076i - Communicating Ratio Solutions

Lesson 76 — Communicating and Justifying Ratio Solutions

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

Learning Intentions

• To communicate a modelling solution so that someone else could act on it.
• To justify the choices made about representation and assumptions.

Success Criteria

I can:

1. Write a modelling report with all four stages visible.
2. Justify my choice of representation and my assumptions.
3. State the limits of my model honestly.
4. Critique another model constructively.

Warmup

(6 minutes — which report is useful? pairs)

Two reports of the same catering model:

Report A: ”, so 68. Answer: 68.”

Report B: “We expect guests each eating about sandwich quarters, giving quarters, i.e. sandwiches. A loaf makes sandwiches, so we need , rounded up to loaves. If attendance runs higher, loaves would be safe. Main assumption: quarters per guest, based on last year.”

1. Which report could the canteen actually use? Why?
2. What is missing from Report A?
3. Is Report B longer than it needs to be, or is everything doing work?

Answers: 1. B — it says what to buy and why; 2. Context, units, assumptions, and what the numbers mean; 3. Every sentence carries either the decision, its basis, or its limits — nothing is padding.

Activities

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

The four-part report frame — one short paragraph or labelled section each:

1. The situation and assumptions. What was asked; what we assumed and why.
2. The mathematics. The representation chosen and the working, clearly set out.
3. The answer in context. The decision or recommendation, with units, after interpretation.
4. Limits and sensitivity. What would change the answer; how confident we are.

I do — writing part 4, the least familiar. For the sandwich model:

“The answer depends most on the -quarters-per-guest assumption. If the true figure is , we would need loaves — so the estimate is sensitive to appetite. Attendance is known within about , which changes the order by at most loaves. We recommend loaves with in reserve.”

Name the move: sensitivity means asking “if my assumption shifts, how much does the answer move?” A model whose answer barely moves is robust; one that swings wildly needs better data.

We do — a sensitivity line for Lesson 75’s fuel problem (, L tank):

“If the mix were mistakenly , the oil needed would rise from mL to about mL — a full extra quarter-bottle. The ratio matters; the tank size is exact, so it contributes no uncertainty.”

Activity 2 — Write the Report (18 min)

Pairs. One full report on a single scenario, using the four-part frame. Allocate scenarios around the room so critiques (Activity 3) are cross-scenario.

Scenario 1 — The school disco drinks. Punch mixes juice : lemonade : ice by volume. Expected attendance ; each attendee drinks about mL. Juice comes in L bottles, lemonade in L bottles; ice in kg bags ( L).

Scenario 2 — The garden beds. A schoolyard project plants herbs : flowers : shrubs across plant spots. Herbs come in punnets of , flowers in punnets of , shrubs singly. Punnets cost 9$8$14$ each.

Scenario 3 — The relay teams. A carnival needs runners : marshals : timers . There are volunteers. Only have running shoes suitable for the track.

Socratic prompts to circulate with:

Prompt	Purpose
Could a stranger buy the right things from your part 3?	Tests whether the answer is actionable.
Which assumption, if wrong, hurts most?	Drives part 4 beyond boilerplate.
Where did you round, and which way? Why?	Interpretation must be visible, not silent.
What did the ratio force, and what did reality force?	Separates the model from the adjustments.

Reference solutions (teacher):

1. Total drink L; parts → one part L: juice L ( bottles), lemonade L ( bottles), ice L ( bags). Sensitivity: mL per guest swings totals by L.
2. One part : herbs ( punnets exactly), flowers ( punnets , spare), shrubs . Cost 402$.
3. One part : runners , marshals , timers . Shoe check: ✓ — the constraint does not bind, but the report should say so ( spare pairs of suitable shoes among non-runners).

Activity 3 — Structured Critique (9 min)

Swap reports with a pair who worked a different scenario.

The critique protocol — two stars and a question:

• Star 1: something the report communicates clearly.
• Star 2: a justified choice (rounding, margin, assumption) done well.
• Question: one thing a reader still cannot tell — phrased as a question, not a correction.

Authors respond to the question in writing — one sentence — and staple it to the report.

Teacher note: the question-not-correction rule keeps the critique about communication (could I follow it?) rather than arithmetic policing, which was Activity 2’s job.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Name the four parts of a modelling report.
2. Write one sensitivity sentence for: “We assumed each guest eats sausages; guests; sausages in packs of ; we ordered packs.”
3. A report says only ” tins”. List two things a reader still needs.
4. Reasoning. Why is “state the limits” part of an honest report rather than an admission of failure?
5. Reasoning. What is the difference between checking arithmetic and critiquing communication?

Answers: 1. Situation & assumptions; mathematics; answer in context; limits & sensitivity; 2. E.g. “If appetite is sausages each, we would need packs — the order is sensitive to the appetite estimate, so we hold packs in reserve”; 3. Tins of what, for what job, and what was assumed (coverage, coats, margin); 4. Every model rests on assumptions; stating them lets the reader judge whether the answer transfers to their situation. Silence would overstate certainty; 5. Arithmetic checking asks “is the working right?”; communication critique asks “could a reader understand, trust and act on it?” — a report can pass one and fail the other.

Common Misconceptions

Misconception	How to pre-empt it
Believing the bare number is the answer.	Report A versus Report B in the warmup.
Padding reports with restated questions instead of decisions.	The “could a stranger act on it?” test.
Treating sensitivity as guesswork.	Model it as a concrete recalculation with a shifted assumption.
Hiding the rounding step.	The circulating prompt asks where and which way, every time.
Critiquing arithmetic when asked to critique communication.	The two-stars-and-a-question protocol constrains the form.
Confusing confidence with certainty.	Part 4’s language: “sensitive to”, “within”, “we recommend”.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Punch is mixed and L is made. How much of the second ingredient is used?

Answer

One part L; the second ingredient L.

E2 (AMC Junior style). A report states: ” guests at mL each is L; one part is L.” What ratio total does this imply?

Answer

parts — consistent with, say, .

E3 (Challenge). A model orders packs of sausages for guests at each, plus a margin. Write a formula for and evaluate for .

Answer

E4 (Challenge). Two reports for the same event recommend and packs. Under what circumstances is each the better recommendation?

Answer

(no margin) suits a tight budget where leftovers are pure waste and a shortfall is tolerable; suits an event where running out is embarrassing or harmful and surplus keeps or can be returned. The choice depends on the costs of each kind of error — which is exactly what part 4 of a report should weigh.

Homework

1. Write the four-part report frame from memory and one sentence describing each part.
2. Convert this working into a two-sentence “answer in context”: ” kg; .” (Context: campers, kg of pasta each, pasta sold in kg bags.)
3. Write a full four-part report for: “Fruit boxes are packed apples : oranges : bananas , pieces per box, boxes needed. Apples come in trays of , oranges in trays of , bananas in hands of .”
4. Write one sensitivity sentence for your Q3 report.
5. Swap-read: take any worked answer from Lesson 75 homework and rewrite it as parts 3 and 4 of a report.
6. Reasoning. A classmate writes “Assumption: the maths is correct.” Explain why this is not an assumption in the modelling sense.
7. Challenge. Design a one-page report template your class could reuse, with prompts under each of the four headings.

Answers: Q2 — e.g. “The camp needs about kg of pasta. Buying in kg bags means bags ( kg), rounded up so no camper goes short; the kg surplus keeps.” Q3 — per box: apples, oranges, bananas; totals: apples ( trays, spare), oranges ( trays, spare), bananas ( hands , spare); report should state rounding direction and spares. Q6 — modelling assumptions are claims about the world (demand, appetite, wastage), checkable against reality; correct arithmetic is a requirement of the solve stage, not a model input.