096i - Modelling with Ratios and Rates Consolidation and Check

Lesson 96 — Problem Solving and Consolidation: Modelling with Ratios and Rates

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

Learning Intentions

• To consolidate the full mathematical modelling cycle — formulate, solve, interpret, review — applied to a ratio or rate problem in a financial context.
• To carry out an independent modelling task from a real-world prompt through to a communicated, reviewed solution.

Success Criteria

I can:

1. Formulate a mathematical model from a real-world ratio or rate scenario.
2. Solve the model accurately, using an appropriate strategy.
3. Interpret the solution in a full sentence, with correct units and sensible rounding.
4. Review the model’s assumptions and refine it where appropriate.

Warmup

(5 minutes — “name the stage”, whole class rapid-fire)

For each statement below, name which stage of the modelling cycle it belongs to: Formulate, Solve, Interpret, or Review.

1. “Let = cost in dollars. Then .”
2. ”.”
3. “The total cost is 49.00600$ MB of data.”
4. “This assumes the flat fee never changes — is that realistic over a -month contract?”

Answers: 1. Formulate. 2. Solve. 3. Interpret. 4. Review.

Teacher note: This warmup is the whole lesson in miniature — today students will move through all four stages independently on a new scenario.

Activities

Activity 1 — Guided Consolidation: a Full Modelling Cycle together (10 min)

Whole class, teacher scaffolds each stage explicitly, moving quickly since all four stages have been taught across Lessons 92–95.

Scenario: “A school is catering a formal dinner for guests. A caterer charges 38$250$8000200$ guests, with no per-guest charge.”

Formulate (I do): Let = cost of Caterer 1 ($), = cost of Caterer 2 ($), = number of guests.

Solve (We do): Together substitute into .

Interpret (We do): Together write the sentence: “For guests, Caterer 1 costs 7090$910$8000$.”

Review (You do, pairs, 3 min): Discuss and note down: what assumption might make this comparison unreliable? (E.g. guest numbers might change closer to the date; Caterer 1’s per-guest rate might not include drinks, while Caterer 2’s flat fee might.)

Activity 2 — Independent Modelling Task: Choose Your Scenario (16 min)

Individually or in pairs. Choose one of the three scenarios below and complete a full four-stage modelling cycle in your workbook, clearly labelling each stage: Formulate, Solve, Interpret, Review.

Scenario A — Best buy. A kg bag of rice costs 5.805$13.509$ kg of rice per month.

Scenario B — Wages. A casual employee earns 26.501843$ Sunday hours.

Scenario C — Currency and travel. A family converts 15001= 0.612%$ service fee on the total AUD amount before converting.

Minimum requirements for each stage:

• Formulate: state knowns, unknowns (with variable names), the relationship (ratio/rate), and at least one assumption.
• Solve: full working, correctly laid out.
• Interpret: a complete sentence with correct units and sensible rounding.
• Review: judge your assumption’s reasonableness, and state one refinement.

Activity 3 — Problem-solving Extension: Comparing Two Models (14 min)

Pairs. For early finishers or as the main task for stronger pairs — a genuine comparison problem requiring all four stages twice.

A gym offers two membership options. Option A: 18$653$ times per week.

Formulate, solve, interpret and review both options, then answer: at what number of monthly visits do the two options cost the same? Which option should the student choose, and why?

Socratic scaffolding:

Prompt	Purpose
Understand: what are the two separate models here?	and , where = visits per month.
Devise a plan: how do you find where two models are equal?	Set and solve for .
Carry it out	.
Interpret	Round appropriately — what does a fractional “visit” mean here?
Looking back — apply to the student’s actual situation	At visits/week visits/month, which option wins, and by how much?

Since visits is the break-even point, and visits/month is far above this, Option B is clearly cheaper for this student: 234C_B = $65$.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Formulate: A taxi charges a 4.20$2.10cn$ km.
2. Solve: Find the cost of a km trip using your equation from Q1.
3. Interpret: Write your answer to Q2 as a full sentence.
4. Review. State one assumption in the taxi model, and judge whether it is reasonable for a trip through heavy traffic.

Answers: 1. . 2. 29.4012$29.40$.” 4. E.g. the model assumes the rate is purely distance-based with no time-based waiting charge — this is unreasonable in heavy traffic, where many real taxi fares include a per-minute waiting charge that this model ignores.

Common Misconceptions

Misconception	How to pre-empt it
Skipping the Formulate stage and jumping straight to a calculation.	Require all four stages to be visibly labelled in every response today.
Treating “Interpret” as just repeating the number with units, without a full sentence answering the original question.	Model the difference between "" and “the cost is 7090$, which answers the caterer comparison question.”
Believing “Review” is optional or just a formality.	Require a stated assumption and a stated refinement, not just “this model might be wrong”.
Comparing two models only at the values given, without checking a break-even point.	Model that “for this many visits/guests/hours, which is cheaper?” is often a more complete answer, when time allows.
Losing track of variable meanings partway through a multi-scenario lesson.	Insist on a defined-variable list at the start of every Formulate stage.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). Two market stalls sell apples. Stall A: 1.20$9.0081010$ apples.

Answer

Options: single apples from Stall A 12.0018$9.00+ 22 \times 1.20 = $2.40= $11.40216$18.00610$11.40$**.

E2 (Kangaroo style). A model states “a photocopier printing at a constant rate of pages per minute will print pages in minutes.” Review this model: is the constant-rate assumption reasonable for a job of this size, and what real-world factor might a reviewer add?

Answer

The assumption is reasonable as a first approximation, but a careful reviewer would add: paper jams, warm-up time, or pauses to refill the paper tray for a -page job — none of which the constant-rate model accounts for. A refined model might add a fixed buffer time, e.g. – extra minutes.

E3 (Challenge). A comparison of two mobile plans finds they cost the same at MB of usage. A reviewer points out that plan usage is rarely known exactly in advance and varies month to month. Explain, using this idea, why a single break-even point is still a useful piece of information even though real usage is uncertain.

Answer

The break-even point acts as a decision rule: a user can estimate whether their typical usage is reliably above or below MB and choose accordingly, even without knowing the exact figure each month. If usage is close to MB and highly variable, the review might recommend the plan with lower risk (e.g. the flat-fee plan, to avoid unpredictable high bills) rather than the one with the lower expected cost.

Homework

1. Formulate: A market stall sells honey at 14500$321.2100$ g) formula for each jar size.
2. Solve: Calculate both unit prices from Q1 and state which jar is the better buy.
3. Interpret: Write your Q2 answer as a full recommendation sentence.
4. Review: State one assumption behind the Q1 model (e.g. about jar quality or use-by dates) and judge its reasonableness.
5. A full modelling task: a family is choosing between two electricity plans. Plan X: c per kWh, no daily charge. Plan Y: c per day plus c per kWh. Complete all four stages (formulate, solve for a household using kWh/day, interpret, review) in your workbook.
6. Reasoning. Explain why the “review” stage of a model is especially important in financial contexts, where a poor assumption could lead to a costly real decision.
7. Challenge. A comparison of two paid options has one with a flat fee and one with a purely usage-based rate. Prove algebraically that there is always exactly one break-even usage value (assuming the usage-based rate is higher than the per-unit rate of the flat-fee option), by setting up and solving against in general terms, where .

Answers: 1. g jar: per g 2.801.2\dfrac{32}{12}100\approx$2.671.2131001.2$2.67100$2.80100500C_X = 0.28\times18=$5.04C_Y=0.95+0.22\times18=$4.91a+bn=cn \Rightarrow a = (c-b)n \Rightarrow n = \dfrac{a}{c-b}c>bn$.