Science: data representation and defensible extrapolation
| English | 中文 | Pinyin |
|---|---|---|
| interpolation/ɪnˌtɜːpəˈleɪʃn/ | 插值 | chā zhí |
| extrapolation/ekˈstræpəleɪʃn/ | 外推 | wài tuī |
A decision before an answer
- A line joining measured points can support interpolation. It does not promise that the trend continues without limit.
- Your goal: Read values with correct axes, units and conditions.
Read the relationship
- Data Representation passages use tables, graphs or diagrams as primary evidence. Start with the variable labels and units, and identify which conditions are held constant. A row may report one trial or an average; a legend may distinguish groups measured under different conditions. Locate the requested value before using prior scientific knowledge.
- Translate a table into a trend and interpolate within a measured interval.
What is the average rise from minute 4 to 6?
Temperature rises 4°C over two minutes.
Use the defining rule
- Translate representations without changing their meaning. A rising quantity can have a falling rate of increase. A steep line is meaningful only relative to axis scales and units. When comparing curves, distinguish an absolute value, a difference and a rate. Use corresponding x-values for a fair comparison unless the stem explicitly asks for another relationship.
- Evaluate an extrapolation against the model’s assumptions.
Which statement fits all four measurements?
Successive equal-duration increments are 8, 6 and 4°C.
Check the conditions
- Interpolation estimates inside observed bounds. For a straight segment between two points, use the fraction of the input interval and apply it to the output difference. Extrapolation extends beyond those bounds and depends more strongly on the proposed model. Negative predicted quantities or physical limits supplied by the passage can reveal a model’s invalid extension.
- Evaluate an extrapolation against the model’s assumptions.
Original trial: identical containers hold a test liquid, starting at 20°C in the same room. Recorded times in minutes are 0, 2, 4, 6; measured temperatures in °C are 20, 28, 34, 38. The liquid warms, but consecutive two-minute rises are 8, 6 and 4°C: its average warming rate decreases. Linear interpolation between 2 and 4 minutes gives T(3)=28+(3−2)/(4−2)×(34−28)=31°C. Extending the first interval’s 4°C/min rate to 6 minutes predicts 44°C, inconsistent with the measured 38°C. New evidence therefore rejects a constant-rate model across all six minutes.
Linear interpolation between 34°C at minute 4 and 38°C at minute 6 gives ____°C at minute 5.
The midpoint of the interval uses half the 4°C rise.
Apply the task format
- Reevaluation asks whether new evidence supports an earlier trend or requires a different explanation. One discordant measurement could reflect noise or a model limitation; use the stated uncertainty and experimental conditions. The dataset here is original and hypothetical, so the task’s given model supplies the relevant relationships. It is not a claim about every real material.
- Evaluate an extrapolation against the model’s assumptions.
A monotonic rise in temperature is not a constant rise per minute. Do not extrapolate an early slope as if later measurements did not exist.
Which answer fits this case?
Read values with correct axes, units and conditions
An estimate outside the measured range is interpolation.
It is extrapolation and needs stronger assumptions.
Keep the distinctions
- interpolation 插值 — An estimate between observed input values using an explicit assumption.
- extrapolation 外推 — An estimate beyond the measured range that relies on model extension.
- Read values with correct axes, units and conditions.
- Translate a table into a trend and interpolate within a measured interval.
- Evaluate an extrapolation against the model’s assumptions.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.