Linear
Constant first difference. A fixed amount is added or removed per equal interval.
What you will learn
- formulate variables, assumptions and constraints for an applied problem
- recognise linear, quadratic and exponential patterns
- choose a model that matches the type of change
- model compound growth and decay, including financial contexts
- interpret a mathematical solution using the original units and situation
- evaluate a model's assumptions, useful range and limitations
- modify or reject a model when evidence does not support it
Prerequisite knowledge
You should be able to calculate percentages, substitute into formulas, use powers, read tables and graphs, recognise linear patterns and work with exponential relations. Familiarity with simple quadratic graphs is helpful when comparing model types.
Concept teaching: the modelling cycle
1. Formulate the problem
Define the quantities, choose variables, identify what is known and state assumptions. For example, a savings model may assume a fixed annual rate and no deposits or withdrawals. Those assumptions are part of the mathematics, not optional commentary.
2. Choose a model from the pattern of change
Quadratic
Constant second difference in equally spaced table values. The rate of change itself changes steadily.
Exponential
Constant ratio or constant percentage change. Each new value depends multiplicatively on the previous value.
Do not choose by appearance alone
Use the situation, table, equation and graph together. Several models may look similar over a short interval.
3. Solve and interpret
Perform the calculation, then translate the answer back into context. Include units, time scale and any whole-number restrictions. A value such as t=4.6 may mean “during the fifth year” if the question asks for the first whole year a threshold is exceeded.
4. Evaluate the model
Ask whether the assumptions remain sensible, whether the prediction is interpolation or extrapolation, how far the model is being extended beyond observed data, and whether real constraints such as capacity, price changes or resource limits have been ignored.
5. Modify and report
If new evidence contradicts the model, change the assumptions, parameters or model family. A strong modelling response reports assumptions, method, result and limitations—not just the final number.
Worked examples
Example 1 — identify a linear model
Values 12, 17, 22, 27 have constant difference +5. A linear rule is appropriate for this pattern.
Example 2 — identify a quadratic model
Values 2, 7, 16, 29 have first differences 5, 9, 13 and constant second difference 4. This is evidence of a quadratic relation.
Example 3 — identify an exponential model
Values 80, 96, 115.2, 138.24 multiply by 1.2. This represents 20% growth per interval, so an exponential model is appropriate.
Example 4 — compound interest
$5000 earns 4% p.a. compounded annually for 3 years. A=5000(1.04)³=$5624.32.
Interpretation: under the stated fixed-rate/no-transaction assumptions, the balance after 3 annual compounding periods is $5624.32.
Example 5 — depreciation
A $24,000 asset loses 15% of its current value each year. It retains 85%, so V=24000(0.85)ᵗ. After 2 years: $17,340.
Example 6 — compare linear and exponential change
Start at 100. Model L adds 20 per year: L=100+20t. Model E grows 20% per year: E=100(1.2)ᵗ. At t=5, L=200 while E≈248.83. Equal-looking early change does not imply equal long-term behaviour.
Example 7 — threshold and whole periods
P=200(1.15)ᵗ. At t=4, P≈349.8; at t=5, P≈402.3. If the question asks when P first exceeds 400 at a whole period, the answer is 5 periods.
Example 8 — half-life model
If a quantity halves every 10 years, Q=Q₀(1/2)^(t/10). After 30 years, three half-lives have passed, so Q=Q₀/8.
Example 9 — model selection from data
A table has outputs 4, 9, 16, 25 for consecutive inputs 1,2,3,4. First differences are 5,7,9 and second differences are 2, so a quadratic model is indicated. Calling it exponential merely because growth accelerates would be incorrect.
Example 10 — critique extrapolation
A wildlife model P=600(1.08)ᵗ assumes 8% growth indefinitely. Habitat capacity, food, disease and management can change the rate. The model may be useful over an evidence-supported interval but unreliable far beyond it.
Common misconceptions and corrections
- “A curved graph means exponential.” Use differences, ratios and context to choose the family.
- “6% growth means multiply by 0.06.” Growth multiplier is 1.06; 0.06 represents only the increase.
- “The model is the real world.” A model is a simplified representation built on assumptions.
- “A good fit now guarantees a good long-term forecast.” Extrapolation can fail when conditions change.
- “A calculation alone completes a modelling question.” Interpretation, evaluation and communication are part of the descriptor.
- “All solutions make sense.” Context may rule out negative time, fractional objects or values outside a realistic domain.
Guided practice
- Classify 5, 8, 11, 14.
- Classify 3, 12, 27, 48 using differences.
- Classify 50, 60, 72, 86.4.
- Calculate $4000 invested at 5% p.a. compounded annually for 2 years.
- A quantity starts at 800 and decays by 12% yearly. Write the model and find its value after 3 years.
Check guided answers
- Linear; constant difference 3.
- Quadratic; first differences 9,15,21 and second difference 6.
- Exponential; ratio 1.2.
- $4410.
- Q=800(0.88)ᵗ; Q(3)≈545.18.
Independent practice
- Decide whether 6, 11, 18, 27 is linear, quadratic or exponential. Justify.
- A town has population 18,000 and grows 2.5% per year. Write an exponential model.
- A laptop worth $1800 depreciates 22% per year. Estimate its value after 3 years.
- Compare S=1000+350t and C=1000(1.25)ᵗ at t=1,2,3,4.
- Find the first whole year when 600(1.09)ᵗ exceeds 1000.
- A model predicts a plant height H=12+4t indefinitely. Give two reasons it should not be extrapolated indefinitely.
- A data set has constant second differences. Explain why a quadratic model is more defensible than an exponential model.
- Write one assumption that should accompany a fixed-rate compound-interest model.
Reasoning and problem-solving task
A new subscription service records 800 subscribers at launch, 960 after one month, 1152 after two months and 1382 after three months (rounded).
- Investigate whether a 20% exponential-growth model is plausible.
- Use your model to predict month 6.
- Explain how rounding in the recorded data affects your confidence in the exact ratio.
- Give at least three real-world reasons the 20% growth rate might not persist.
- Design one piece of new data you would want before using this model for a 2-year business forecast.
A high-quality response separates mathematical fit from real-world validity.
Important questions and answers
- What makes a model linear?
- A constant additive change over equal input intervals.
- What makes a model exponential?
- A constant multiplicative ratio or percentage change over equal intervals.
- How can a table suggest a quadratic model?
- For equally spaced inputs, constant second differences are a key indicator.
- What is extrapolation?
- Using a model beyond the range of observed data. It usually carries more risk than interpolation.
- Why state assumptions?
- They define the conditions under which the model's reasoning is intended to apply.
- When should a model be modified?
- When evidence, constraints or changing conditions show its assumptions or predictions are no longer adequate.
Assessment-style questions
- 3 marks: Determine whether 40, 52, 67.6, 87.88 represents linear or exponential change. Justify numerically.
- 4 marks: A car is purchased for $38,000 and depreciates 18% per year. Write a model and estimate its value after 4 years.
- 5 marks: A savings account starts with $6500 and earns 4.6% p.a. compounded annually. Find the balance after 5 years and state two assumptions of the model.
- 6 marks: Compare a linear model P=1200+180t with an exponential model Q=1200(1.12)ᵗ. Identify a time when their predictions meaningfully diverge and explain why.
- 7 marks: A data set appears exponential over 6 observations. Describe a modelling process to fit, test, interpret and evaluate the model before using it to forecast 20 intervals ahead.
Review hint: In a modelling question, structure your response as assumptions → model → calculation → interpretation → evaluation. That prevents the common mistake of stopping at a calculator answer.
Exit ticket: mastery check
- What pattern suggests a linear model?
- What pattern suggests a quadratic model?
- What pattern suggests an exponential model?
- Write the multiplier for 13% decay.
- Give one reason a mathematically accurate fitted model may still be a poor long-term predictor.
Teacher and parent guidance
For teachers
Assess the full modelling cycle, not only substitution into formulas. Give students data where more than one model looks plausible over a short range and ask them to defend a choice. Financial modelling is valuable because assumptions are concrete; population/environmental contexts are valuable for discussing limits and extrapolation. Use authentic data only when its source and units are clear.
For parents and carers
After a student calculates a prediction, ask three questions: “What does that number mean?”, “What did your model assume?” and “Would you trust it far into the future?” Those questions target the Year 10 modelling skill more directly than doing extra arithmetic.
Curriculum alignment
Australian Curriculum: AC9M10A04, Year 10 Algebra — mathematical modelling of growth and decay, including financial contexts, with selection of linear, quadratic or exponential models and evaluation/reporting of assumptions, methods and findings.
Victoria: VC2M10A15, Level 10 Algebra is a very strong alignment. It covers modelling inverse proportion, growth and decay including financial contexts, choosing linear/quadratic/exponential models, interpreting solutions and evaluating/modifying models. The Victorian descriptor contains some additional inverse-proportion scope not required by this Australian lesson.
NSW: Stage 5 distributes the lesson across outcomes rather than one equivalent. MA5-FIN-C-02 covers compound interest and depreciation. MA5-NLI-C-01 and MA5-NLI-C-02 support algebraic/graphical exponential relationships and curve features. MAO-WM-01 supports reasoning, technique choice and communication. Relevant Path content can extend the modelling where a school's program requires it.
Alignment explanation: Financial examples directly support the Australian modelling descriptor, Victorian A15 and NSW finance outcome. Model selection and growth/decay comparisons align strongly with Australian/Victorian expectations and NSW non-linear relationships. The evaluation sections deliberately teach assumptions and limitations because modelling is not complete when the numerical calculation ends.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Choose linear/quadratic/exponential model | AC9M10A04 | VC2M10A15 | MA5-NLI-C-01/C-02 supporting |
| Compound interest/depreciation | AC9M10A04 | VC2M10A15 | MA5-FIN-C-02 |
| Growth and decay | AC9M10A04 | VC2M10A15 | MA5-NLI-C-01/C-02 |
| Interpret/evaluate/modify/report | AC9M10A04 + proficiency | VC2M10A15 | MAO-WM-01 + relevant content outcomes |
Practice and teaching resources
Official curriculum references
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Exponential growth and decay word problems
Khan Academy — Interpreting a repeated percentage change as an exponential model.
As you watch: How does the multiplication factor show growth or decay?
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Mathematical modelling to solve applied problems involving growth and decay...
Mapped skill: use mathematical modelling to solve applied problems involving growth and decay, including financial contexts; formulate problems, choosing to apply linear, quadratic or exponential models; interpret solutions in terms of the situation; evaluate and modify models as necessary and report assumptions, 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.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10A04 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10A15 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-FIN-C-02 + MA5-NLI-C-01 + MA5-NLI-C-02 + MAO-WM-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
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.
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Topic reference: AC9M10A04 — Growth, Decay and Mathematical Modelling — AC9M10A04
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