051i - Formulating Linear Models

Lesson 51 — Formulating Linear Models

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

Learning Intentions

• To understand the stages of the mathematical modelling cycle: formulate, represent, solve, interpret, evaluate.
• To formulate real-world financial scenarios as linear functions by identifying variables, constants and assumptions.

Success Criteria

I can:

1. Name and describe the five stages of the mathematical modelling cycle.
2. Identify the independent and dependent variable in a real-world scenario.
3. Translate a worded financial scenario into a linear function of the form .
4. State the assumptions underlying a linear model and explain why they matter.

Warmup

(5 minutes — quick-fire, mini whiteboards)

For each scenario, identify (a) what changes, (b) what stays fixed, and (c) which quantity depends on which.

1. A plumber charges a 70$55$ for every hour on the job.
2. A rideshare trip costs 3.50$1.20$ per kilometre.
3. A school fete stall sells raffle tickets at 2$ each, with no other costs.
4. A gym charges a 50$18$ per week.

Discussion: In each case, name the independent variable (what you choose or control — e.g. hours, kilometres, tickets, weeks) and the dependent variable (what results — the cost). Introduce today’s big idea: real situations like these can be captured by a linear function, but turning words into mathematics is itself a skill — this is called formulating a model.

Activities

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

Draw the modelling cycle on the board as a loop with five stages:

Stage	What happens
1. Formulate	Translate the real situation into mathematics — identify variables and write a function.
2. Represent	Choose a table, graph, equation or description that suits the purpose.
3. Solve	Use algebra or graphing to answer the mathematical question.
4. Interpret	Translate the mathematical answer back into the real situation, with units.
5. Evaluate	Check whether the model is reasonable, and refine it if needed.

Today we focus only on Stage 1: Formulate. (Lessons 52–55 each take one further stage; Lesson 56 puts the whole cycle together.)

I do: Mobile phone plan — “Bolt Mobile charges a 205$ cents for every megabyte of data used.”

Talk aloud: “What can the customer choose? The amount of data, megabytes — that’s my independent variable. What results? The monthly cost, dollars — that’s my dependent variable, since it depends on . The fixed 2055= $0.05$.”

We do: Together formulate: “FitZone gym charges a 50$18wM$ = total cost.

You do: Formulate a linear function for each. Define your variables first.

1. A plumber charges a 70$55h$.
2. A printing shop charges 412p$.
3. A car hire company charges 45n$, with no other fee.

Activity 2 — Identifying Assumptions and Constraints (12 min)

Explain: every linear model rests on assumptions — simplifications that make the real world mathematically tractable. Naming them is part of formulating.

I do: For , the assumptions are: (a) the rate per MB never changes no matter how much data is used; (b) there is no cap on data; (c) , since you cannot use negative data. State this constraint as a domain restriction: .

We do: For : the assumptions are that the weekly fee never changes and membership can be cancelled at any whole number of weeks. Domain: , and is a whole number (weeks are billed in whole units).

You do: For each function formulated in Activity 1, state one assumption being made and the domain restriction on the independent variable.

Activity 3 — Inquiry Task: Formulating from an Ambiguous Scenario (11 min)

Pairs, then whole-class share.

A community group is running a car wash to raise money for new sports equipment costing 150$8$25$ on soap and sponges before they start.

Formulate a linear model for the group’s net profit, , after washing cars.

Socratic scaffolding:

Prompt	Purpose
Understand: what quantity are we modelling?	Net profit, not just income — costs must be subtracted.
What is the independent variable?	Number of cars washed, .
What is fixed, and is it positive or negative in the model?	The 25$ soap cost is spent before any income — it’s a negative constant.
What is the rate of change?	8$ for every car washed.
Can you write an expression now?	.
Look back — does the model make sense for small ?	At , — they start 25$ in the red, which matches the story.

Extend: the group also wants to know how many cars they need to wash to cover both the soap cost and the 150$ equipment cost — but do not solve this yet; just discuss which stage of the modelling cycle that question belongs to (Solve — Lesson 53).

Checks for Understanding

(5 minutes — exit ticket, collected)

1. A taxi charges a 4.20$2.10F$.
2. In your model from Q1, state the domain restriction on the independent variable.
3. A market stallholder pays 60$15Ps$ scarves.
4. Explain, in one sentence, the difference between the independent and dependent variable in a linear model.

Answers: 1. Let = kilometres travelled, = fare in dollars: ; 2. ; 3. ; 4. The independent variable is the quantity you choose or control; the dependent variable is the resulting quantity that depends on it.

Common Misconceptions

Misconception	How to pre-empt it
Any two numbers in a worded problem can go into in either order.	Always ask “what changes when what changes?” before assigning and .
The fixed fee is always added, never subtracted.	Use the car-wash example, where the setup cost reduces profit — a negative constant.
Formulating means writing any equation that uses the numbers given.	Insist on defining variables in words first, then translating each phrase into a term.
The independent variable can take any real-number value.	Explicitly state the domain (e.g. ) as part of formulating, not as an afterthought.
Rate language (“per”, “each”, “every”) always means multiplication by the coefficient.	Confirm with a worked check: does the total go up by the same amount each time? If not, it may not be linear.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). A courier charges a flat fee plus an amount per kilogram. A kg parcel costs 117$19Cm$.

Answer

The rate of change is dollars per kg. Using : .

E2 (Kangaroo style). A tank starts with L of water and drains at a constant rate. After minutes it holds L. Formulate a linear function for the volume after minutes, and state a sensible domain.

Answer

Rate of change L/min.

Domain: , since the tank cannot hold a negative volume.

E3 (Challenge). Two variables are connected by a rule that is linear. When , ; when , . Formulate the rule, without being told the context.

Answer

Homework

1. A locksmith charges a 50$30C$.
2. State the domain restriction for your model in Q1.
3. A caterer charges 120$22Tg$.
4. A savings account starts with 80$15Bw$ weeks.
5. Reasoning. Explain why the constant term in a cost model is usually the value of the function when the independent variable is zero, and give a real-world meaning for this in the caterer example (Q3).
6. Challenge. A parking station charges a flat entry fee plus an hourly rate. Parking for hours costs 135$25Ch$.

Answers: 1. Let = number of locks, ; 2. ( a whole number); 3. ; 4. ; 5. When the independent variable is , all the “rate” terms vanish, leaving only the constant — in Q3, this is the cost with guests, i.e. the fixed setup fee of 120= \dfrac{25-13}{5-2} = 4(2,13)13 = 4(2) + c \Rightarrow c = 5C = 4h + 5$.