To choose a simple model, look at how the output changes when the input moves in equal steps. A constant change points to a linear model, and a steadily growing change points to a quadratic one.
This lesson follows defining variables and assumptions in the SPM Mathematics modelling sequence. The next lesson is explaining limitations and refining a model.
How do I pick a model from a table?
Write the differences in the output column. Then decide using the pattern.
| Pattern in the differences | Model |
|---|---|
| the same each time | linear |
| the same each time, and output is 0 when input is 0 | proportional |
| growing by a constant amount | quadratic |
Always check the fit by substituting values back into your equation.
Worked example: a tank filling with water
A tank holds 10 litres at the start. The volume V (litres) is measured every 2 minutes.
| t (minutes) | 0 | 2 | 4 | 6 |
|---|---|---|---|---|
| V (litres) | 10 | 16 | 22 | 28 |
Step 1. The differences are 6, 6 and 6. They are constant, so the model is linear.
Step 2. The gradient is 6 litres per 2 minutes, which is 3 litres per minute.
Step 3. The intercept is 10, because V = 10 at t = 0. The model is V = 10 + 3t.
Step 4. Check: t = 4 gives 10 + 12 = 22, which matches the table. t = 6 gives 10 + 18 = 28, which also matches.
The fit is confirmed with two values that were not used to find the gradient.
The mistake that costs marks
The common slip is choosing a proportional model, V = 3t, because the rate is constant. It ignores the 10 litres already in the tank.
| Step | Wrong | Right |
|---|---|---|
| Model | V = 3t | V = 10 + 3t |
| Test at t = 0 | 0 litres, but the table says 10 | 10 litres, which matches |
| Test at t = 4 | 12 litres, but the table says 22 | 22 litres, which matches |
One test at t = 0 exposes the missing constant.
Check yourself
Decide which model fits: x = 1, 2, 3, 4 and y = 2, 8, 18, 32. Write the equation.
Answer
The differences in y are 6, 10 and 14, which are not constant. The differences of those are 4 and 4, which are constant, so the model is quadratic.
Try y = 2x². At x = 1 and x = 2 it gives 2 and 8.
At x = 3 and x = 4 it gives 18 and 32. All four values match.
What to study next
Continue to explaining limitations and refining a model. If you want to test a model against a full table, read testing a model against supplied data.
For a teacher to go through data tables with you, see online one-to-one Mathematics tuition. The algebra step repair trainer is useful if rearranging the equation is what slows you down.