077i - Consolidation Modelling with Ratios

Lesson 77 — Consolidation and Check: Modelling with Ratios

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

Learning Intentions

• To carry out a complete modelling task independently.
• To demonstrate all four stages of the modelling cycle with justification.

Success Criteria

I can:

1. Formulate a real situation as a ratio problem with stated assumptions.
2. Solve accurately and interpret with correct rounding.
3. Communicate a recommendation someone could act on.
4. Assess the sensitivity and limits of my model.

Warmup

(6 minutes — cycle diagnosis, pairs)

Each student below is stuck at a different stage. Name the stage and give one piece of advice.

1. Priya has pages of correct arithmetic but cannot say what the numbers mean for the canteen.
2. Jack wrote ”, guests” and froze — he doesn’t know what to calculate.
3. Mei has a clear answer ( tins) but her teacher asks, “what if your coverage estimate is wrong?”
4. Tom’s poster shows only ” buses” in large letters.

Answers: 1. Interpret — go back to the situation and translate each number; 2. Formulate — state assumptions and choose the representation first; 3. Limits/sensitivity — recalculate with a shifted assumption; 4. Communicate — add the situation, basis and rounding so a reader can trust and act.

Activities

Activity 1 — The Main Task (25 min)

Individual or pairs. One complete modelling task, produced as a four-part report (Lesson 76 frame). This is the assessable core of the unit.

The Sports Carnival Sausage Sizzle.

Your class runs the sausage sizzle at the athletics carnival. From the school:

• Expected attendance: about people (students, staff, families).
• Last year roughly in every attendees bought a sausage sandwich, and buyers averaged sandwiches each.
• Each sandwich needs sausage, bread slice, and onion. Onion is used with sausages in the mass ratio sausages : onion .
• Sausages: packs of , average sausage g, 9.50$ per pack.
• Bread: loaves of usable slices, 3.20$ per loaf.
• Onions: sold loose, 3.40$ per kg.
• Sauce, napkins and the barbecue are donated.

Produce a report recommending the shopping list and estimating the cost. Include your assumptions, all working, correct rounding with reasons, and a sensitivity note.

Reference solution (teacher):

Marking emphases:

• Assumptions stated (buying fraction and average hold from last year; one sausage per sandwich; no wastage or a stated margin).
• Rounding directions justified (bread up; onion is loose so nearest sensible amount, with the choice named).
• A sensitivity line, e.g. “if in attendees buy, sandwiches rise to — packs and loaves; cost 318$. The buying fraction is the most sensitive assumption.”
• A usable shopping list with total cost.

Extension for early finishers: the P&C will sell sandwiches at 3.50$. Estimate the profit, and find the break-even number of sandwiches.

(Revenue 1470\approx $1189.50280.50 \div 3.50 = 80.1\ldots \to 81$ sandwiches.)

Activity 2 — Peer Audit (8 min)

Swap reports. Audit against a printed checklist — tick, cross, or “can’t tell”:

Check	✓ / ✗ / ?
Assumptions stated before any arithmetic
Ratio used correctly ( by mass, not by count)
Every rounding has a direction and a reason
The shopping list is complete and costed
A sensitivity note names the weakest assumption
A stranger could shop from this report

The onion trap — flag it explicitly in the audit debrief: the ratio is by mass, so it applies to kg of sausages, not to sausages. Reports that computed “onions” have mixed a count with a mass — the most instructive error this task produces.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. In the carnival task, why was the bread rounded up but the sausage packs not rounded at all?
2. State the onion calculation and why it uses kg rather than .
3. Which assumption in the task is most sensitive, and how would you test it?
4. Reasoning. Your model’s cost came to 280.50$318$. Give two legitimate reasons the reports could differ.
5. Reasoning. What makes a modelling answer “good”, given that there is no single correct number?

Answers: 1. Bread: loaves must round up or sandwiches run short; sausages divided exactly ( packs), so no rounding arose; 2. The ratio is by mass; sausage mass is g kg, so onion kg; 3. The buying fraction (); recompute with and compare; 4. Different margins, or a different assumed buying fraction/average — both legitimate if stated; 5. Stated assumptions, correct mathematics, context-appropriate rounding, actionable communication, and honest limits.

Common Misconceptions

Misconception	How to pre-empt it
Applying a mass ratio to a count.	The onion trap, addressed in the audit debrief.
Rounding without a stated direction or reason.	The audit checklist makes it a named criterion.
Omitting assumptions because “everyone knows them”.	The checklist’s first row; the warmup’s Jack.
Reporting arithmetic instead of a recommendation.	The “stranger could shop from it” test.
Treating a differing classmate’s answer as an error.	Exit ticket Q4 legitimises assumption-driven variation.
Sensitivity as an afterthought sentence with no numbers.	Require a recalculated figure, not just “it might change”.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A sizzle sells sandwiches using bread loaves of slices. How many loaves were needed?

Answer

loaves exactly.

E2 (AMC Junior style). Sausages and onions are used in mass ratio . If kg of sausages are cooked, how much onion?

Answer

kg.

E3 (Challenge). A stall’s profit is , where is sandwiches sold. (a) Find the break-even . (b) Find the profit at . (c) What does the model assume about costs?

Answer

(a) . (b) 1189.50n$.

E4 (Challenge). Attendance estimates for the carnival are . Recompute the sandwich estimate at both ends and state the range of loaves needed.

Answer

At : sandwiches → loaves . At : sandwiches → loaves . Range: – loaves — attendance uncertainty alone moves the order by five loaves.

Homework

1. Complete any unfinished sections of your carnival report, incorporating one audit comment you received.
2. Recompute your shopping list assuming in attendees buy (other assumptions unchanged). Present old and new lists side by side.
3. A drinks stand at the same carnival mixes cordial and expects to serve cups of mL. Concentrate comes in L bottles at 4.20$. Write a mini-report (all four parts, one sentence each is acceptable) for the concentrate order.
4. Reasoning. Explain, using the carnival task, the difference between an exact quantity (sausage packs) and an estimated one (attendance), and how each affects confidence in the final cost.
5. Challenge. Extend the profit model: sauce actually costs c per sandwich and gas costs 15P = 3.5n - 280.50$ to include these, and find the new break-even.

Answers: Q2 — sandwiches (or work with ; state the choice): packs , loaves , onion kg; cost 317.78= 50= \tfrac18 \times 50 = 6.25\to 7= $29.40200P = 3.5n - 0.02n - 295.50 = 3.48n - 295.50n = 295.50 \div 3.48 = 84.9 \to 85$ sandwiches.