056i - Modelling with Linear Relations Consolidation

Lesson 56 — Consolidation and Check: Modelling with Linear Relations

Strand: Algebra | Descriptor: AC9M8A03 | Duration: 45 minutes

Learning Intentions

• To consolidate all five stages of the mathematical modelling cycle: formulate, represent, solve, interpret, evaluate.
• To run a complete modelling cycle, end-to-end, on a new financial scenario.

Success Criteria

I can:

1. Formulate a linear function from a worded financial scenario, defining variables and stating a domain.
2. Represent the model using a table, graph or equation, as suits the purpose.
3. Solve the model algebraically, including finding a break-even point.
4. Interpret the solution in a full sentence, with correct units and appropriate rounding.
5. Evaluate whether the model is reasonable and suggest a refinement if needed.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. Every real-world cost situation can be modelled by a linear function.
2. A linear model’s -intercept represents the cost when the independent variable is zero.
3. Solving and evaluating a linear equation are the same process.
4. A model should be trusted equally at every value of the independent variable, however large.

Answers: 1. Sometimes — many are (fixed fee + constant rate), but others involve discounts, caps or compounding that break linearity; 2. Always — substituting leaves only the constant term; 3. Never — evaluating goes input→output; solving goes output→input; 4. Never — extrapolation far beyond the data used to build the model is risky.

Discussion: Recall all five stages of the modelling cycle from Lesson 51. Today we use every stage, in order, on one new problem.

Activities

Activity 1 — Mixed Fluency Review of All Five Stages (13 min)

Rapid-fire, then pair check.

I do: Recap the five stages on the board with a one-line description each (Formulate, Represent, Solve, Interpret, Evaluate), and a one-word memory hook for each (“translate”, “display”, “calculate”, “explain”, “check”).

We do: For the scenario “A dog-walking business charges a 15$8$ per dog walked,” work through each stage together, briefly:

Stage	Working
Formulate	Let = number of dogs, = total cost. , .
Represent	Table for : .
Solve	Given , solve .
Interpret	”A 717$ dogs.” (Whole number, no rounding needed.)
Evaluate	Reasonable for small groups; likely breaks down for very large groups if a bulk discount applies.

You do: Students complete the same five-stage table independently for: “A car wash charges a 20$6$92$. Swap and check with a partner.

Activity 2 — Full Modelling Cycle: the Solar Panel Decision (20 min)

Structured problem, pairs, then whole-class share.

A family is considering installing solar panels. The system costs 4200$35$ on the electricity bill every month.

(a) Formulate. Define variables and write a linear model for the family’s net cost, (installation cost minus savings so far), after months. State a sensible domain.

(b) Represent. Build a table of for .

(c) Solve. Find, algebraically, the number of months until the system pays for itself (i.e. — the break-even point).

(d) Interpret. State the break-even point in a full sentence, rounding appropriately, and convert to years.

(e) Evaluate. The panels come with a -year performance warranty, after which output (and therefore monthly savings) is expected to decline. Discuss whether this linear model remains appropriate for the full years, and suggest one refinement.

Guided working:

(b) Sample table:

(months)	0	20	40	60	80	100	120
($)	4200	3500	2800	2100	1400	700	0

(d) “The system pays for itself after exactly months, which is years.”

(e) The domain of validity is roughly (the -year warranty period, in months). Since , the break-even prediction itself falls safely within the reliable range. However, extending the model beyond to predict long-term savings would be extrapolation — output declines after the warranty period, so a refined model would need a reduced monthly saving (or a piecewise rule) for .

Checks for Understanding

(6 minutes — exit ticket, collected)

A community fun-run charges 8$300$ in fixed costs (venue, marshals, first-aid) to cover before any profit is made.

1. Formulate. Write a linear model for the organiser’s net profit, , after runners register.
2. Solve. Find the number of runners needed to break even ().
3. Interpret. Write your answer to Q2 as a full sentence.
4. Evaluate. Give one reason the linear model might become unreasonable for very large numbers of runners (e.g. ).

Answers: 1. , ; 2. round up to runners (since runners would still leave a loss); 3. “At least runners are needed for the event to break even.”; 4. Any reasonable answer, e.g. a venue capacity limit, extra marshals or facilities needed for very large crowds, or a possible bulk-booking discount on venue hire — the fixed cost of 300$ may not stay fixed at very large scale.

Common Misconceptions

Misconception	How to pre-empt it
The five stages must be treated as isolated, unconnected tasks.	Use one running scenario across all five stages so students see the cycle as a single connected process.
Break-even always means “profit is zero and nothing else.”	Confirm what zero represents in each specific context (e.g. net cost of zero means the outlay has been fully recovered).
Once a model is solved, the task is finished.	Reinforce that Interpret and Evaluate are still required — a bare number is not a complete answer.
A model that works well within its data range is automatically safe to extrapolate.	Always state the domain of validity explicitly, and ask “what happens just outside it?”
Formulating, representing, solving, interpreting and evaluating must happen in a strict, unrepeatable order.	Note that real modelling often loops back — evaluation may prompt re-formulating (this previews Lesson 61’s proof-style problem solving).

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A community garden costs 600$25$ per month in grocery costs. Formulate a net-cost model and solve for the break-even number of months.

Answer

E2 (Kangaroo style). Two water tanks are being filled: Tank A starts at L and fills at L/min; Tank B starts at L and fills at L/min. Solve for when the tanks hold equal amounts of water.

Answer

E3 (Challenge). A solar system saves 3510120$28$4200150$? Use a piecewise model to justify your answer.

Answer

By month : — already broken even exactly at months. From to months (a further months) the system continues saving at 2815028 \times 30 = $840150$.

Homework

Run the full modelling cycle for the following scenario: A school is fundraising to buy sports equipment worth 980$60$4$ profit per sausage sold.

1. Formulate. Define variables and write a linear model for net profit after selling sausages, including a domain.
2. Represent. Build a table of for .
3. Solve. Find algebraically how many sausages must be sold to fully fund the equipment (i.e. cover both the 60$980$ equipment cost).
4. Interpret. State your answer to Q3 as a full sentence, rounding appropriately.
5. Reasoning. Explain why the constant term in your model from Q1 is negative.
6. Challenge. After selling sausages, the school realises the sausage price needs to rise, since real sales data show profit is actually only 3.50Pss>200$.

Answers: 1. , ; 2. ; 3. Total needed (or solve directly): ; 4. “The school needs to sell at least sausages to fully fund the equipment.”; 5. The 60s=0$60P=4s-600 \le s \le 200P = 4(200)-60+3.5(s-200) = 740+3.5(s-200)s>200$.