Algebra describes the real world
A mathematical model uses algebra to represent a real situation, so you can explain, predict and make decisions.
The modelling cycle
| Step | Question |
|---|---|
| 1. Understand | What’s the real problem? |
| 2. Assume | What can we simplify? (These are assumptions.) |
| 3. Formulate | Which equation or function fits? |
| 4. Solve | Calculate or analyze the model. |
| 5. Interpret | What does the answer mean in context? |
| 6. Validate | Does it match reality? Revise if needed. |
Choosing a model type
| Pattern in data | Model |
|---|---|
| Constant change per unit | linear model: y = mx + b |
| Constant percent change | exponential model: y = a(b)ˣ |
| Rises then falls (or falls then rises) | quadratic: y = ax² + bx + c |
| Inverse relationship | rational: y = k/x |
| Repeating cycles | sinusoidal: y = a sin(b(x − c)) + d |
Linear example: phone plan
A plan costs $30/month plus $10 per GB of data.
C = 10g + 30
- Parameters: slope 10, intercept 30.
- Prediction: 4 GB → $70.
Exponential example: population
A town of 12,000 grows 2.5% per year.
P = 12,000(1.025)ᵗ
- In 10 years: P ≈ 15,361
- Doubling time: solve 1.025ᵗ = 2 → t ≈ 28 years
Assumption: the growth rate stays constant.
Quadratic example: projectile
A ball thrown up: h = −4.9t² + 15t + 2 (metres, seconds).
- Max height at t = 15/(2 × 4.9) ≈ 1.53 s; h ≈ 13.5 m
- Lands when h = 0: t ≈ 3.19 s
Assumption: no air resistance.
Finding parameters from data
Two points can determine a linear or exponential model.
Exponential example: A car worth $30,000 is worth $24,000 a year later.
- b = 24,000/30,000 = 0.8
- V = 30,000(0.8)ᵗ
Limits of models
- Models work only within a sensible range (domain).
- Real data has variation; models give approximations.
- Assumptions may fail over time (growth may slow).
Worked examples
Example 1: A candle 25 cm tall burns 1.5 cm/h. Model and burn-out time? h = 25 − 1.5t; t ≈ 16.7 h.
Example 2: $2,000 invested at 4% compounded yearly. Model? A = 2,000(1.04)ᵗ.
Example 3: Data: (0, 50), (1, 60), (2, 72), (3, 86.4). Linear or exponential? Ratios are 1.2 each → exponential, y = 50(1.2)ˣ.
Common mistakes
- Choosing a model type without checking the pattern.
- Extrapolating far beyond the data.
- Ignoring units and context in interpreting answers.
Try it yourself
- A plumber charges $90 plus $65/h. Write a model.
- A culture of 500 bacteria doubles every hour. Write a model.
- Data: (0, 8), (1, 12), (2, 16). Which model type?
- Name one assumption in the bacteria model.
Answers: 1) C = 65h + 90 2) N = 500(2)ᵗ 3) linear (y = 4x + 8) 4) e.g. unlimited food and space
Quick recap
- A mathematical model represents a real situation.
- Choose a linear model, exponential model or another type from the data pattern.
- State assumptions and validate with real data.