Applications only. The families themselves are 2.5.
Fit the quadratic with them, confirm it reproduces all four measurements, and let them feel pleased about it. Then ask for the height after a year. The answer is 14.84 m, and the room will immediately want to know what went wrong with the fitting.
Nothing did. That is the point, and it is worth saying in those words: extrapolation does not make a model wrong, it asks the model a question it was never fitted to answer. Step the figure and watch the thick green measured section shrink to almost nothing as the view widens. The picture carries the argument.
Then make the domain a number rather than a feeling. The plant grows to 120 cm, the model passes that at week 13.06, so the domain ends at about week 13. A student who writes "the model is only valid for small w" gets nothing; one who writes "about 0 to 13 weeks, because the plant only grows to 120 cm" gets the mark.
| Question | Answer |
|---|---|
| 1. Week 10 | 50 + 25 + 2 = 77 cm. |
| 2. Week 52 | 1352 + 130 + 2 = 1484 cm. |
| 3. When it reaches 120 cm | w² + 5w − 236 = 0, so 13.06 weeks. |
| 4. Week −2 | 2 − 5 + 2 = −1 cm. |
| 5. The comment that earns the mark | C. A number, a judgement and a limit. |
1 markThe equations, set up from the data.
1 markSolving them for the parameters.
1 markA reasonable domain, with a reason.
1 markThe prediction, with a comment on whether it can be trusted.
Two of the four are sentences. Give the class the shape of each one and make them write it even when the answer is obvious: a number, then whether you believe it, then where you would stop believing it.
| They wrote | What happened |
|---|---|
| 52 on question 1 | Dropped the 2.5w term. |
| 127 | Used w² rather than 0.5w². The leading coefficient is a half and it is easy to lose. |
| 75 | Dropped the +2, which is the height at week 0 and therefore the one parameter the data hands you directly. |
| 14.84 on question 2 | Answered in metres when the question said centimetres. Right work, wrong unit. |
| 1352 | Only the first term. Worth noting that at week 52 the first term dominates, so this is nearly right and still wrong. |
| −18.06 on question 3 | The other root. A negative week is before planting, so the positive one is the answer. |
| 15.49 | Solved 0.5w² = 120, dropping the other terms. |
| 13 instead of 13.06 | Right value, wrong precision. Two decimal places were asked for. |
| 9 on question 4 | Treated −2 as +2. The middle term is 2.5 × (−2). |
| "The model is wrong" | On question 5. The misunderstanding to correct: it reproduces all four measurements exactly, so it is not the model that is wrong. |
"How do I know what a reasonable domain is?" From the context, every time. Something in the situation gives you a limit: a year group has 180 students, a coach holds 45, a plant grows to 120 cm, a month has 31 days. Find the limit and solve for where the model reaches it.
"Should I check backwards too?" Yes, and almost nobody does. Here w = −2 gives a height of −1 cm. A negative output in a quantity that cannot be negative is the easiest domain argument available and it is free.
"Why not use regression?" At Standard Level on this course you are not expected to do non-linear regression, and the syllabus says so. What you are expected to do is set up and solve up to three linear equations in three variables. A student reaching for QuadReg has not done anything forbidden, but they have skipped the step that carries the mark.
| Stage | What to do |
|---|---|
| Demonstrate | Solve the three equations, then define the function and ask it for week 52 in one keystroke. The speed is the point: the fitting and the predicting are seconds of work, so all the time in a modelling question belongs to the two sentences. |
| Where they stick | On the Casio, c = 2 has to be entered as 0, 0, 1 | 2 and students put the 2 in the first column. The solver then answers a different system without complaining, which is the worst kind of wrong. Make them read the three rows back before pressing SOLVE. |
| The check | Substitute the point you did NOT use to fit. Here week 3 should give 14, and it does. A model that fails its own held-back point has been fitted wrongly or the family is wrong, and either way you want to know before you write a conclusion. |
Teach the held-back point as routine. Four data points and a three-parameter model means one point is spare, and using it as a check costs one substitution and catches almost every setup error.
| Step | What |
|---|---|
| 1 | The four measurements. Fit the quadratic together, three equations on the board. |
| 2 | Check it against week 3, the point not used. It works. |
| 3 | "How tall after a year?" Let 14.84 m land. |
| 4 | Establish that the fit is still perfect. The model is not the problem. |
| 5 | Find the domain as a number: 120 cm at week 13.06. |
| 6 | Check backwards. A negative height at week −2. |
| 7 | Write the comment sentence together, then have them write one for a different model. |
Do not say "the model breaks down". It does not break; it carries on doing exactly what it was fitted to do. What ends is the range over which it describes a plant. The difference matters because students who think models break look for a better model instead of stating a domain.
Do not accept "it is not realistic" on its own. It is the right instinct and it is not an answer. Push for the number: not realistic because the plant only reaches 120 cm, which the model passes at week 13.