095i - Reviewing Ratio and Rate Models

Lesson 95 — Mathematical Modelling: Reviewing the Appropriateness of a Model

Strand: Measurement | Descriptor: AC9M8M07 | Duration: 45 minutes

Learning Intentions

• To review a ratio or rate model by testing its assumptions against real-world conditions.
• To refine a model when its assumptions prove unreasonable, and explain the effect of the refinement.

Success Criteria

I can:

1. Identify the assumptions underlying a ratio or rate model.
2. Judge whether an assumption is reasonable, and explain why or why not.
3. Refine a model by adjusting or replacing an unreasonable assumption.
4. Compare the original and refined models’ solutions, and explain the difference.

Warmup

(5 minutes — “reasonable or not?”, pairs)

For each modelling assumption, decide whether it is reasonable or questionable for the situation described, and justify briefly.

1. “A car’s fuel consumption is constant at L/ km” — for a trip entirely on a flat highway.
2. “A car’s fuel consumption is constant at L/ km” — for a trip through mountains with a loaded trailer.
3. “The exchange rate stays fixed at AUD USD” — for a transfer completed within the next minutes.
4. “The exchange rate stays fixed at AUD USD” — for a transfer completed over the next months.

Answers: 1. Reasonable — steady conditions match the constant-rate assumption well. 2. Questionable — hills and load both increase fuel use above the flat-road average. 3. Reasonable — exchange rates barely move in minutes. 4. Questionable — exchange rates fluctuate meaningfully over months.

Discussion: The same assumption can be reasonable in one context and unreasonable in another — reviewing a model means checking assumptions against the specific situation, not judging them in the abstract.

Activities

Activity 1 — Explicit Instruction: what “reviewing a model” Means (10 min)

Recap the modelling cycle: Formulate → Solve → Interpret → Review. Today’s focus is Review: checking whether the model’s assumptions hold, whether the solution is sensible, and refining the model if not.

Three review questions to ask of any model:

1. Are the assumptions realistic for this specific situation?
2. Is the solution sensible — does the size and sign of the answer make sense?
3. What would happen to the answer if an assumption changed?

I do: “A model assumes a constant fuel-consumption rate of L/ km for a km trip that is entirely highway driving.” Review:

Assumption check: reasonable — highway driving is close to steady-state, matching manufacturer test conditions. Sensibility check: model predicts L for the trip; a typical car’s tank is – L, so this is plausible. Sensitivity check: if the real rate were L/ km instead (a km mountain leg), the model would underestimate fuel use by L — worth adding a safety margin for planning.

We do: Together review: “A model assumes a recipe scales perfectly and evenly, even to fractional chicken breasts, for people using chicken breasts.” Discuss: is “fractional chicken breasts” realistic? What refinement would a real cook make? (Round up to breasts, and note the model now slightly over-provides — is that a problem, or a sensible buffer?)

You do: For each model, state (i) whether the key assumption is reasonable for the stated situation, (ii) one refinement that would improve the model:

(a) “A currency conversion model uses today’s rate to estimate the AUD value of a donation platform’s USD pledges, which will actually be received and converted in 3 months.”

(b) “A concrete-mixing model assumes cement, sand and water combine in an exact ratio with no wastage, for a large outdoor pour in windy conditions.”

(c) “A best-buy model compares unit price only, ignoring that the cheaper option requires a -minute extra drive to a different shop.”

Activity 2 — Guided Practice: Refining a Model and Comparing Solutions (10 min)

I do: Refine the fuel model from Activity 1. Original model: L. Refined model, accounting for a increase due to mixed terrain: L.

State the review conclusion: “The original model likely underestimates fuel needed by around L on this route; a driver should budget for the refined estimate.”

We do: Together refine the best-buy model from (c): if the extra drive costs 0.3520$40$1.20$, discuss and calculate whether the “cheaper” shop is still the better overall choice once the drive cost is included.

The refined model shows a smaller but still positive net saving of 0.8520$0.85$, which is a judgement beyond the mathematics.

Activity 3 — Inquiry Task: Reviewing the Fundraiser Model (14 min)

Pairs. Return to the fundraiser model (Lessons 92–94): raffle 35.00$36.00$488.40$559.40$.

Conduct a full review:

1. List every assumption made across the three parts of the model (raffle, bake sale, currency conversion).
2. For each assumption, judge whether it is reasonable, and explain why or why not, referencing the specific fundraiser context (e.g. is a fixed exchange rate reasonable if pledges are converted immediately versus over several weeks of a campaign?).
3. Choose the one assumption you judge weakest, and refine the model to account for it — recalculate the affected part.
4. Write a short concluding statement: by how much (in dollars, and as a percentage) does your refinement change the total raised?

Socratic scaffolding:

Prompt	Purpose
Understand: what assumption are you testing?	Name it precisely before judging it.
What do you know about how this quantity actually behaves in reality?	E.g. exchange rates fluctuate day to day; bake sale trays rarely sell out exactly evenly.
Devise a plan: how would you adjust the formula to reflect reality more closely?	E.g. apply a range of exchange rates instead of one fixed value, or model of the tray selling rather than .
Carry it out	Recalculate the affected part with the refined assumption.
Looking back	Compare original and refined totals — is the change large enough to matter for a real report to the principal?

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A model assumes ” hour of tutoring costs exactly 45$, with no minimum booking.” State one reason this assumption might be unrealistic in practice.
2. A model predicts L of paint needed, based on a constant coverage rate of per litre. If the real wall surface is rough (reducing coverage to per litre), recalculate the refined paint requirement. (Total area .)
3. Explain, in one sentence, the difference between interpreting a solution and reviewing a model.
4. Reasoning. A currency conversion model used a fixed exchange rate for pledges collected over a -month campaign. Explain why this is a reasonable simplification for a Year 8 maths model, even though it is not perfectly realistic.

Answers: 1. E.g. many tutoring services require a minimum -hour booking, so the “no minimum” assumption may not hold. 2. L, so about L is needed — roughly L more than the original estimate. 3. Interpreting translates a solved number into a meaningful real-world statement; reviewing checks whether the model’s assumptions and solution are actually appropriate for the situation, and refines them if not. 4. A single fixed rate keeps the model manageable and gives a reasonable estimate, since exchange rate movements over 2 months are usually small relative to the total — full realism (a fluctuating rate) would require calculus/statistics tools beyond this course, so a simplifying assumption is an appropriate modelling choice, as long as it is stated.

Common Misconceptions

Misconception	How to pre-empt it
Believing a model is either “right” or “wrong” rather than more or less appropriate.	Frame review as a spectrum: models are judged by how well their assumptions fit the specific situation, not by a pass/fail standard.
Refining a model by guessing a new number with no justification.	Require every refinement to state why the new value/assumption is more realistic.
Reviewing only the arithmetic, not the assumptions.	Separate “is the calculation correct?” (solving) from “is the model appropriate?” (reviewing) explicitly.
Assuming more complex models are always better.	Discuss that a simple model can be entirely appropriate if its assumptions are reasonable for the context — added complexity is only useful if it improves realism meaningfully.
Refining every assumption at once, losing track of which change caused which effect.	Insist on changing and testing one assumption at a time.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A model estimates delivery cost as a flat 8$210$1514$ kg parcel under both the original and refined models, and state the difference.

Answer

The refined model gives 15}$ more, entirely due to the handling fee the original model omitted.

E2 (Kangaroo style). A model assumes a train travels at a constant km/h for an entire km journey, giving a travel time of hours. A reviewer notes the train actually stops for minutes at each of intermediate stations. Find the refined total journey time.

Answer

3 hours 12 minutes, minutes longer than the original model predicted.

E3 (Challenge). A model for the cost of tiling a floor assumes tiles can be cut with zero wastage. A reviewer argues real tiling always wastes some material at the edges, typically extra tiles. If the original model estimates tiles are needed, find the refined estimate, and explain why the refined number must be rounded in a particular direction.

Answer

81 tiles — this happens to be a whole number already, but in general the result must be rounded up, since a tiler cannot purchase a fractional tile, and under-ordering would halt the job.

Homework

1. State one assumption underlying the model “wages 24 \times$ hours worked”, and judge whether it is reasonable for a casual retail job with penalty rates on Sundays.
2. A model predicts a 40158$ minutes. Recalculate the refined total time.
3. A concrete model assumes cement:sand:water with no wastage for a kg pour. A reviewer suggests a realistic wastage allowance. Recalculate the refined total mass of materials needed.
4. Explain, using the exchange-rate example, why “the assumption is unrealistic” is not by itself a strong enough review — a good review must also state what to do about it.
5. Reasoning. A model for classroom furniture cost assumes every chair costs exactly the listed price, ignoring bulk discounts. A school orders chairs at 3510%30$ or more. Compare the original and refined total costs, and explain the size of the difference.
6. Challenge. A model estimates the time to photocopy pages at a constant rate of pages per minute, giving minutes. A reviewer notes the photocopier jams on average once every pages, costing minutes per jam to clear. Refine the model and find the total realistic time.

Answers: 1. The assumption ignores Sunday penalty rates, which are common in retail awards — it is only reasonable if no Sunday shifts are worked. 2. minutes. 3. Total original mass — reinterpreting: original ratio parts sum to , so cement kg, sand kg, water kg; with wastage, multiply each by : cement kg, sand kg, water kg. 4. A review must recommend a specific refinement (e.g. use a rate range, or add a buffer), not just flag the issue, otherwise the model cannot actually be improved. 5. Original: 14001400\times0.90=$1260=$14010%\approx500\div150\approx3.33\to3=3\times2=6\approx12.5+6=18.5$ minutes.