What is a mathematical model?
A model is a simple copy of something real. A toy car is a model of a car. A mathematical model uses numbers, equations, graphs or tables instead of plastic.
Examples:
- Taxi fare: y = 12x + 50
- Area of a square garden of side s: A = s²
- Money in a bank at 5% a year: A = P × 1.05ᵗ
A model has variables (things that change), parameters (fixed numbers such as 12 and 50) and assumptions (what we choose to ignore to keep it simple).
The modelling cycle
- Understand the problem. What do we want to know?
- Make assumptions and choose variables. Decide the input (independent variable) and output (dependent variable). Write what you ignore.
- Build the model. Use a table, graph or equation. Look at the shape of the data.
- Solve. Use the model to calculate or predict.
- Interpret. Turn the maths answer back into words with units.
- Validate (check). Compare with real data. If the error is too big, change the assumptions or the model and go round again.
This is a loop, not a straight line. Scientists and engineers go round it many times.
Choosing the right type of model
Linear: y = mx + c
Equal steps in x give equal differences in y. Example: fares, phone plans, a candle burning down.
Quadratic: y = ax² + bx + c
The graph is a parabola. The second differences are equal. Example: height of a thrown ball, area of a square.
Exponential: y = a × bˣ
Equal steps in x give equal ratios in y (the value is multiplied each time). Example: bacteria doubling, compound interest, a cooling cup of tea.
Periodic (trigonometric)
Repeats in cycles. Example: tides, hours of daylight in a year.
How to decide
Plot the data. Check differences (linear), second differences (quadratic) or ratios (exponential). Choose the simplest model with a small error.
Checking a model: error, fitting and limits
The error (residual) is real value − model value. Small errors mean a good fit.
A line of best fit is the line that makes the errors as small as possible overall (computers use "least squares": they make the sum of squared errors smallest).
Interpolation (predicting inside the data range) is usually safe. Extrapolation (predicting far outside it) is risky: a child's height grows about 6 cm a year, but the model fails at age 40!
Always state the domain where the model works and the assumptions you made. Models are used in science, economics, medicine and technology, and each one has limits.
Key formulas and definitions
- Linear: y = mx + c (m = rate of change, c = starting value)
- Quadratic: y = ax² + bx + c
- Exponential: y = a × bˣ (b > 1 growth, 0 < b < 1 decay)
- Error (residual) = real value − model value
- Percent error = |error| ÷ real value × 100%
Worked examples
1. A plumber charges ₹200 to visit plus ₹150 per hour. Write a model and find the cost for 3 hours.
Let h = hours, C = cost. C = 150h + 200. For h = 3: C = 450 + 200 = ₹650.
2. A table: x = 0, 1, 2, 3 gives y = 5, 8, 11, 14. Which model fits?
Differences are 3, 3, 3 (equal), so it is linear: y = 3x + 5.
3. A table: x = 0, 1, 2, 3 gives y = 3, 6, 12, 24. Which model fits?
Ratios are 2, 2, 2 (equal), so it is exponential: y = 3 × 2ˣ.
4. A ball's height is modelled by h = −5t² + 20t + 1 (metres, seconds). Find the maximum height.
The top is at t = −b/(2a) = −20/(−10) = 2 s. h = −20 + 40 + 1 = 21 m.
5. A model predicts 480 visitors; the real number was 520. Find the error and percent error.
Error = 520 − 480 = 40. Percent error = 40 ÷ 520 × 100 ≈ 7.7%.
6. A town of 50 000 people grows 4% a year. Model it and predict the population after 5 years.
P = 50 000 × 1.04ᵗ. For t = 5: 50 000 × 1.2167 ≈ 60 800 people (assuming the rate stays 4%).
Common mistakes
- Forgetting to write the assumptions. Every model ignores something; say what.
- Using a straight line for data that curves. Check the differences and ratios first.
- Trusting predictions far outside the data (extrapolation) without warning.
- Giving the maths answer without units or meaning, like "230" instead of "a 15 km ride costs about ₹230".