Real-world linear models with digital exploration | Year 8

01 / LEARN

Understand the idea.

Separate a fixed amount from a rate

A linear model combines a starting amount with a constant change per input unit. Define both variables and their units before writing the rule, so the constant charge and the repeated charge cannot be confused.

For a fee of AUD $5$ plus AUD $3$ per kilometre, $C = 5 + 3 d$ , where $C$ is dollars and $d$ is kilometres. Distance can vary continuously, but a count of chairs needs whole-number inputs.

Explore without trusting every prediction

Build a spreadsheet by placing inputs in one column and calculating each output from the input cell. Filling the formula down produces a table; check initial rows by hand before trusting the graph or further predictions.

A rule remains valid only while its assumptions hold. A tank draining at a constant rate eventually empties; extending the same line below zero gives an impossible volume. State the allowed input interval as part of the model.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Write a cost rule

Delivery costs AUD 4 plus AUD 2 per kilometre. Find the rule and cost for 7 km.

Delivery model
Distance (km)	0	1	3	7
Cost (AUD)	4	6	10	18

Worked solution

1
Define the quantities and starting cost.
$d \geq 0, C (0) = 4$
Let $d$ be distance in kilometres and $C$ be cost in AUD; the fixed fee applies before any distance charge.
2
Add the distance-dependent charge.
$C = 4 + 2 d$
Each kilometre contributes two dollars in addition to the fixed fee.
3
Substitute the requested distance.
$C = 4 + 2 (7) = 4 + 14 = 18$
Seven kilometres contribute fourteen dollars, giving a total cost of AUD 18.

Answer

$C = 4 + 2 d$ ; AUD 18.

EXAMPLE 02

Limit a draining model

A tank has 20 L and loses 4 L/min. Give a rule and its valid time interval.

Worked solution

1
Write the decreasing-volume rule.
$V = 20 - 4 t$
Let $V$ be litres and $t$ be minutes since draining began; each minute removes four litres.
2
Find when the tank is empty.
$\begin{aligned} 20 - 4 t & = 0 \\ 4 t & = 20 \\ t & = 5 \end{aligned}$
The constant-rate phase ends when the remaining volume reaches zero.
3
Restrict the input interval.
$0 \leq t \leq 5$
Time starts at zero, and later inputs would make this rule predict negative volume.
4
Check the two endpoints.
$V (0) = 20, V (5) = 0$
The endpoints match the full starting tank and the empty finishing tank.

Answer

$V = 20 - 4 t$ for $0 \leq t \leq 5$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

A hall costs AUD 30 plus AUD 8/hour. Find the cost of 4 hours.

Show answer & reasoning

AUD 62.

The fixed charge is AUD $30$ , and four hours add AUD $8 \times 4 = 32$ . Therefore $C = 30 + 8 (4) = 62$ , so the total is AUD 62.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

A taxi charges AUD 6 plus AUD 3/km. Write a rule and find the 5 km charge.
3 marks
A container fills at 5 L/min from empty. Give the rule and volume at 6 minutes.
3 marks
A table gives time $t$ in minutes and volume $V$ in litres: $(t, V) = (0, 40), (1, 34), (2, 28)$ . Suggest a linear rule and predict $V$ at $t = 5$ .
3 marks
For $C = 10 + 2 n$ , make a table for $n = 0, 1, 2, 3$ .
3 marks
A 30 L tank drains at 3 L/min. A student predicts -6 L at 12 minutes. Explain the problem.
3 marks
Inputs 0, 1, 2 and 3 are in spreadsheet cells A2:A5. Give a B2 formula for $C = 5 + 4 d$ and the expected outputs.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A taxi charges AUD 6 plus AUD 3/km. Write a rule and find the 5 km charge.
3 marks
Identify the fixed fee and rate.
Substitute $d = 5$ .
$C = 6 + 3 d$ ; AUD 21.
A container fills at 5 L/min from empty. Give the rule and volume at 6 minutes.
3 marks
The starting volume is zero.
Multiply the rate by elapsed time.
$V = 5 t$ ; 30 L.
A table gives time $t$ in minutes and volume $V$ in litres: $(t, V) = (0, 40), (1, 34), (2, 28)$ . Suggest a linear rule and predict $V$ at $t = 5$ .
3 marks
The starting volume is 40 and each step loses 6.
At 5, $40 - 30 = 10$ .
$V = 40 - 6 t$ ; 10 L.
For $C = 10 + 2 n$ , make a table for $n = 0, 1, 2, 3$ .
3 marks
Substitute each input.
The output rises by 2 each time.
10, 12, 14 and 16.
A 30 L tank drains at 3 L/min. A student predicts -6 L at 12 minutes. Explain the problem.
3 marks
It empties when $30 - 3 t = 0$ , at 10 minutes.
After that the physical volume remains zero, not negative.
The model is valid only for $0 \leq t \leq 10$ minutes.
Inputs 0, 1, 2 and 3 are in spreadsheet cells A2:A5. Give a B2 formula for $C = 5 + 4 d$ and the expected outputs.
3 marks
The A2 reference supplies the input.
Filling down changes the row reference.
$=5+4*A2$ ; outputs 5, 9, 13 and 17.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Multi-step equations; brackets and variables on both sides; equations with rational solutions; substitution checks; one-variable inequalities and number-line solutions; tables and straight-line graphs; horizontal/vertical lines; how coefficients change a graph; graphical solutions and intersection checks; real-world linear models with digital exploration.

Official curriculum source ↗

In this lesson

Understand the idea.

Separate a fixed amount from a rate

Explore without trusting every prediction

See the method in action.

Write a cost rule

Define the quantities and starting cost.

Add the distance-dependent charge.

Substitute the requested distance.

Limit a draining model

Write the decreasing-volume rule.

Find when the tank is empty.

Restrict the input interval.

Check the two endpoints.

Try it yourself.

Check your understanding

Keep practising