A model is built from data in a certain range, and it may fail outside that range. Before you trust a prediction, check whether the value is possible and how far it lies from the data.
This lesson is part of choosing a model from imperfect information. It sharpens the last step of explaining limitations and refining a model.
What do I check before trusting a prediction?
Ask three questions, in this order.
- Is the predicted value physically possible?
- Is the rule still true at that input?
- How far is the input from the data I used?
A “no” to the first two, or a large distance in the third, means the prediction needs a warning.
Worked example: a burning candle
A candle is 20 cm long when lit. Its length L (cm) is measured each hour.
| t (hours) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| L (cm) | 18 | 16 | 14 | 12 |
Model. The length drops 2 cm each hour, so L = 20 − 2t. Check: t = 3 gives 20 − 6 = 14, which matches.
Predict t = 12. L = 20 − 24 = −4 cm. A candle cannot have a negative length, so the prediction fails the first question.
Find the limit. The candle is gone when L = 0, so 20 − 2t = 0 and t = 10. The model holds for 0 ≤ t ≤ 10.
Statement. The model is valid for 0 ≤ t ≤ 10. At t = 12 the candle has burnt out, and the real length is 0 cm.
The data covered 4 hours, and the valid range is 10 hours. Even so, predictions at 8 hours lean on the assumption that the burn rate stays constant.
The mistake that costs marks
The common slip is to report L = −4 without comment. The arithmetic is right, and the answer is impossible.
| Step | Wrong | Right |
|---|---|---|
| Calculate | L = 20 − 2(12) = −4 | L = −4, which is impossible |
| Check | not done | a length cannot be negative |
| Conclusion | length is −4 cm | the candle has burnt out by t = 10, so L = 0 |
| Range | not stated | 0 ≤ t ≤ 10 |
A quick look at the sign of the answer would have caught the slip.
Check yourself
A tank holds 10 litres at the start and fills at 5 litres per minute, so V = 10 + 5t. The tank’s capacity is 60 litres. What is the valid range, and what happens at t = 12?
Answer
The tank is full when 10 + 5t = 60, so 5t = 50 and t = 10. The model is valid for 0 ≤ t ≤ 10.
At t = 12 the model gives V = 10 + 60 = 70 litres, which exceeds the capacity. The real volume is 60 litres, and the extra water overflows.
What to study next
Go on to explaining two defensible models when a question leaves an assumption open. For a graph-based way to compare a model with data, try the graph evidence comparison lab.
The integrated practice set mixes all four skills. For a teacher to work through range checks with you, see online one-to-one Mathematics tuition.