Applications only. The modelling process itself is 2.6.
Give them the three numbers and ask for hour ten. Just 2, 4, 8, and "how many by hour ten?". Almost everyone doubles: 2048 or thereabouts. A few will spot a quadratic. Write both up.
Then step the figure. The second view is the important one, not the third: by hour five the two models already differ by a factor of two, and nothing in the data said which was right. The third view makes it dramatic and the second view makes it true.
The fourth view is the answer. There are 180 students in the year group, the exponential predicts 256 sign-ups by hour seven, and that settles it without any appeal to which curve looks nicer. The context chose the model, and the data could not.
Point out the square-root vertical scale out loud. On a linear scale the quadratic would be flat against the axis beside 2048, which would be its own kind of lie, and a student who notices the scale and says so is doing exactly what 2.6 asks for.
| Question | Answer |
|---|---|
| 1. Quadratic at hour 10 | 100 + 10 + 2 = 112. |
| 2. Exponential at hour 10 | 2 × 1024 = 2048. |
| 3. Coffee after 10 minutes | 6.44 + 28 = 34.44°C. |
| 4. Period of the tide model | 360/30 = 12 hours. |
| 5. What to write | B. Either fits; the context decides. |
1 markNaming a family, with a reason from the context.
1 markSetting up the equations for the parameters.
1 markSolving them, usually with technology.
1 markUsing the model, and saying where it stops being believable.
The first and last are sentences rather than arithmetic, and they are the two most often blank. Train the two sentences explicitly: "I have chosen an exponential model because the growth is proportional to the number already signed up", and "this model is not reasonable beyond hour 7, because the year group has only 180 students in it".
| They wrote | What happened |
|---|---|
| 102 on question 1 | Dropped a term from x² + x + 2. |
| 1000 | Cubed. Worth checking they know which family they are in. |
| 1024 on question 2 | Left out the k. At hour 0 the count was 2, and that 2 is the k. |
| 20 | Multiplied 2 by 10. An exponent read as a multiplier, which is the error 1.3 and 1.4 exist to stop. |
| 6.44 on question 3 | Left off the +28. The commonest exponential-model error: the vertical shift is dropped because it is not in the growth. |
| 28 | Gave the asymptote as the answer. Worth asking whether the coffee ever actually reaches the room. |
| 30 on question 4 | Gave b rather than the period. |
| 1.2 or 2.5 | Gave the amplitude or the principal axis. All three parameters are in the question and only one was asked for. |
| 6 | Half a period, which is high tide to low tide. A good answer to a different question. |
| "r² = 1 so it is the right model" | The belief to break. Three points determine a quadratic exactly, so a perfect fit is arithmetic. |
"So which model is right?" Neither, for ever. Both are wrong by hour 13. The useful question is which is defensible over the range you need, and that is a judgement you have to write down rather than a number you can find.
"Can't I just use the one with the better fit?" Here they both fit perfectly, so there is nothing to compare. Say plainly that fit is how you eliminate families, not how you choose between the survivors, and the straight line is the one the data did eliminate: it predicts 5 at hour 1 where the data says 4.
"Why is there a +28 in the cooling model?" Because coffee cools to the room, not to zero. Ask what 60(0.8)ₜ on its own would predict after an hour: essentially 0°C, in Bangkok. The constant is the room and the 60 is how far above it the drink started.
| Stage | What to do |
|---|---|
| Demonstrate | Solve the three simultaneous equations for the quadratic in front of them, then fit an exponential to the same points, then put both on a graph with the three data points. Seeing both curves through all three dots is the demonstration; everything else on this page is commentary on it. |
| Where they stick | The Casio's simultaneous solver wants coefficients, not equations, so c = 2 has to be entered as 0, 0, 1 | 2. Students enter 2 in the wrong column and get nonsense. Also: at SL they are not expected to do non-linear regression, so if a student reaches for ExpReg, that is fine as a check and not as the method. |
| The check | Before solving anything, ask what the model should give at hour 0. The answer is in the data: 2. A model that does not reproduce the point you fitted it to has been set up wrongly. |
Keep the angle in DEGREES for the tide model. The syllabus uses degrees for sinusoidal models at Standard Level, and a machine in radians turns a 12-hour period into one of about 0.2 hours without complaining.
| Step | What |
|---|---|
| 1 | "2, 4, 8. How many by hour ten?" Collect answers. Write both models up. |
| 2 | Fit both, in front of them, and confirm both go through all three points. |
| 3 | Step to hour 5, then hour 10. The second step is the one that matters. |
| 4 | The 180 cap. Let the context do the choosing. |
| 5 | The straight line, as the family the data DID eliminate. |
| 6 | The families table, built from "what is the story?" rather than read out. |
| 7 | The three worked models: cooling, tide, inverse square. One each for asymptote, period and negative power. |
Do not say "exponential growth" about anything with a ceiling. Sign-ups, populations and infections all have one, and the honest model for those is logistic, which is AHL 2.9. At Standard Level the right move is an exponential plus a stated limit on where it applies, not an exponential pretending the limit does not exist.
Do not let a good fit stand as a justification. If "it fits well" is accepted once in class, it will be the whole of their reasoning in an exam, and it earns nothing.