QCE Maths Methods IA1 strategy: 2026 guide

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. 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

QCAA's four IA1 criteria:

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

Top band requires excellence in all four.

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?).

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 outline: drug concentration in the bloodstream

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.

Assumptions. The drug is absorbed instantaneously into the bloodstream (immediate); elimination follows first-order kinetics with half-life 2 hours; the volume of distribution is constant.

Variables. Time t (hours). Concentration C(t) (mg/L). Dose 200 mg. Half-life 2 h.

Model. First-order decay between doses: $C(t) = C_0 e^{-kt}$ where $k = \ln 2 / 2 = 0.347$ per hour.

Solve. After the nth dose, just-before-next-dose concentration:

where $r = e^{-6k} = e^{-2.08} = 0.125$.

Steady state: $C_{\infty} = C_0 / (1 - r) = 200 / (1 - 0.125) = 229$ (in mg units in the relevant volume).

Time to reach 95 percent of steady state: $r^n < 0.05$, so $n > \log 0.05 / \log 0.125 = 1.44$. Steady state reached in about 2 dosing intervals (12 hours).

Interpret. Within 12 hours of the first dose, the patient is at near-steady-state concentration.

Evaluate. Real absorption is not instantaneous; the model overstates the peak. The constant elimination assumption is reasonable for paracetamol within therapeutic dose. Volume of distribution varies between individuals (about 0.9 L/kg of body mass).

Refine. Replace instantaneous absorption with a single-compartment model with absorption rate constant $k_a$. Solve the differential equation. Compare predictions.

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.

In one sentence

QCE Maths Methods IA1 rewards a clear problem definition with stated assumptions, a justified model with correct mathematics, an evaluation that identifies limitations, a refinement that addresses one of them, and clear scientific writing throughout the 10-page report.