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Linear Modelling
Bridge the gap between algebra and the real world. Use linear equations to model costs, distances, temperatures, and more. Make predictions from data.
Worksheet
Download or print the worksheet to work through this lesson.
Before we begin: A mobile phone plan charges $20 per month plus $0.10 per text message. How would you write an equation for the total monthly cost? What does each part of your equation represent in the real world?
Type your initial response below. You will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this after you have worked through the lesson.
Learning Intentions
Know
- Linear models have the form $y = mx + c$ where $m$ is the rate of change and $c$ is the initial value.
- Interpolation uses data within the range; extrapolation predicts outside the range.
Understand
- How gradient and y-intercept have real-world meaning.
- The limitations of linear models and when they break down.
Can Do
- Create linear models from word descriptions.
- Create linear models from data points.
- Use models to interpolate and extrapolate values.
- Interpret gradient and intercept in context.
Success Criteria
- I can write a linear equation to model a real-world situation.
- I can identify the real-world meaning of the gradient and y-intercept.
- I can use a linear model to make predictions.
- I can distinguish between interpolation and extrapolation.
Key Terms
Common Mistakes to Avoid
Wrong: Assuming a linear model works forever. Extrapolating a plant's growth linearly for 50 years would predict a tree taller than a skyscraper.
Right: Linear models are approximations valid within a certain range. Always question whether the relationship stays linear beyond your data.
Wrong: Swapping gradient and intercept meanings. Thinking the y-intercept of a cost model is the "rate per unit" and the gradient is the "starting cost."
Right: In $C = 50t + 20$, the gradient $50$ is dollars per hour (rate), and the intercept $20$ is the fixed fee at $t = 0$.
Building Models from Words
Building Models from Word Descriptions
Many real-world situations involve a fixed amount plus a variable amount. This structure maps directly to $y = mx + c$.
Common contexts:
- Cost models: $C = \text{rate} \times t + \text{fixed fee}$
- Distance: $d = \text{speed} \times t + \text{initial distance}$
- Temperature: $T = \text{rate of change} \times \text{time} + \text{initial temperature}$
- Depreciation: $V = V_0 - \text{rate} \times t$
Building Models from Data
Building Models from Data Points
When given data, treat two points as you would for finding any line's equation. The resulting model can then make predictions.
The process:
- Identify the two variables and which is independent ($x$) and dependent ($y$).
- Choose two data points and calculate the gradient.
- Find the equation using one of the points.
- Use the equation to predict unknown values.
Interpolation vs Extrapolation
Interpolation vs Extrapolation
Not all predictions are equally reliable. Predicting within your data range is safer than predicting beyond it.
| Type | Definition | Reliability |
|---|---|---|
| Interpolation | Predicting within the data range | More reliable |
| Extrapolation | Predicting outside the data range | Less reliable |
Why extrapolation is risky: Real-world relationships rarely stay perfectly linear forever. A model based on data from ages 5–15 won't reliably predict height at age 50.
Interactive: Linear Modeller
Your Turn
Question 1: A gym membership costs $25 per week plus a $50 joining fee. Write a linear equation for the total cost $C$ after $w$ weeks, and find the cost after 10 weeks.
Question 2: A car's value decreases from $24,000 to $18,000 over 3 years. Assuming linear depreciation, find the equation for value $V$ after $t$ years, and predict the value after 5 years.
Copy Into Your Books
Revisit Your Thinking
Look back at your Think First response about the phone plan ($20 + $0.10 per text). The equation is $C = 0.10n + 20$ where $n$ = number of texts. Now, if you sent 150 texts, what would your bill be? If your bill was $35, how many texts did you send? Show your working.
Multiple Choice
Select the best answer for each question.
1 mark A plumber charges $80 call-out plus $60 per hour. The cost for $h$ hours is:
1 mark In the model $d = 5t + 10$, the number $10$ represents:
1 mark A plant is 5 cm tall and grows 2 cm per week. Its height after $w$ weeks is:
1 mark Using data from hours 0–8 to predict a value at hour 12 is:
1 mark A car travels at 80 km/h with a 20 km head start. The distance from the starting point after $t$ hours is:
Short Answer
Show all working and justify your answers.
Question 6
An electricity bill has a fixed daily supply charge of $1.20 and a usage charge of $0.28 per kWh.
(a) Write a linear equation for the total daily cost $C$ in terms of $u$, the number of kWh used.
(b) Calculate the cost for 25 kWh used in one day.
Question 7
The population of a small town was recorded:
2018: 5,200 2022: 6,400
(a) Assuming linear growth, find the equation for population $P$ in terms of $t$ years after 2018.
(b) Predict the population in 2025. Is this interpolation or extrapolation?
(c) Explain one reason why this prediction might not be accurate.
Question 8
A train journey has the following data:
After 1 hour: 120 km from the station
After 3 hours: 280 km from the station
(a) Find the equation for distance $d$ from the station in terms of time $t$ in hours.
(b) What is the train's average speed, and what does the y-intercept represent?
(c) A second train leaves the same station at the same time travelling at 100 km/h with no head start. Write its equation and determine when the two trains will be the same distance from the station.
Model Answers
(a) $C = 0.28u + 1.20$
(b) $C = 0.28(25) + 1.20 = 7.00 + 1.20 = \$8.20$
Marking guidance: 1 mark for correct equation with variables identified, 1 mark for substitution, 1 mark for correct cost.
(a) $m = \frac{6400 - 5200}{2022 - 2018} = \frac{1200}{4} = 300$ people/year.
$P = 300t + 5200$
(b) 2025 is $t = 7$: $P = 300(7) + 5200 = 2100 + 5200 = 7300$. This is extrapolation (2025 is beyond the data range 2018–2022).
(c) Population growth may not stay linear. Factors like economic changes, migration, or birth rate changes could affect the trend.
Marking guidance: 1 mark for (a), 1 mark for prediction in (b), 1 mark for identifying extrapolation, 1 mark for valid explanation in (c).
(a) $m = \frac{280 - 120}{3 - 1} = \frac{160}{2} = 80$ km/h.
$120 = 80(1) + c$ → $c = 40$
$d = 80t + 40$
(b) Speed = $80$ km/h. The y-intercept ($40$ km) is the head start — the train was already $40$ km from the station when timing began.
(c) Second train: $d = 100t$
Set equal: $80t + 40 = 100t$ → $40 = 20t$ → $t = 2$ hours.
They meet after 2 hours, at $d = 200$ km from the station.
Marking guidance: 1 mark for equation in (a), 2 marks for (b) with interpretation, 2 marks for (c) with correct working and answer.
Consolidation Game
Test your knowledge from this lesson and previous lessons.