What is QCE Maths Methods IA1?

IA1 is the Problem-Solving and Modelling Task (PSMT), the first internal assessment in Unit 3. Students work on a real-world modelling problem over a 4-week period, producing a written report of up to 10 pages plus appendices. Worth 20 percent of the subject result.

Four QCAA criteria on the ISMG (Instrument-Specific Marking Guide): Formulate (translate context into mathematical model), Solve (apply correct mathematics), Evaluate and verify (assess reasonableness, refine), Communicate (clear scientific writing). Each criterion has a marking grid with descriptors.

What kind of context appears in IA1?

An open-ended real-world scenario requiring modelling with Unit 3 mathematics: optimisation problems, growth and decay (exponential or logistic), periodic phenomena (sinusoidal), rate of change applications. Recent prompts have used population dynamics, drug dosage, traffic flow, climate data.

What is the modelling cycle?

Define the problem, make assumptions, formulate a model, solve, interpret in context, evaluate (test against the question, consider limitations), refine if needed. QCAA expects evidence of the full cycle, including a refinement step where the initial model is critiqued and improved.

What are common errors in IA1?

Insufficient justification of model choice. Missing or unjustified assumptions. Solution presented without context (just numbers, no units, no interpretation). No refinement: presenting one model only. Poor scientific writing (run-on sentences, no headings, missing diagrams).

QLDMath Methods

QCE Maths Methods IA1 strategy: 2026 guide

A 2026 guide to QCE Maths Methods IA1 (Problem-Solving and Modelling Task). The four ISMG criteria (Formulate, Solve, Evaluate and verify, Communicate), the modelling cycle, step-by-step PSMT-style worked examples, common pitfalls, and a four-week timeline for a top-band response.

Generated by Claude Opus 4.814 min readQCAA-MM-IA1Updated 2026-06-20

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What IA1 is
QCAA marking criteria (ISMG)
The mathematical modelling cycle
Worked example: drug concentration in the bloodstream
Additional worked example: optimisation, closed cylinder
Additional worked example: sinusoidal model for tidal water depth
Common pitfalls and how to avoid them
Mathematics deployed
Communication standards
Four-week timeline
Check your knowledge

What IA1 is

IA1 is the Problem-Solving and Modelling Task (PSMT), one of three internal assessments in QCAA Maths Methods. It runs across about 4 weeks in Unit 3 (Year 12). Worth 20 percent of the subject result.

The deliverable is a written report of up to 10 pages plus appendices, presenting a real-world modelling problem, its mathematical model, the solution, and evaluation.

QCAA marking criteria (ISMG)

The QCAA Instrument-Specific Marking Guide for IA1 has four criteria:

Formulate. Translate the real-world problem into a mathematical model. Includes problem definition, assumptions, model choice with justification.
Solve. Apply correct mathematics. Includes calculation, technology use, intermediate results.
Evaluate and verify. Assess the reasonableness of the solution in the original context. Identify model limitations. Refine the model.
Communicate. Clear scientific writing with appropriate use of mathematical notation, diagrams, tables.

Top band requires excellence in all four. The "mathematical reasoning" expectation runs across all four criteria.

The mathematical modelling cycle

Step 1: Define the problem. State precisely what is being asked. Identify the real-world question.

Step 2: Make assumptions. List explicit assumptions about the context. Justify each one as reasonable.

Step 3: Identify variables. Independent (input), dependent (output), and parameters (constants in the context).

Step 4: Formulate the model. Express the relationship between variables. Justify why this form (linear, exponential, sinusoidal, etc.).

Step 5: Solve. Apply mathematics: differentiation, antidifferentiation, equation solving, simulation.

Step 6: Interpret. Express results in the original context with units. Answer the original question.

Step 7: Evaluate. Compare predictions against reality (data, intuition, extreme cases). Identify limitations.

Step 8: Refine. Modify the model to address a limitation. Re-solve. Compare.

Worked example: drug concentration in the bloodstream

PSMT-style: paracetamol steady-state concentration with first-order kinetics

Step 1. Formulate, define the problem: A patient is prescribed a 200 mg dose of paracetamol every 6 hours. Predict the steady-state concentration and the time to reach it.
Step 2. Formulate, state assumptions: (i) The drug is absorbed instantaneously into the bloodstream. (ii) Elimination follows first-order kinetics with half-life $2$ hours. (iii) The volume of distribution is constant. Each assumption is plausible at therapeutic doses for healthy adults; they will be revisited in the evaluation.
Step 3. Formulate, identify variables: Time $t$ (hours). Concentration $C(t)$ (in dose-equivalent mg units). Decay constant $k = \ln 2 / 2 \approx 0.347$ per hour.
Step 4. Solve, between-dose decay: First-order decay: $C(t) = C_0 e^{-kt}$ . Over one $6$ -hour interval the multiplier is $r = e^{-6k} = e^{-2.08} \approx 0.125$ .
Step 5. Solve, sum the geometric series: Just before the $n$ th dose, the accumulated concentration is the partial sum of a geometric series:

C_n = C_0 \cdot \frac{1 - r^n}{1 - r}

Step 6. Solve, take the limit for steady state: As $n \to \infty$ , $r^n \to 0$ (since $|r| < 1$ ), giving steady-state $C_\infty = \frac{C_0}{1 - r} = \frac{200}{1 - 0.125} \approx 229$ dose-equivalent mg.
Step 7. Solve, time to 95 percent of steady state: Require $r^n < 0.05$ . Taking logs: $n > \frac{\ln 0.05}{\ln 0.125} \approx 1.44$ . So $n = 2$ doses ( $12$ hours) brings the patient within $5\%$ of steady state.
Step 8. Evaluate and verify in context: Within $12$ hours of the first dose, concentration is close to its long-run value. Check the limit: $C_\infty \approx 1.14 \times C_0$ , sensible because $r \ll 1$ (most drug has cleared by the next dose).
Step 9. Evaluate, identify limitations: Real absorption is not instantaneous, so the model overstates the peak. Constant elimination is reasonable at therapeutic dose. Volume of distribution varies between individuals (about $0.9$ L/kg of body mass).
Step 10. Refine: Replace instantaneous absorption with a single-compartment model with absorption rate constant $k_a$ , giving $C(t) = \frac{F D k_a}{V (k_a - k)} (e^{-kt} - e^{-k_a t})$ . Re-solve numerically and compare peak and steady-state values. Use CAS to plot both models on one axis.
Step 11. Communicate: Report sections: Introduction, Formulate, Solve, Evaluate and verify, Refine, Conclusion, Appendix. Each figure labelled with axes, units, source; each table with headers and consistent significant figures.
Answer: Steady-state concentration $\approx 229$ dose-equivalent mg; $95\%$ of steady state reached within $12$ hours (two dosing intervals).

Additional worked example: optimisation, closed cylinder

PSMT-style: minimise the surface area of a closed cylinder of volume

500

^3

Step 1. Formulate, define the problem: A manufacturer wants to make a closed cylindrical can of volume $500$ cm $^3$ using as little sheet metal as possible. Find the radius and height that minimise surface area, and report the minimum surface area.
Step 2. Formulate, identify variables: Radius $r$ (cm), height $h$ (cm), volume $V = 500$ cm $^3$ , surface area $S$ (cm $^2$ ).
Step 3. Formulate, assumptions: The can is a right circular cylinder with rigid wall (sheet thickness ignored), the lid and base are flat circular discs, no seam allowance.
Step 4. Formulate, constraint: $V = \pi r^2 h = 500$ , so $h = \frac{500}{\pi r^2}$ .
Step 5. Formulate, objective in one variable: Closed cylinder surface area:

S(r) = 2 \pi r^2 + 2 \pi r h = 2 \pi r^2 + \frac{1000}{r}

Step 6. Solve, differentiate.

S'(r) = 4 \pi r - \frac{1000}{r^2}

Step 7. Solve, stationary point: $S'(r) = 0$ gives $4 \pi r^3 = 1000$ , so $r^3 = \frac{250}{\pi}$ and $r = \left(\frac{250}{\pi}\right)^{1/3} \approx 4.30$ cm.
Step 8. Solve, second-derivative test: $S''(r) = 4 \pi + \frac{2000}{r^3} > 0$ for $r > 0$ , confirming a local minimum (and the only stationary point on $r > 0$ , so a global minimum).
Step 9. Solve, compute $h$ and $S$: $h = \frac{500}{\pi (4.30)^2} \approx 8.60$ cm. $S(4.30) \approx 2 \pi (4.30)^2 + \frac{1000}{4.30} \approx 348$ cm $^2$ .
Step 10. Evaluate and verify: Note $h = 2r$ : the optimal closed cylinder has height equal to diameter. This is a known textbook result and supports reasonableness of the answer.
Step 11. Evaluate, limitations: Real cans include a seam allowance and rolled lip; lids are usually thinner sheet than walls. A refined model could weight wall and lid metal differently.
Step 12. Communicate: Present a sketch of $S$ versus $r$ over $r \in (0, 10]$ showing the minimum; state results to three significant figures with units.
Answer: $r \approx 4.30$ cm, $h \approx 8.60$ cm, $S \approx 348$ cm $^2$ .

Additional worked example: sinusoidal model for tidal water depth

PSMT-style: tidal depth modelled by a sinusoidal function

Step 1. Formulate, define the problem: The water depth $D$ (metres) at a wharf is modelled as $D(t) = 5 + 2 \sin\!\left(\frac{\pi t}{6}\right)$ where $t$ is hours after midnight. A boat needs at least $6$ metres of water. For what fraction of a $24$ -hour day can the boat enter?
Step 2. Formulate, identify model parameters: Mean depth $5$ m, amplitude $2$ m, period $T = 2 \pi / (\pi / 6) = 12$ hours (semi-diurnal tide), so two high tides per day.
Step 3. Formulate, assumptions: Tidal motion is purely sinusoidal (no weather effect, no wind, no seasonal range change). The instrument reads to $\pm 0.05$ m.
Step 4. Solve, set up the inequality: Need $D(t) \ge 6$ , i.e. $5 + 2 \sin\!\left(\frac{\pi t}{6}\right) \ge 6$ , so $\sin\!\left(\frac{\pi t}{6}\right) \ge \frac{1}{2}$ .
Step 5. Solve, find boundary times within one period: $\sin \theta = 1/2$ at $\theta = \pi/6$ and $\theta = 5 \pi / 6$ . With $\theta = \pi t / 6$ , this gives $t = 1$ and $t = 5$ .
Step 6. Solve, find admissible interval in one period: $\sin \theta \ge 1/2$ for $\theta \in [\pi/6, 5\pi/6]$ , hence $t \in [1, 5]$ hours each period.
Step 7. Solve, total in 24 hours: Two periods per day, each contributing $5 - 1 = 4$ hours of access. Total: $8$ hours.
Step 8. Evaluate and verify: As a fraction of the day: $8 / 24 = 1/3 \approx 33.3\%$ . Sanity check: the model exceeds $6$ m on the upper third of its range, so a third-of-cycle access window is consistent.
Step 9. Evaluate, limitations: Real tides are influenced by weather (storm surge) and lunar cycle. The model uses a single fixed amplitude.
Step 10. Refine: Replace constant amplitude with $A(t) = 2 + 0.3 \sin(2 \pi t / (14 \times 24))$ to capture the spring-neap fortnightly variation, and re-solve numerically.
Step 11. Communicate: Plot $D(t)$ over $t \in [0, 24]$ with the $6$ m threshold marked; shade the admissible windows.
Answer: The boat can enter for about $8$ hours per day, roughly one-third of the time.

Common pitfalls and how to avoid them

Pitfall 1: insufficient justification of model choice. Solution: state explicitly why this functional form fits. "Exponential decay because the rate of change of drug concentration is proportional to the current concentration."

Pitfall 2: unstated or unjustified assumptions. Solution: list every assumption in a numbered list. Justify each with one sentence.

Pitfall 3: pure calculation without context. Solution: at every step, restate what the number means in the original problem.

Pitfall 4: no refinement. Solution: identify at least one limitation in your first model. Modify the model. Compare predictions of the refined model with the original.

Pitfall 5: poor scientific writing. Solution: use numbered sections (Introduction, Assumptions, Model, Solution, Evaluation, Refinement, Conclusion). Use clear topic sentences. Use mathematical notation correctly.

Mathematics deployed

Calculus. Derivatives for rates of change. Integrals for accumulated quantities. Optimisation by finding stationary points.

Algebra. Geometric series for periodic doses. Quadratic and cubic root-finding. Logarithms for solving exponential equations.

Functions. Exponential and logarithmic models. Sinusoidal models for periodic phenomena (tides, daylight hours). Logistic models for constrained growth.

Numerical methods. Newton-Raphson where required. Numerical integration (trapezoidal, Simpson's).

CAS use. Document CAS commands. The CAS handles the algebra; you specify what to compute.

Communication standards

Use formal scientific writing. Numbered sections. Mathematical notation correctly (correct use of equation editor, proper variable names).

Diagrams. Graphs of model predictions with axes labelled, units, scale, gridlines.

Tables. Header row, units in headers, consistent sig fig.

References. Cite data sources. Use QCAA-prescribed style.

Four-week timeline

Week 1. Receive task. Read carefully. Define the problem in your own words. Identify two or three possible modelling approaches. Get teacher feedback.

Week 2. Build initial model. Make assumptions. Formulate. Solve. Get teacher draft feedback.

Week 3. Evaluate the model. Identify a limitation. Refine. Re-solve. Compare.

Week 4. Write the report. Revise. Proofread. Submit.

Check your knowledge

For the modelling cycle, list the eight steps in order.
Name the four QCAA ISMG criteria for IA1 and one descriptor for each.
For the cylinder problem (volume $500$ cm $^3$ ), what is the ratio $h : r$ at the optimum, and why is this a useful "reasonableness" check?
A logistic population model $P(t) = \frac{1000}{1 + 9 e^{-0.2 t}}$ has carrying capacity $1000$ . At what value of $P$ does the population grow fastest? Justify briefly.
For the tidal model $D(t) = 5 + 2 \sin(\pi t / 6)$ , write the equation you would solve to find the times in a day when $D = 4.5$ .
Give one realistic refinement to a drug-concentration model that assumes instantaneous absorption.
State one assumption you would have to make in modelling daily traffic flow on a single road with a sinusoidal function. Justify briefly.
Sketch (or describe) the structure of a strong PSMT report, naming each section.

Solutions

Q1: (1) Define the problem, (2) make assumptions, (3) identify variables, (4) formulate the model, (5) solve, (6) interpret, (7) evaluate, (8) refine.
Q2: Formulate (translate context into a mathematical model with justified assumptions); Solve (apply correct mathematics with appropriate technology); Evaluate and verify (assess reasonableness and refine); Communicate (organise the report and use mathematical notation correctly).
Q3: Optimum occurs at $h = 2 r$ (height equals diameter), i.e. $h : r = 2 : 1$ . It is a known textbook result: a quick reasonableness check that the calculation was set up correctly.
Q4: Logistic growth is fastest at the inflection point, where $P = L / 2 = 500$ . Justification: the per-capita rate $\frac{dP}{dt} = k P (1 - P/L)$ is a downward parabola in $P$ , maximised at $P = L/2$ .
Q5: Solve $5 + 2 \sin(\pi t / 6) = 4.5$ , i.e. $\sin(\pi t / 6) = -1/4$ , on $t \in [0, 24)$ .
Q6: Replace the instantaneous-absorption assumption with a first-order absorption phase characterised by rate constant $k_a$ , giving the two-exponential single-compartment model $C(t) = \frac{F D k_a}{V (k_a - k)} (e^{-kt} - e^{-k_a t})$ .
Q7: Example: assume daily traffic flow is periodic with period $24$ hours. Justification: morning and evening commuting peaks repeat each weekday, broadly consistent with sinusoidal behaviour; the model would not capture weekend or holiday patterns.
Q8: Strong structure: (1) Introduction (problem statement and approach), (2) Formulate (variables, assumptions with justifications), (3) Solve (mathematical work with CAS evidence), (4) Evaluate and verify (sensitivity, alternative scenarios, limitations), (5) Refine (modified model, comparison), (6) Conclusion (direct answer with units), (7) Appendix (CAS screenshots, supplementary calculations), (8) References.