092i - Formulating Ratio and Rate Models

Lesson 92 — Mathematical Modelling: Formulating Problems with Ratios and Rates

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

Learning Intentions

• To understand the mathematical modelling cycle — formulate, solve, interpret, review — as applied to ratio and rate problems.
• To formulate real-world ratio and rate problems as mathematical expressions or equations, identifying knowns, unknowns and assumptions.

Success Criteria

I can:

1. Describe the stages of the mathematical modelling cycle: formulate, solve, interpret, review.
2. Identify the quantities, units and relationship (ratio or rate) embedded in a real-world scenario.
3. Translate a worded scenario into a mathematical ratio, rate, or equation.
4. State the assumptions I am making when formulating a model.

Warmup

(5 minutes — mini whiteboards, rapid-fire)

For each statement, decide whether it describes a ratio (same units) or a rate (different units), and name the two quantities involved.

1. Concrete is mixed part cement to parts sand.
2. A car uses litres of petrol per km.
3. Paint is mixed in the ratio , blue to white.
4. An exchange rate of AUD USD.
5. A recipe uses g flour for every eggs.

Answers: 1. Ratio (volume:volume). 2. Rate (L:km). 3. Ratio. 4. Rate (this is a borderline, interesting case — worth discussing: both quantities are “money”, but in different currencies, so it behaves as a rate). 5. Rate (mass:count).

Activities

Activity 1 — Explicit Instruction: the Mathematical Modelling Cycle (10 min)

Introduce the 4-stage cycle used throughout this unit: Formulate → Solve → Interpret → Review. Today’s focus is entirely on Formulate: turning a real situation into mathematics before any calculating begins.

I do: “A juice company mixes concentrate and water in the ratio to make ready-to-drink juice. A canteen needs L of ready-to-drink juice for a school event.”

Think aloud: What’s known? Ratio , total volume L. What’s unknown? Litres of concentrate, litres of water. What assumption am I making? That the ratio stays exactly constant regardless of batch size. Formulate:

We do: Formulate (knowns, unknowns, assumption, expression) for: “A currency exchange booth converts AUD to NZD at AUD NZD. A tourist wants to convert 250\text{NZD} = 250 \times 1.08$ before evaluating.

You do: Formulate (do not solve) the following, writing knowns/unknowns/assumption and an expression or equation for each:

(a) A car’s fuel consumption is L per km. How much fuel for a km trip?

(b) Paint colour “Sunset” is mixed red:yellow:white . A painter needs L. How much of each colour?

(c) A recipe for people uses g rice and chicken breasts. How much of each for people?

Activity 2 — Guided Practice: Writing Formal Equations and Naming Assumptions (10 min)

Discuss why assumptions matter — they determine whether a model’s answer can be trusted. For part (a) above, model writing a formal equation with a defined variable: let = fuel required (L).

We do: Together write formal equations, with a defined variable, for parts (b) and (c).

You do: For each of (a), (b), (c), write one sentence stating the key assumption the model relies on (e.g. constant fuel-consumption rate; recipe scales perfectly and evenly, even to fractional chicken breasts).

Activity 3 — Inquiry Task: Formulate, Don’t Solve (14 min)

Pairs. The class fundraiser. You are given the following information:

The Year 8 fundraiser is selling raffle tickets at for 52:3:42.41= 0.66$ USD to know the total raised.

Without solving anything, produce a complete formulation: list every known quantity with its unit, identify each relationship as a ratio or a rate, define variables for each unknown, write an expression or equation for each part of the problem, and state at least one assumption.

Socratic scaffolding:

Prompt	Purpose
Understand: what is the real-world question being asked?	Separate the story from the mathematics — what final quantities must be found?
What quantities are given, and in what units?	List every number with its unit; flag which pairs are ratios (same unit) vs rates (different units).
What is unknown?	Name each unknown and assign it a variable.
Devise a plan: which relationship connects the known and unknown quantities?	Identify whether a ratio, a unit rate, or a combination applies to each part.
Can you write the relationship as an equation or proportion?	Translate words into symbols, e.g. .
What assumption are you relying on?	State explicitly what must stay constant/true for the model to work.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Classify as ratio or rate: “A tank drains at L per minute.”
2. A garden bed mixes topsoil and compost in the ratio . Write an equation to find the compost, , needed for L of mix.
3. State one assumption you would need when formulating a model for “the fuel cost of a km trip, given a car uses L per km.”
4. Reasoning. Explain the difference between formulating and solving a problem, using the fundraiser task as an example.

Answers: 1. Rate (L : min). 2. Total parts ; , giving L. 3. E.g. fuel consumption stays constant across the whole trip, and fuel price per litre doesn’t change. 4. Formulating translates the situation into maths — identifying knowns, unknowns, relationships and assumptions, and writing an expression — without calculating; solving carries out the arithmetic to reach a number.

Common Misconceptions

Misconception	How to pre-empt it
Treating every “X per Y” phrase as a rate without checking units.	Use the concrete/paint ratio examples where “parts” share units to contrast with true rates.
Jumping straight to calculating before identifying knowns/unknowns.	Insist on a written “known / unknown / assumption” list before any arithmetic begins.
Believing a model requires no assumptions.	Require one explicit, written assumption per formulation.
Confusing “one part” of a ratio with a fixed real-world unit.	Clarify that a “part” is a scaled unit found by dividing the total by the sum of the ratio.
Writing an equation without units attached to each quantity.	Require units on every term of a formulated equation.

Enrichment — Competition-Style Problems

E1 (Fluency/formulate-and-solve). A recipe requires flour, sugar and butter in the ratio for a batch of g. Formulate an equation for the mass of sugar needed for a batch totalling g, then solve.

Answer

Total parts ; sugar .

E2 (Kangaroo style). A map has scale . Formulate the relationship between map distance (cm) and real distance (cm), then find the real distance, in km, represented by cm on the map.

Answer

E3 (Challenge). A shop offers “buy , get free” on a 4.5030%3$ items at each shop, then determine which is the better deal.

Answer

Shop 1 is cheaper.

Homework

1. Classify each as ratio or rate: (a) mL cordial to mL water (b) km walked in hours (c) red counters to blue counters (d) a heart rate of beats per minute.
2. Formulate (knowns, unknown, equation) then solve: “A school oval is fertilised using fertiliser and water in the ratio . How much fertiliser is needed for L of mixture?”
3. A currency converter uses the rate AUD GBP. Formulate an equation to convert an amount AUD to GBP, then find the GBP value of 180$.
4. A trail mix combines nuts, dried fruit and seeds in the ratio for a g batch. Formulate equations for the mass of each ingredient, then solve.
5. Reasoning. A model assumes a car’s fuel consumption is a constant L/ km for a trip that includes both highway and mountain driving. Explain why this assumption might make the model inaccurate.
6. Challenge. Plan A charges 0.08$15$0.05mm$ at which both plans cost the same.

Answers: 1(a) ratio (b) rate (c) ratio (d) rate. 2. Total parts ; fertiliser L. 3. ; for , GBP . 4. Total parts ; nuts g, dried fruit g, seeds g. 5. Real fuel use is likely higher than modelled on mountain roads (climbing costs more fuel) and possibly lower on steady highway driving — the constant-rate assumption ignores terrain. 6. Cost; Cost; setting equal gives , so MB.