Mathematical Modelling with Linear Relations — AC9M8A03

• Identify starting values and constant rates in contexts.
• Form linear rules and choose tables, graphs or equations strategically.
• Interpret solutions using units and contextual meaning.
• Review assumptions, domains and limitations of linear models.

Recall rates, coordinates, substitution, linear graphs, percentages and the meaning of a fixed amount plus a per-unit amount.

A common linear model has the structure output = rate × input + starting value. The rate describes constant change; the starting value is the output when the input is zero, if zero is meaningful in the situation.

A modelling cycle is: formulate the question and assumptions → choose variables and representation → calculate → interpret with units → review whether the answer and model are reasonable.

The Australian descriptor includes financial contexts, pay rates, fares, motion, trade quotes and other applied situations. A correct equation is not enough: a model can become invalid outside its meaningful domain.

• The intercept is just a graph feature. In context it often represents a fixed or initial amount.
• A straight-line rule can be extrapolated forever. Real constraints can limit its domain.
• Units do not matter after calculation. Units are essential to interpretation.
• A correct calculation proves the model is good. Assumptions and appropriateness must also be reviewed.

1. Write a model for $8 entry plus $3 per ride.
2. Interpret the rate and intercept in C=2.4d+6.
3. Choose a table, graph or equation to compare two mobile plans and explain why.
4. State one domain restriction for a tank-draining model.

1. Form a rule for a $20 call-out plus $65 per hour.
2. Calculate the cost at 3.5 hours.
3. Compare C=15+4n and C=31+2n and find the break-even n.
4. Explain the meaning of 15 and 4 in the first rule.
5. Give a reason a pay-rate model may have a restricted domain.
6. Choose the best representation for showing when two plans cross.
7. Critique a model that predicts a negative quantity after a resource is exhausted.

A delivery model C=8+1.80d is advertised only for 0≤d≤40 km. A student substitutes d=75 and reports the result as a valid quote. Explain the mathematics, the modelling error and how the answer should be communicated.

1. What does the coefficient represent? A constant rate of change per unit of input.
2. What does the intercept represent? The initial/fixed output when input is zero, where meaningful.
3. Why choose different representations? Equations support exact calculation, tables selected cases and graphs comparison/trends.
4. When is a linear model inappropriate? When rate, assumptions or domain no longer match the real situation.

1. A taxi fare is $5 flag fall plus $2.40/km. Form a model and calculate 18 km.
   Identify fixed fee and per-kilometre rate before substituting.
2. Plans P=18+3n and Q=30+2n are offered. Find and interpret the break-even point.
   Equate the rules, then explain the result in context.
3. A tank model is V=600−30t. State its meaningful domain and explain why extrapolation is limited.
   Find when volume reaches zero and connect the restriction to the physical situation.

• I identify rate and starting value.
• I form and interpret linear rules.
• I choose representations for a purpose.
• I review assumptions and domains.

Exit ticket: Why can a mathematically correct linear prediction still be a poor real-world answer?

Australian Curriculum v9.0 — AC9M8A03: use mathematical modelling to solve applied problems involving linear relations, including financial contexts; formulate, represent, interpret, communicate and review.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8A03: Exact relationship.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-LIN-C-01; MAO-WM-01: Partial. NSW supports linear relationships and mathematical reasoning, but does not mirror the full national modelling cycle and financial-context descriptor as one Stage 4 outcome.

• AC9M8A02 — linear relations and equations
• AC9M8A04 — experimenting with linear functions
• AC9M8N05 — percentage and financial modelling

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

Curriculum equivalents for Mathematical modelling to solve applied problems involving linear relations, including...

Mapped skill: use mathematical modelling to solve applied problems involving linear relations, including financial contexts; formulate problems with linear functions, choosing a representation; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model

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.