Mathematical Modelling with Ratios and Rates — AC9M8M07

• Formulate practical problems using ratios or rates.
• Choose tables, double number lines, equations or graphs strategically.
• Solve constant-speed and financial rate problems.
• Interpret solutions and review model assumptions/limits.

Recall equivalent ratios, unit rates, percentages, conversion factors, distance=speed×time and how units support interpretation.

A modelling cycle is more than calculation: formulate → represent → solve → interpret → review. State what quantities vary, their units, and whether a proportional/constant-rate assumption is justified.

Equivalent ratios multiply or divide every part by the same non-zero factor. Rate models may be direct (e.g. constant speed) or include fees/brackets that break simple proportionality. If a phone plan has a fixed fee, “cost per unit” alone may not predict total cost for every usage level.

Preserve the authored elaborations: map/plan scales and mixtures, magnification, exchange-rate travel budgeting, taxation/pay/phone-plan comparisons, radiocarbon ratio contexts used to study long-term human habitation, and First Nations string/cordage ratio applications. Present cultural contexts respectfully and as mathematical applications, not decorative references.

• Adding the same number to ratio terms. Equivalent ratios require multiplicative scaling.
• Assuming every rate model passes through zero. Fixed fees or thresholds create non-proportional models.
• Ignoring units/fees. A model can be numerically correct but contextually incomplete.
• Not reviewing the assumption. Constant speed, exchange rate or material strength may vary in reality.

1. Scale 3:5 to an equivalent ratio with total 64.
2. Use a 1:25,000 map scale to convert 6 cm to real distance.
3. Find distance travelled at 80 km/h for 1.75 h.
4. State one assumption in a currency-conversion model.

1. Divide 96 in the ratio 2:3:7.
2. A recipe ratio flour:sugar=5:2 uses 750 g flour. Find sugar.
3. A journey of 315 km is travelled at constant 90 km/h. Find time.
4. Convert AUD 480 at a stated exchange rate of 0.61 foreign units per AUD.
5. Compare two phone plans with a fixed fee plus per-unit rate.
6. Explain why a tax-bracket model is not a single constant tax rate over all income.
7. Describe what would make a rope-strength ratio model unsuitable outside its measured range.

A travel budget uses today’s exchange rate with no fees and assumes a constant daily spending rate. Explain which parts are proportional, which assumptions could fail, and how you would build a more realistic range of possible costs without using senior mathematics.

1. What is the modelling cycle? Formulate, represent, solve, interpret and review.
2. How are equivalent ratios formed? Multiply/divide every term by the same non-zero factor.
3. When is d=vt appropriate? When speed v is treated as constant over time t.
4. Why review a rate model? Real situations may include fixed fees, changing rates, limits or uncertainty.

1. A map uses scale 1:80,000. Two points are 9.5 cm apart. Find real distance in km.
   80,000 cm=0.8 km per map centimetre.
2. Plan A costs $18+$0.09x and Plan B costs $30+$0.05x. Find break-even usage and interpret which plan is better below/above it.
   Solve equal-cost equation, then interpret the two regions.
3. A car model assumes 75 km/h constant speed for a 260 km trip. Predict travel time and evaluate one limitation.
   Use t=d/v, then discuss stops, traffic or speed variation.

• I formulate ratio/rate models.
• I choose useful representations.
• I solve and keep units meaningful.
• I review assumptions and limitations.

Exit ticket: Why can two mathematically correct rate calculations support different decisions once fixed fees are included?

Australian Curriculum v9.0 — AC9M8M07: use mathematical modelling to solve practical problems involving ratios and rates, including distance-time problems for travel at a constant speed and financial contexts; formulate problems; interpret and communicate solutions in terms of the situation, reviewing the appropriateness of the model.

Victorian Curriculum F–10 Version 2.0 — Level 8, VC2M8M07: Exact.

NSW Mathematics K–10 Syllabus (2022) — Stage 4, MA4-RAT-C-01; MA4-FRC-C-01; MAO-WM-01: Partial. NSW distributes ratio/rate, distance-time, percentage/financial and Working mathematically content across outcomes rather than mirroring the national modelling descriptor one-to-one.

• AC9M8M05 — rates and unit rates
• AC9M8N05 — percentage/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 practical problems involving ratios and rates...

Mapped skill: use mathematical modelling to solve practical problems involving ratios and rates, including financial contexts; formulate problems; 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.