data. | A condition accepted by the modeller when reality is uncertain or too complex. |
| What does it do? | Identifies a feature that changes how the problem should be approached. | Simplifies, limits or defines the model so calculations are possible. |
| How is it justified? | Cite the evidence and explain why the feature is important. | Explain why the condition is reasonable, necessary and consequential. |
| What is the common mistake? | Restating a task fact without explaining its relevance. | Writing “assume” before an obvious fact without explaining its effect. |
The syllabus treats them as a set. Describing the modelling approach, it says that important assumptions, variables and observations are identified and justified, based on the logic of a proposed solution or model. Variables are the third member of that group and are marked in the same criterion, so name the quantities you chose to represent as well as the facts you noticed and the conditions you accepted.
What the Formulate ISMG actually says
The wording below is the Formulate criterion from the instrument-specific marking guide in the current QCAA maths syllabuses. It is identical in General Mathematics, Mathematical Methods and Specialist Mathematics, so it applies whichever of the three you are sitting. Essential Mathematics also sets problem-solving and modelling tasks, but it is an Applied subject marked against different standards, so work from your own task sheet rather than the marks below.
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3–4
justified, important, plural
Justified statements of important assumptions. Justified statements of important observations. Justified mathematical translation of important aspects of the task.
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1–2
stated, relevant, singular
Statement of a relevant assumption. Statement of a relevant observation. Mathematical translation of an aspect of the task.
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0
no match
The response does not match any of the descriptors above.
Three words separate the bands, and each one is a decision you make while drafting.
- Statement becomes justified. The lower band only asks you to state. The top band asks you to explain why the statement is reasonable and what it does to the model.
- Relevant becomes important. A relevant fact is connected to the task. An important one changes a mathematical choice — remove it and something in your model has to change.
- Singular becomes plural. The lower band reads “a relevant assumption” and “a relevant observation”. The top band reads “assumptions” and “observations”. Take the plural literally: at least two justified statements of each.
How many observations and assumptions do you need?
At least two of each, justified, is the safe reading of the top-band descriptor. There is no upper limit in the syllabus, and there is no credit for length — a PSMT is capped at 10 A4 pages and 2000 words, and every sentence you spend defending an unnecessary assumption is a sentence not spent on Solve or Evaluate.
The practical target is two to four of each, chosen because they materially affect the model. Then check the cheapest quality test there is: for each statement, ask what would change if it were false or ignored. If nothing changes, it is decoration, and decoration reads to a marker as padding rather than as evidence.
Practise: observation or assumption?
The boundary becomes clearer when you classify statements in context. Commit to an answer before opening the explanation; the useful skill is being able to explain the source and purpose of the statement, not memorising a keyword.
Formulate field test
Observation or assumption?
Classify each PSMT statement, then compare your reasoning with the worked explanation.
The supplied site plan labels the garden bed as 12 m long and 8 m wide.
The garden bed can be modelled as a perfect rectangle.
One travel-time value is much higher than the rest of the supplied data.
The bus travels at a constant average speed between each stop.
Demand rises sharply between 3 pm and 4 pm in the provided graph.
The surveyed group represents the behaviour of the whole school.
Worked PSMT example: modelling a garden bed
Imagine a task asking you to estimate how much soil is needed for a school garden. The supplied site plan gives a nominal length of 12 metres and width of 8 metres, but the photographed boundary is slightly uneven.
Observation
Statement: The supplied plan labels the garden bed as 12 m by 8 m.
Why it matters: These dimensions provide the initial values used to calculate the modelled surface area.
Assumption
Statement: The garden bed is modelled as a perfect rectangular prism with a uniform soil depth.
Why it matters: This permits one length, width and depth to represent the whole bed, but may overestimate or underestimate soil near the irregular edges.
The dimensions are not an assumption merely because you use them in a model: they were supplied as evidence. The rectangular shape and uniform depth are assumptions because the real garden does not establish those ideal conditions. This source test prevents a common error where every number is labelled an observation and every sentence containing “assume” is accepted as a useful assumption.
Worked PSMT example: modelling travel time
Now imagine a task using recorded journey times to recommend a school-bus timetable. The dataset shows most trips taking 22–27 minutes, with one trip taking 46 minutes.
The 46-minute journey is substantially higher than the remaining recorded times. This observation is important because retaining the value increases the estimated travel time and may change the recommended departure time.
Why this works: It identifies evidence, describes the pattern and connects that pattern to the modelling decision.
The 46-minute journey is assumed to reflect an unusual disruption rather than normal traffic conditions. Excluding it produces a timetable based on typical journeys, although the model may underestimate delays when disruptions occur.
Why this works: It names the accepted condition, explains why the model uses it and acknowledges the effect on the solution.
The same data point can generate both an observation and an assumption. First you observe that the value is unusual. Then you may assume a cause or decide how the model should treat it. Keep those reasoning steps separate so the marker can follow the move from evidence to modelling choice.
Example statements, by task context
PSMT contexts repeat far more than the task sheets suggest. The table below pairs a typical observation with a typical assumption for the situations that come up most often, so you can see the same source test applied across different mathematics. Treat them as patterns to adapt, not sentences to copy — an assumption borrowed from someone else's task is exactly the kind that cannot be justified.
| Task context | Typical observation | Typical assumption |
|---|---|---|
| Area, volume or packaging | The plan gives the container's stated internal dimensions. | Material thickness is negligible, so internal and external dimensions can be treated as equal. |
| Finance and loan repayment | The advertised rate is quoted as an annual percentage compounded monthly. | The interest rate stays fixed for the full term of the model. |
| Sampling and surveys | The sample contains 60 responses from a cohort of 800 students. | The sample is representative of the whole cohort, so its proportions can be generalised. |
| Rates, speed and travel | Recorded journey times cluster between 22 and 27 minutes. | Average speed is constant between stops, so time can be modelled as a linear function of distance. |
| Growth and decay | Recorded values roughly double every three years. | The growth rate remains unchanged beyond the range of the supplied data. |
| Scheduling and optimisation | The venue is available for a maximum of six hours. | Changeover time between sessions is fixed and does not vary with group size. |
How to justify an important observation
A strong observation is more than a description. Use a three-part sentence:
Identify the evidence
Name the supplied dimension, pattern, constraint, outlier or relationship. Use a value, source or visible feature where possible.
Explain its relevance
State how the feature influences your selection of a variable, mathematical procedure, representation or decision rule.
Connect it to the model
Show what would change if the observation were ignored or interpreted differently.
A useful sentence frame is: “The supplied data shows ___; this is important because ___, so the model ___.” Treat the frame as a reasoning check, not wording that must appear in every paragraph.
How to justify an important assumption
An assumption needs a different chain: accepted condition, reason and consequence. “It is assumed the speed is constant” only names the condition. A justification explains why constant speed is a workable approximation and how variation in traffic could affect the result.
- State the condition precisely. Avoid vague claims such as “all values are accurate.”
- Explain why it is reasonable. Use task information, research, scale or the purpose of the model.
- Explain why it is needed. Name the calculation or representation that depends on it.
- Trace the effect. State whether the assumption could overestimate, underestimate or limit the solution.
- Return to it in Evaluate. Judge reasonableness against the assumption rather than abandoning it after Formulate.
Evaluate marks the same statements again
This is the part students miss, and it is where the 4-mark criterion turns into a 9-mark one. The top band of the Evaluate criterion, worth 4–5 marks, asks for verified results, justified statements about the reasonableness of the solution by considering the assumptions, justified statements about the reasonableness of the solution by considering the observations, and justified statements of relevant strengths and limitations.
Read that carefully: assumptions and observations are named separately, and both must be considered. So the statements you choose in Formulate are the statements you have to answer for in Evaluate. Two consequences follow for how you draft.
- Choose statements you can revisit. An assumption you cannot say anything useful about later is an assumption that will cost you marks twice.
- Return to each one by name. Do not write a general paragraph about limitations. Take each assumption and each observation in turn and say whether the solution still holds given it.
Four common PSMT mistakes
- Calling instructions observations. “Use technology” is a task direction, not a feature you noticed about the problem.
- Calling supplied facts assumptions. A dimension explicitly printed on the task sheet is evidence unless you introduce a further condition about its accuracy.
- Justifying with convenience alone. “This makes the maths easier” does not show why the simplification is reasonable for the real situation.
- Forgetting evaluation. Important assumptions and observations should reappear when you judge whether the solution is reasonable and identify its limitations.
The ISMG reading guide helps you distinguish a descriptor from a checklist, while the QCAA cognitive verb guide explains what justify and evaluate require. Before you submit, check the QCAA response-length rules rather than moving reasoning into an appendix to fit the page limit.
