A limitation is an assumption that may fail, plus a statement of which way the answer would change. A refinement is a new version of the model that removes that limitation.
This lesson follows choosing a simple model for a real situation in the SPM Mathematics modelling sequence. It returns to the van-hire model from defining variables and assumptions.
How do I write a limitation that earns marks?
Use two parts: the assumption that fails, and the effect on the answer. “The model ignores tolls, so the real cost could be higher” has both parts.
- Pick the assumption that is least likely to hold.
- State what the real situation might contain instead.
- Say whether the answer would rise or fall.
Avoid “the model is not exact”, which names no assumption.
Worked example: adding a waiting charge
The van model is C = 120 + 0.8d, where C is the cost in RM and d is the distance in km. For d = 150, C = RM240.
Limitation. The model assumes the driver never waits. On a school trip the van may wait while students visit a museum, and the real cost would be higher if the company charges for waiting time.
Refinement. Suppose the company charges RM15 for each hour of waiting beyond 2 hours, and let w be the total waiting hours. The refined model is:
C = 120 + 0.8d + 15(w − 2), for w ≥ 2.
For d = 150 and w = 5, the cost is 120 + 120 + 15 × 3 = 120 + 120 + 45 = RM285.
The refinement answers the limitation directly. The range w ≥ 2 is stated, because the formula would give a negative charge for w below 2.
The mistake that costs marks
The common slip is saying only that the model “may not be accurate”. It names no assumption and gives no direction.
| Step | Wrong | Right |
|---|---|---|
| Limitation | the model may not be accurate | the model ignores waiting time |
| Effect | not stated | the real cost could be higher |
| Refinement | none | add 15(w − 2) for w ≥ 2 |
| Range | not stated | w ≥ 2 |
Name the assumption and the direction, and the comment is complete.
Check yourself
A plant height model is h = 4 + 2.5t, where h is in cm and t is in weeks, based on measurements for the first 6 weeks. Predict h at t = 30 and give one limitation.
Answer
At t = 30, h = 4 + 2.5 × 30 = 4 + 75 = 79 cm.
Limitation: the model assumes the plant grows at the same rate forever, but the data covers only 6 weeks. A plant stops growing at a certain height, so the real value at 30 weeks would be lower than 79 cm.
What to study next
Read testing whether a model still works outside the supplied data range for a fuller treatment of predicting beyond the data. Then try the modelling practice set.
For a teacher to help you write limitations in the wording a marker expects, see online one-to-one Mathematics tuition. The algebra step repair trainer can help with the algebra of a refined model.