Common content, so this runs for both courses.
Put the first view up and ask the class to mark it out of three. Most will say two or three: the shape is right, the curve is smooth, it looks like work. It earns one.
Then step to the second. Same curve, labels added, full marks. The sentence to land is that the curve was never the thing being marked. Three marks for a sketch means three things to find, and a student who hands in only a shape has done one of them.
The third view is there to settle the other half: a draw is a different instruction, not a better one. If a question wants a draw it will give you a grid, and if it has not given you a grid it does not want one.
| Question | Answer |
|---|---|
| 1. Largest root | x(x² − 3) = 0, so √3 = 1.73. |
| 2. y at the local maximum | (−1)³ − 3(−1) = 2. |
| 3. Number of x-intercepts | 3. |
| 4. An accurate plot with no labels | 1 mark of three. |
| 5. The label on (−1, 2) | B. Local maximum. |
1 markThe shape, with the right end behaviour.
1 markThe intercepts, marked and valued.
1 markThe turning points, with coordinates.
Axes labelled is not usually its own mark but it is a condition on the other two: a coordinate cannot be read off an unlabelled pair of lines, so an unlabelled sketch loses the feature marks even when the features are drawn in the right places.
| They wrote | What happened |
|---|---|
| 3 on question 1 | Stopped at x² = 3. |
| 1.5 | Halved 3 instead of taking its root. Worth a word: halving and rooting feel similar and are not. |
| −1 on question 2 | Gave the x-coordinate. The commonest confusion in the whole topic. |
| −4 | Computed −1 − 3 rather than −1 + 3. A sign inside a substitution. |
| −2 | Gave the local minimum. Ask which one is on the left. |
| 1 on question 3 | Thinks a cubic crosses once. True when both turning points are the same side of the axis; here they are at +2 and −2. |
| 3 on question 4 | Believes accuracy is what earns marks. The belief this page exists to remove. |
| "Maximum" | On question 5, a false claim about a cubic. At x = 10 the function is 970. |
| "Vertex" | Imported from parabolas. A cubic has two turning points and neither is a vertex. |
"How accurate does a sketch have to be?" Not at all, and that is not permission to be careless. The relationships have to be right: the local max above the axis, the local min below it, the curve crossing three times, the left end going down. Those are checkable without a ruler.
"Do I have to label things the question did not ask about?" Label what the question is about plus the intercepts and turning points. That covers almost every mark scheme and costs ten seconds.
"My calculator drew it differently." It drew a different window. Ask what the x and y ranges were. On the default window this cubic is a near-vertical line and has no visible features at all, which is worth showing once so they distrust the default for ever.
| Stage | What to do |
|---|---|
| Demonstrate | Graph the cubic on the default window first, deliberately, and ask what the features are. There are none to see. Then set x from −3 to 3 and y from −4 to 4 and ask again. Doing it in that order is what makes the point land. |
| Where they stick | Both machines find one feature per go, with bounds. Students bound the whole screen and get whichever extremum the algorithm meets first, then report it as the only one. |
| The check | Before using any tool, ask how many roots and how many turning points a cubic can have. A student expecting three and two will notice when they have found one and one. |
Make them write the window down beside the sketch. It is the single habit that stops a screen being copied as though it were the graph.
| Step | What |
|---|---|
| 1 | First view of the figure. "Mark this out of three." Collect the guesses. |
| 2 | Second view. The same curve scores full marks. Ask what changed. |
| 3 | Build the feature list from the curve rather than handing it over. |
| 4 | The word "local", and why "maximum" is false here. |
| 5 | Default window on the calculator, then a chosen one. In that order. |
| 6 | Third view: a draw, and when a question actually wants one. |
Do not say "a sketch is a rough graph". It gives permission for exactly the answer that scores one mark of three. Say "a sketch is a labelled graph that does not need to be to scale".
Do not accept an unlabelled sketch in class because the shape is right. If it passes in the room it will be handed in, and the shape is one mark of three.