Real-world linear models with digital exploration | Year 8

01 / LEARN

The idea, step by step

Separate a fixed amount from a rate

A fixed charge plus a constant cost per unit produces a linear model. Define the variables and units: a AUD 5 fee plus AUD 3/km is $C = 5 + 3 d$ , with C in dollars and d in kilometres.

The context controls the allowed input values. Distance may be continuous and non-negative; a number of chairs must be a non-negative whole number.

Explore without trusting every prediction

A table or spreadsheet can compare inputs or coefficients. Put inputs in a column and use a formula referring to the input cell. Fill it down and check several results by hand.

A model must match its conditions. A tank cannot contain a negative volume; a constant rate may cease when a container fills. State such limits rather than extrapolating blindly.

A common trap

Extending a linear formula to impossible inputs.

Instead: State the domain and check conditions such as empty/full containers and whole-number counts.

02 / WORKED EXAMPLES

See the thinking

Example 1

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

The starting charge is 4.
Each kilometre adds 2.
At d = 7, $C = 4 + 14 = 18$ .

C = 4 + 2 d

; AUD 18.

Example 2

Limit a draining model

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

The starting value is 20 and the volume falls by 4 each minute.
Set $20 - 4 t = 0$ to find the empty time t = 5.
After 5 minutes this draining rule would predict an impossible negative volume.

V = 20 - 4 t

for

0 \leq t \leq 5

Check your understanding

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

Show answer

AUD 62.

Use $C = 30 + 8 h$ , giving $30 + 32$ .

03 / WORKSHEET

Your turn

6 questions · 18 marks · Work at your own pace.

Worksheet PDF 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 and 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 and 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 ↗

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