Higher Level only. It is the tool 2.5 said did not exist at Standard Level.
Put the four points up and ask which family they come from. (1, 6), (2, 12), (4, 48), (8, 768). Let them argue. Someone will notice the doubling of x and quadrupling of y between 2 and 4 and say power; someone else will see the 768 and say exponential. Both have evidence, because 3 × 2ₓ and 3x² agree exactly at x = 2 and x = 4.
That coincidence is the lesson, so dwell on it. Two different families through the same two points, and the scatter cannot separate them.
Then the second view. Exactly straight, both gradients 0.693. Then the third, where the gradient goes 1, then 4. One plot is straight and one is not, and that is the whole method.
Finish on the conversion, because it is where the marks go and where students stop. Gradient 0.693 is ln a, not a.
| Question | Answer |
|---|---|
| 1. The base a | e0.693 = 2. |
| 2. The coefficient k | e1.099 = 3. |
| 3. y at x = 8 | 3 × 256 = 768. |
| 4. Gradient 2 on ln y against ln x | A power, with n = 2. |
| 5. Why the scatter cannot decide | A. Both families hit two of the points exactly. |
1 markChoosing the right pair of variables, with a reason.
1 markThe gradient and the intercept.
1 markConverting them into the model's parameters.
1 markStating the model, ideally checked on a point.
The third is the mark this sub-topic exists for, and it is the one most often missing. A straight line with r² = 1 and no conversion is half an answer.
| They wrote | What happened |
|---|---|
| 0.693 for a | Gave ln a. The error the sub-topic is about. |
| 1.099 for k | Gave ln k, same error on the other parameter. |
| 3 for a, or 2 for k | Swapped the two: gradient gives the base, intercept gives the coefficient. |
| 6 for k | Took the first y value. k is the value at x = 0, which this data does not contain. |
| 256 on question 3 | Forgot the factor of 3. |
| 192 | Used the power model. A good diagnostic: they straightened the wrong plot, or ignored the result. |
| 7.39 on question 4 | Applied e to the gradient. For a POWER relationship the gradient is already the answer, and that asymmetry is worth naming. |
| r² = 1 quoted as the answer | Stopped at the fit. Ask what the question asked for. |
"Why does the power one not need converting?" Because ln y = ln k + n ln x has n sitting as the gradient already, where the exponential has ln a there. Write both derivations side by side once and the asymmetry stops being arbitrary.
"Can I use log base 10?" Yes, throughout. The gradient is then log a instead of ln a, and you undo it with 10 to the power rather than e. What you cannot do is use one base for one list and another for the other, which produces a gradient that is right for neither.
"What if neither plot is straight?" Then it is neither family, which is a legitimate and markable conclusion. It is also worth saying that real data is never exactly straight: you are comparing which is straighter, and r² is the tool for that comparison rather than for proving a model right.
| Stage | What to do |
|---|---|
| Demonstrate | Build four lists: x, y, ln x, ln y. Then run the same linear regression twice, swapping only which list is the x variable, and put the two r² values side by side: 1 against 0.92. The method is one decision made twice, and seeing it done twice is what teaches it. |
| Where they stick | Making the log columns. On the Nspire a column formula is entered in the grey cell under the name; on the Casio the formula goes on the list heading itself. Students type the logs in by hand, get them wrong, and blame the method. |
| The check | Substitute a point that was not used to fit. Here x = 8 should give 768, and it does. If it gives 192, the power model got used somewhere. |
Have them write the two derivations, ln y = ln k + x ln a and ln y = ln k + n ln x, at the top of the page before touching the machine. Every decision in the sub-topic is read off those two lines.
| Step | What |
|---|---|
| 1 | The four points. "Which family?" Let both answers be defended. |
| 2 | Show that both candidate models hit x = 2 and x = 4 exactly. The scatter cannot decide. |
| 3 | Derive the two log identities on the board, before any plotting. |
| 4 | Step the figure. Straight, then curved. Gradients measured at both ends. |
| 5 | Convert: e to the gradient, e to the intercept. Both lines written out. |
| 6 | Check on x = 8, a point not used. |
| 7 | The scaling half: why a log axis is worth using even when the family is known. |
Do not say "r² = 1 proves it is exponential". It is evidence, and on four points it is weak evidence. The honest statement is that ln y against x is straight and ln y against ln x is not, so of these two families the exponential is the one that fits.
Do not let the gradient be reported as the parameter. If "a = 0.693" passes once in class, it will be in the exam, and it is the one thing this sub-topic is marking.