074i - Formulating Ratio Models

Lesson 74 — Mathematical Modelling: Formulating Ratio Problems

Strand: Measurement | Descriptor: AC9M7M06 | Duration: 45 minutes

Learning Intentions

• To understand the mathematical modelling cycle.
• To formulate a real situation as a ratio problem, making assumptions explicit.

Success Criteria

I can:

1. Describe the four stages of the modelling cycle.
2. Extract the mathematically relevant information from a messy situation.
3. State the assumptions my model makes.
4. Choose a ratio representation that fits the situation.

Warmup

(6 minutes — what’s missing? pairs)

“Sam is making pancakes for a class breakfast.”

This is a real situation, but not yet a maths problem.

1. What questions would you need answered before you could calculate anything?
2. List at least four.

Expected questions: How many people? How many pancakes each? What does the recipe make, and with what ingredients? What ingredients are already available? Any dietary variations?

The point: real situations arrive incomplete. Turning one into a solvable problem — deciding what matters, what to assume, and what to represent mathematically — is called formulating, and it is the first stage of modelling.

Activities

Activity 1 — Explicit Instruction: the Modelling Cycle (12 min)

The four stages — display for the whole unit:

with an arrow looping back from Interpret to Formulate if the answer does not fit the real situation.

Stage	What happens	Key question
Formulate	Choose what matters, make assumptions, represent mathematically	What am I assuming?
Solve	Do the mathematics	Is my working correct?
Interpret	Translate the answer back into the situation	Does this make sense here?
Communicate	Report the answer, the method and its limits	Could someone act on this?

Worked demonstration — formulating only. “A school fete stall sells fruit cups. Last year, for every cups of watermelon they used, they used cups of pineapple, and they served about customers. This year they expect half as many again.”

Model the formulation aloud:

• Relevant: ratio , customer count , growth “half as many again” ().
• Irrelevant (for now): prices, weather, cup sizes — unless the question asks.
• Assumptions: each customer buys one cup; this year’s mix should taste the same (ratio preserved); last year’s quantities were right.
• Representation: expected customers ; fruit in ratio scaled to serves.

Emphasise: none of this is calculation yet. Formulating is deciding what to calculate.

We do — formulate (do not solve): “A youth club is planning a bus trip. Buses seat . Last time, about of the members came.”

• Relevant: seat capacity, expected attendance.
• Assumptions: same attendance fraction applies; everyone needs a seat; no other transport.
• Representation: expected attendance ; buses , rounded up.

Activity 2 — Formulation Practice (14 min)

Pairs. For each scenario, produce a formulation card: relevant facts, assumptions, representation. Solving is quick once formulated — allow it, but mark the formulation.

Scenario 1 — The mural. An art class will paint a mural using green paint mixed from blue and yellow in ratio . The wall is about and L of paint covers roughly . Paint is sold in L tins.

Scenario 2 — The fundraiser trail mix. Students will sell bags of trail mix made from nuts, sultanas and pretzels in ratio . Each bag holds g, and they hope to sell bags.

Scenario 3 — The sports day drink station. Sports drink concentrate is diluted with water. Each runner drinks about mL, and runners are expected.

Socratic scaffolding for Scenario 3:

Prompt	Purpose
What is the final quantity needed?	Total drink: mL L.
What assumptions did that step make?	Every runner drinks, and drinks mL on average; no spillage or seconds.
Would you add a safety margin?	Sensible — say extra. That is a modelling choice, to be stated.
Now the ratio. What does mean for L?	parts total; concentrate of the volume.
Represent the problem.	Concentrate L, water L (before any margin).
Looking back	Different pairs will choose different margins — and that is fine if stated. Models differ because assumptions differ.

Sample formulations:

1. Paint needed L (assume one coat, stated coverage). Green L in ratio → blue L, yellow L → buy tins blue… careful: paint is mixed from tins, so buy blue and yellow ( and round up per colour), noting leftover.
2. Total mix kg; parts → one part kg; nuts kg, sultanas kg, pretzels kg. Assumptions: all bags sell or are made anyway; bag mass is exact.
3. As scaffolded: L concentrate, L water, plus stated margin.

Activity 3 — Inquiry: whose Model is Better? (8 min)

Whole class, using Scenario 3.

Pair A assumed each runner drinks mL and added no margin: L concentrate. Pair B assumed mL each and added : L concentrate.

1. Whose model is “right”?
2. What is the cost of over-catering? Of under-catering?
3. What extra information would sharpen the model?

Discussion targets: neither is wrong — they differ in assumptions. Under-catering fails runners on a hot day (high cost); over-catering wastes some concentrate (low cost). The asymmetry justifies B’s margin. Actual data — last year’s consumption, the weather forecast — would sharpen the estimate. Models are judged by fitness for purpose, not by being “the” answer.

Checks for Understanding

(5 minutes — exit ticket)

1. Name the four stages of the modelling cycle.
2. A cafe mixes coffee blend A : B in ratio and expects to use kg this week. State one assumption this model makes.
3. Formulate (do not solve): “Rope is sold in m coils. A camp needs rope for tents, each using about m.”
4. Reasoning. Why must assumptions be stated rather than left silent?
5. Reasoning. Two correct models give different answers. How is that possible?

Answers: 1. Formulate, solve, interpret, communicate; 2. E.g. demand equals last week’s; the blend ratio stays fixed; no wastage; 3. Total m (assume no wastage/knots — or add a margin); coils rounded up ; 4. The answer depends on them — a reader cannot judge or reuse the model without knowing what was assumed; 5. They made different (reasonable) assumptions, e.g. different margins or usage rates.

Common Misconceptions

Misconception	How to pre-empt it
Rushing to arithmetic before deciding what to calculate.	The warmup and formulation cards force the formulate stage into the open.
Believing every problem has one right answer.	The whose-model-is-better inquiry directly targets this.
Leaving assumptions implicit.	Assumptions are a marked component of every formulation card.
Ignoring practical constraints (tin sizes, seat counts).	Scenarios 1 and the bus example build them in.
Treating a safety margin as cheating.	It is a modelling choice — legitimate when stated and justified.
Confusing irrelevant detail with relevant data.	The relevant/irrelevant sort in the worked demonstration.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cordial is mixed . How much cordial concentrate is needed for L of drink?

Answer

Concentrate is of the drink: L.

E2 (AMC Junior style). A school expects of its students at a concert, with families adding half as many again. Seats come in rows of . Formulate and solve: how many rows?

Answer

Students ; with families ; rows rows. (Assumptions: attendance fraction holds; every attendee needs one seat.)

E3 (Challenge). A muesli recipe uses oats : fruit : seeds . A batch must fill jars of g. Oats come in kg bags. Formulate, solve, and state one assumption.

Answer

Total kg; one part kg; oats kg → bags exactly; fruit kg; seeds kg. Assumption: jars are filled exactly with no spillage.

E4 (Challenge). Two models of the same bake sale: Model 1 assumes customers spending 445$5.50$ each. Which predicts more revenue, and what would you tell the organisers?

Answer

Model 1: 240$247.50\approx $244 \pm 4$240$250$ — and note that the similarity of two different models increases confidence in the range.

Homework

1. State the four stages of the modelling cycle and what each involves, in your own words.
2. For each situation, list two relevant facts you would need and one assumption you would make: (a) Catering pizzas for a class party. (b) Buying fencing for a chicken run. (c) Mixing two-stroke fuel (petrol : oil ) for a mower.
3. Formulate and solve: “A fruit stall makes juice from oranges and apples in ratio . They expect to sell L of juice.”
4. Formulate and solve: “A minibus seats . About of a squad of players will attend training.”
5. Formulate and solve: “Concrete is mixed cement : sand : gravel , and is needed. Cement is sold in bags.”
6. Reasoning. A model for a sausage sizzle assumed each visitor buys sausages. Is a non-whole number a sensible assumption? Explain.
7. Reasoning. Give an example where under-catering is worse than over-catering, and one where the reverse holds.
8. Challenge. Design your own modelling scenario involving a ratio, write its formulation card (relevant facts, assumptions, representation), and solve it.

Answers: Q3 — one part L: oranges’ juice L, apples’ L; assumes demand of L and fixed ratio. Q4 — expected players; minibuses. Q5 — one part : cement , sand , gravel ; bags exactly. Q6 — yes: it is an average across many visitors, not a claim about one person. Q7 — e.g. medicine or water at a sports event (under-catering dangerous) versus perishable catering (over-catering wasteful). Q8 — student’s own.