Nonlinear investigations and new evidence
Work backwards from a target response, justify a hypothesis, and test how a changed procedure affects a scientific model.
What you will tackle
- Read a nonlinear temperature-response dataset
- Use several reasoning steps and qualified interpolation
- Identify conditions for a target outcome
- Predict and evaluate a changed-procedure hypothesis
A maximum is limited to tested conditions
The invented gas-production experiment below holds reaction time, liquid volume and substrate concentration fixed while varying temperature and material amount. Dataset A uses one unit of active material; B uses two. A's response rises to 30°C and then falls, so a single straight increasing rule would fail across the whole range. A target response can correspond to more than one temperature on a nonlinear curve.
Use and test a model
A hypothesis is a testable proposed relationship, such as doubling active material doubles the measured gas under these conditions. It predicts B = 2A at corresponding temperatures. New evidence can support, limit or contradict that relationship. A changed procedure can also change other factors, so justify the controls and treat interpolation or extrapolation as assumptions rather than observations.
| Temperature (°C) | A: gas (mL) | B: gas (mL) |
|---|---|---|
| 10 | 2 | 4 |
| 20 | 5 | 10 |
| 30 | 9 | 18 |
| 40 | 6 | 12 |
Build your notes, then practise
Work on paper before opening an explanation. Guided tasks provide a starting point; independent tasks remove the scaffolding; application tasks ask you to justify a method or interpretation in context.
Guided practice
Guided 1
Use Dataset A: temperatures 10,20,30,40°C produce 2,5,9,6 mL. Find the largest tested response and estimate a temperature giving 7 mL between 20 and 30°C under local linear interpolation. Would an always-increasing model fit all points?
Show a starting hint
Use only the two points bracketing the specified estimate.
Compare with the explained answer
- The largest observed response is 9 mL at 30°C.
- 7 mL is halfway from 5 to 9, so the local estimate is 25°C; it is not a measured value.
- The fall from 9 to 6 mL between 30 and 40°C contradicts an always-increasing rule.
Answer / response criteria: 30°C and 9 mL; estimated 25°C; no.
Guided 2
A supplied ideal-light model gives intensity values 36,9,4 units at distances 1,2,3 m. Use products to identify an inverse-square relationship, predict at 4 m, and distinguish the prediction from an observation.
Compare with the explained answer
- Compute Id²: 36·1² = 9·2² = 4·3² = 36.
- The model is I = 36/d²; at 4 m it predicts 36/16 = 2.25 units.
- The 4 m result is outside the measured distance range and depends on the stated ideal model; it was not observed.
Answer / response criteria: I = 36/d²; predicted 2.25 units, qualified extrapolation.
Independent practice
Independent 1
Dataset B at the same temperatures gives 4,10,18,12 mL. State a hypothesis connecting A and B and check every paired value. In A, estimate another temperature giving 7 mL between 30 and 40°C. Explain why the target does not determine one unique temperature.
Compare with the explained answer
- The hypothesis B = 2A matches all four supplied pairs: 4/2,10/5,18/9,12/6 are 2.
- From 9 down to 7 is two-thirds of the fall from 9 to 6; the local estimate is 30 + (2/3)10 = 36⅔°C.
- The rising and falling branches can reach the same response, so 7 mL has two local estimates in the shown range.
Answer / response criteria: B = 2A fits the supplied data; about 36.7°C on the falling branch, as well as 25°C on the rising branch.
Independent 2
An invented experiment records gas V = 2,8,18 mL at t = 1,2,3 seconds. Test a model V = kt², derive k, and use it to predict gas at 4 seconds and the time for 32 mL. Convert that predicted time into minutes.
Compare with the explained answer
- V/t² is 2 for each observation, so k = 2 mL/s² fits the data.
- At 4 seconds the model gives 2·16 = 32 mL. Solving 2t² = 32 gives positive time 4 seconds.
- Four seconds is 1/15 minute; both uses extend beyond the measured three-second interval and require the model to continue.
Answer / response criteria: k = 2; 32 mL; 4 seconds = 1/15 minute, subject to extrapolation.
Application practice
Application 1
A new repeat of B at 40°C produces 10 mL rather than 12. Does this prove that material amount has no effect? Design a controlled follow-up and explain what additional evidence is needed before extending the doubling model to 50°C.
Compare with the explained answer
- The result differs from the exact 2A prediction at 40°C, but does not prove zero effect; repetition and measurement variation matter.
- Repeat both material amounts at matched temperatures with the same time, volume and concentration, recording variability and checking material consistency.
- 50°C lies outside the tested range. Measurements there, and checks on whether temperature changes the mechanism, are needed before extending the model.
Answer / response criteria: The exact doubling claim needs qualification and replicated checks; no-effect and 50°C claims are unsupported.
Application 2
Under the supplied ideal-light relationship I = P/d² (P is a normalized source-strength parameter with geometry absorbed), initial source-strength parameter P = 36 gives I = 9 at d = 2. Predict the consequence of doubling source strength without changing distance, and of doubling distance without changing source strength. Design a check that separates those systematic effects from repeat variation.
Compare with the explained answer
- Doubling P at fixed d doubles intensity to 18; doubling d at fixed P reduces it to 36/16 = 2.25.
- Vary only source strength in one comparison and only distance in another; keep detector, alignment and environment consistent.
- Repeat each condition, report variability and compare measured means with both predictions. A single unexpected reading cannot identify its cause.
Answer / response criteria: Predicted 18 and 2.25 units; controlled repeated comparisons are required to test the model.
What can you explain now?
Under the supplied ideal-light relationship I = P/d² (P is a normalized source-strength parameter with geometry absorbed), initial source-strength parameter P = 36 gives I = 9 at d = 2. Predict the consequence of doubling source strength without changing distance, and of doubling distance without changing source strength. Design a check that separates those systematic effects from repeat variation.
Compare with the explanation
Predicted 18 and 2.25 units; controlled repeated comparisons are required to test the model.
Explain the reasoning before comparing. This review records your self-assessment rather than an official score or mastery decision.
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Source and review status
Coverage follows the named requirements and examples in the 2026–2027 ACT test guide and subject descriptions. That guide explicitly does not list every possible topic. The broader coded College and Career Readiness Standards are also mapped to conceptual teaching and practice; this does not calibrate a learner to a score band. The separate scientific-background context list and reading-complexity rubric have their own stated limits.
Original teaching and explained self-check practice are available. Specialist review and difficulty calibration remain pending. Essays require self-check or teacher review and are not automatically marked or awarded official scores.
Official source specification / guide