094i - Interpreting and Communicating Modelling Solutions

Lesson 94 — Mathematical Modelling: Interpreting and Communicating Solutions

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

Learning Intentions

• To interpret a numerical solution to a ratio or rate model in terms of the original real-world situation.
• To communicate a modelling solution clearly, including units, context, and a recommendation where relevant.

Success Criteria

I can:

1. Translate a bare numerical answer back into a sentence that answers the original question.
2. Identify what a solution means practically, including rounding decisions that make sense in context.
3. Communicate a recommendation or decision based on a modelling solution, with justification.
4. Distinguish between the mathematical answer and the practical answer to a modelling problem.

Warmup

(5 minutes — pairs, “translate the number”)

For each bare numerical answer, write one full sentence that communicates it as the answer to the question shown.

1. Question: “How many buses are needed to transport students if each bus holds ?” Answer: .
2. Question: “What is the best-buy price per g for a g box costing 6.300.84$.
3. Question: “How much concentrate is needed for L of juice, mixed concentrate to water?” Answer: .

Answers: 1. ” buses are needed” (round up — a partial bus still requires a whole bus). 2. “The unit price is c per g.” 3. ” L of concentrate is needed.”

Discussion: Q1 shows that the mathematical solution ( buses) and the practical solution ( buses) can differ — today’s focus is making that translation correctly, every time.

Activities

Activity 1 — Explicit Instruction: from Number to Sentence (10 min)

Recap the modelling cycle: Formulate → Solve → Interpret → Review. Today’s focus is Interpret: turning a solved number into a meaningful statement about the real situation, including sensible rounding and units.

I do: “A car uses fuel at L per km. How much fuel is needed for a km trip?” Solved value: .

Interpretation is not just restating the number — it requires context:

“The car requires 34.5 litres of fuel for the 460 km trip. Since fuel is purchased in any amount, no rounding is needed here — the exact decimal value is meaningful.”

We do: Together interpret: ” raffle tickets are bought at for 5= $35.00$.” Write the full sentence together, and discuss: does this number need rounding? (No — it is already an exact currency value.)

You do: Write a full interpreting sentence, with correct rounding reasoning, for each:

(a) A recipe scaling calculation gives chicken breasts needed for people. (Can you buy a quarter of a chicken breast? What would you actually do?)

(b) A fertiliser mixing calculation gives L of fertiliser needed. (How precisely can this be practically measured?)

(c) A data plan cost calculation gives 25 + $0.04 \times 600 = $49.00$.

Activity 2 — Guided Practice: Rounding Decisions and Communicating a Recommendation (10 min)

Discuss: not every rounding direction is “up” or “to the nearest whole” — it depends on the practical constraint in the situation.

Situation	Rounding rule	Why
Buses, rooms, containers needed	Round up	A partial unit still requires a whole extra one.
Complete rows/groups from a fixed total	Round down	You cannot form a partial group.
Money	Round to the nearest cent	Currency has a smallest unit.
Continuous quantities (fuel, paint, liquid)	Keep as calculated, or round to a sensible measuring precision	No natural “whole unit” constraint.

I do: “Plan A costs 0.08$15$0.05m = 500$ MB.” Interpretation with recommendation:

“For usage below 500 MB, Plan A is cheaper (no flat fee to cover). For usage above 500 MB, Plan B is cheaper, since its lower per-MB rate outweighs the flat fee. A student who typically uses about MB per month should choose Plan B.”

We do: Together interpret and recommend for: “Booth A: AUD EUR, no fee. Booth B: AUD EUR, 5a = 105$105$).

Activity 3 — Inquiry Task: Interpreting the Fundraiser Solution (14 min)

Pairs. Return to the fundraiser model, now fully solved:

Raffle sales: 35.00$36.00$488.40$740$559.40$.

Write a short paragraph (as if reporting to the school principal) that:

• States the total amount raised, with correct units and rounding.
• Identifies which revenue stream contributed most, and by how much more than the next-largest.
• Notes any figure in the model that depended on an assumption (recall Lesson 92’s assumptions), and explains what would happen to the total if that assumption were wrong — e.g. if the exchange rate had moved before the funds were actually converted.

Socratic scaffolding:

Prompt	Purpose
Understand: who is this report for, and what do they need to know?	A non-technical reader needs the headline number first, not the working.
What does each solved number actually represent in the real situation?	Reconnect to “pledges converted to Australian dollars”, not just “an answer”.
Devise a plan: what order should the paragraph go in?	Headline total, then breakdown, then caveats/assumptions.
Carry it out	Draft the paragraph.
Looking back	Would a reader with no maths background understand every sentence? Remove any bare unexplained numbers.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A calculation gives as the number of tables needed to seat guests, per table. Write the practical interpretation.
2. A best-buy comparison gives Brand A at 0.42100$0.39100$ g. Write a one-sentence recommendation.
3. A simple interest calculation gives for interest earned in dollars. Write the correctly rounded, unit-labelled interpretation.
4. Reasoning. Explain why “the answer is ” is not a complete response to a real-world modelling question.

Answers: 1. ” tables are needed” (round up — a partial table still requires a full extra table). 2. “Brand B is the better buy, at c per g cheaper than Brand A.” 3. “The interest earned is 144.38$ (rounded to the nearest cent).” 4. It gives no units, no context, and no indication of whether/how to round for the practical situation — a modelling answer must be translated back into a sentence about the real quantity being asked for.

Common Misconceptions

Misconception	How to pre-empt it
Leaving the raw decimal as the final answer (e.g. ” buses”).	Insist every applied answer ends in a full interpreting sentence, not a bare number.
Rounding every answer up “to be safe”.	Use the rounding-rule table — direction depends on what the quantity represents.
Ignoring units in the final sentence.	Require units in every interpreted answer, matching the original question’s units.
Treating the mathematical solution as automatically the final practical answer.	Contrast buses with the practical requirement of whole buses.
Giving a recommendation without justification.	Require “because…” in every recommendation sentence.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A school hires buses that each seat students for an excursion of students. How many buses must be hired, and how many empty seats will there be?

Answer

7 buses are needed (round up). Seats available ; empty seats .

E2 (Kangaroo style). A recipe calculation gives eggs needed for a scaled batch. A baker can only use whole eggs. Explain the two sensible practical choices available, and which is generally preferred in baking.

Answer

Round down to eggs (batch will be very slightly less rich/moist) or round up to eggs (batch will be slightly richer). In baking, rounding up to whole eggs is generally preferred, since eggs cannot be practically split without affecting the recipe’s structure, and a slightly richer result is usually less noticeable than a slightly drier one.

E3 (Challenge). A currency model gives a converted amount of 488.401$488$ raised from pledges.” A classmate says this rounding loses money that was actually raised. Evaluate this claim.

Answer

The classmate is incorrect about the money — no money is lost by rounding a report figure; the actual bank transaction still processes to the nearest cent (488.40$488.40$) is more appropriate than rounding to the nearest dollar, since financial reporting conventions require cent-level accuracy.

Homework

1. A modelling calculation gives as the number of vans needed to carry students, per van. Write the full practical interpretation.
2. A modelling calculation gives as the largest number of complete rows of trees that can be planted from trees. Write the full practical interpretation, including any trees left over.
3. Two phone plans cost the same at MB. Write a two-sentence recommendation covering both usage regimes (below and above MB).
4. A simple interest calculation gives dollars. Write the correctly formatted final-answer sentence.
5. Reasoning. A model for “number of pizza boxes needed” gives exactly (no remainder). Explain why this case needs no special rounding discussion, unlike a value such as .
6. Challenge. A group solves a best-buy model and finds Brand X at 2.10$2.085$50$ total more than the family’s budget allows for Brand X’s smaller bottles. Write a short recommendation paragraph that goes beyond the raw unit price to include this practical constraint.

Answers: 1. ” vans are needed” (round up; vans would leave students without a seat). 2. ” complete rows can be planted, with trees left over” (). 3. “Below MB, the plan with no flat fee is cheaper. Above MB, the plan with the flat fee plus lower per-MB rate becomes cheaper, since the flat fee is outweighed by the rate saving.” 4. “The interest earned is 382.50$.” 5. An exact whole-number solution requires no rounding decision at all — the mathematical and practical answers coincide. 6. Recommendation should note that although Brand Y has the lower unit price, the family’s budget constraint may make Brand X the only affordable practical choice — the “best buy” by unit price is not always the best real-world decision.