Linear and Quadratic Mathematical Modelling — AC9M9A05

Mathematical modelling turns a real situation into a linear or quadratic relationship, solves it, interprets the result and checks whether the model is reasonable.

By the end of this lesson, you should be able to:

• formulate applied problems using variables and assumptions
• choose between linear and quadratic models
• solve and interpret outputs including financial contexts
• evaluate assumptions, limitations and reasonableness

Recall linear graphs, quadratics, substitution, simple-interest ideas, units and contextual interpretation.

Start with the situation, not the formula

Define variables, units and assumptions before choosing a function.

Linear means constant rate of change

A model y=a+bx has constant first differences and a straight graph. This suits many cost, distance and simple-interest contexts.

Quadratic change has changing rate

Area and products of changing lengths may be quadratic; tables can reveal constant second differences.

Modelling includes evaluation

Check domain, units, assumptions, unrealistic extrapolation and whether another model would be more suitable.

• Choosing a quadratic because a tiny graph window looks curved: Use structure and context.
• Stopping at an algebraic solution: Interpret with units and context.
• Treating assumptions as mistakes: Assumptions simplify reality; state and evaluate them.
• Extrapolating indefinitely: Models have domains where they are reasonable.

1. Create a linear model for a plan with a fixed fee and per-use charge.
2. Use simple interest for $2000 at 4.5% p.a. for 3 years.
3. A rectangle has width x and length x+6. Form its area model.
4. For each model, state one assumption and a realistic domain.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Model a taxi fare with $5.20 flagfall and $2.10 per kilometre.
2. Compare two linear phone plans and find break-even usage.
3. Model simple interest on $3500 at 3.8% p.a. for t years.
4. A rectangle has sides x and x+4; form its area model.
5. Choose linear or quadratic for a supplied table and justify.
6. Explain why a negative time solution can be algebraically valid but contextually impossible.

Two different models fit the first three data points almost equally well. Explain what additional evidence you would collect before choosing, and why fit is not the only criterion.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

What is the first modelling step?

Define variables, relationships and assumptions.

How do I recognise a linear model?

It has constant rate of change.

What makes a model useful?

It fits the relevant context well enough for its purpose and has stated limitations.

1. [6 marks] Form and solve a linear financial model, interpret the result and state two assumptions.
2. [7 marks] Choose a linear or quadratic model for a dataset, justify using structure and evaluate one limitation.
3. [8 marks] Build a quadratic area model, solve a stated condition and decide which mathematical solutions are realistic.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

• I can define variables and units before modelling.
• I can distinguish constant-rate linear change from quadratic change.
• I can interpret solutions in context.
• I can reject impossible contextual solutions.
• I can state assumptions and limitations.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

Australian Curriculum v9.0 — AC9M9A05: use mathematical modelling to solve applied problems involving change including financial contexts; formulate problems, choosing to use either linear or quadratic functions; interpret solutions in terms of the situation; evaluate the model and report methods and findings

Victoria: VC2M9A06 — Level 9 Algebra. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Core — Number and finance / Linear and non-linear relationships; Paths — Functions and graphs / Variation and rates of change. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9A04 — Quadratic Functions, Graphs and Equations
• AC9M9M05 — Direct Proportion, Rates, Ratio and Scale Modelling

• Australian Curriculum v9.0 — Mathematics
• Victorian Curriculum F–10 Version 2.0 — Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Mathematical modelling to solve applied problems involving change including financial...

Mapped skill: use mathematical modelling to solve applied problems involving change including financial contexts; formulate problems, choosing to use either linear or quadratic functions; interpret solutions in terms of the situation; evaluate the model and report methods and findings

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.