Year 9 Mathematics · AC9M9A05

Linear and Quadratic Mathematical Modelling

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

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Learning goalsSay it simply

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
Key conceptTeach from the board

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.

Worked examplesWe do

Worked examples

AC9M9A05 - Linear and Quadratic Mathematical Modelling
Example 1

Linear cost A hire cost of $25 plus $8 per hour is C=25+8h.

Example 2

Simple interest For principal P, annual simple rate r and t years, A=P(1+rt).

Example 3

Quadratic area Sides x+2 and x+5 give A=x²+7x+10.

Example 4

Limitation A short-term linear population model can become unrealistic when resource limits matter.

Curriculum examplesCopied content

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.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9A05VC2M9A06 — Level 9 AlgebraStage 5 Core — Number and finance / Linear and non-linear relationships; Paths — Functions and graphs / Variation and rates of change
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
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.
Practice and reviewReady for practice
  • 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. [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.

Curriculum alignmentStart here

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
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