Answers, where the four marks sit, and the forty seconds that changes how a class reads a correlation coefficient for good.
Ask them to predict, then drag one point a long way out. Most will say the line moves. Few will predict how far r falls.
A convincing 0.989 becomes a number they would never quote, from moving a single student. Forty seconds, and afterwards a class treats a correlation coefficient as something that can be broken rather than a fact handed down by the calculator.
Means are exactly 5 and 62, chosen so the hand calculation agrees with the readout to the digit rather than inviting a rounding argument.
| Starting values | Sxx = 60, Sxy = 276, so b = 4.6 exactly and a = 39. The line is y = 4.60x + 39.0, with r = 0.989 and r² = 0.978. |
| 1. Sales at 31 degrees | 136. 8.2 × 31 − 118 = 136.2, and coffees are whole. |
| 2. Why refuse at 5 degrees | B. Outside the range of the data, and the model returns minus 77 coffees. |
| 3. Same r, different gradients | B. Equally tight fit; B simply changes faster per unit of x. |
2 marksThe regression line. Almost nobody loses these, because the calculator produces them. Do not spend the lesson here.
1 markInterpreting the gradient in context, with units. "Each extra hour of revision is associated with about 4.6 more marks." Writing "the gradient is 4.60" restates the answer and earns nothing.
1 markKnowing the limit. 4.6 × 40 + 39 = 223, a mark above 100. The impossible value is the strongest form of the answer, stronger than naming extrapolation alone.
| They choose | What it means |
|---|---|
| 372 (Q1) | Added 118 rather than subtracting. Sign slip, not a method problem. |
| 136.2 (Q1) | Correct arithmetic, and the page accepts it while noting coffees are whole. Do not mark this as wrong; it is a presentation point. |
| "Too cold" (Q2) | The one that matters. This is reasoning about coffee rather than about mathematics and earns nothing, yet it is the first thing most of the class writes. The mathematical answer is extrapolation beyond the data range. |
| "r would differ" (Q2) | Close, and worth praising. They have the right instinct and the wrong vocabulary. Steer to the word extrapolation. |
| "B is stronger" (Q3) | Confusing gradient with strength, which is the central confusion of the sub-topic. r says how tightly, the gradient says how much. Draw both on the board. |
"97.8% correlation." They have quoted r² as if it were r. Keep r for strength. r² needs the phrase "proportion of the variation explained", said carefully or not at all at this level.
"Revising more causes higher marks." Do not just reply with the slogan. Make them produce a rival explanation, which they can: a student who revises nine hours is probably different in several other ways too. The slogan labels the problem; producing the alternative is what teaches it.
Rounding the gradient then using it. Harmless with this data because 4.6 is exact, which is why it was chosen. Flag it anyway: with real data it loses accuracy marks.
| What is happening | |
|---|---|
| 1 | Let them conclude "revise more, score more" out loud and leave it standing. You come back for it at the end. |
| 2 | The widget. Predict, drag, watch r collapse, then talk about outliers in bivariate data. |
| 3 | What r measures and does not. The parabola point is worth a quick sketch: r near zero means not linear, not unrelated. |
| 4 | The hand calculation, checked against the readout. Insist on the check. |
| 5 | Questions 1 to 3, with question 2 as the one that matters. |
| 6 | Return to the opening claim. Does the data support it? This is where the lesson finishes rather than stops. |
Do not give them the calculator sequence before they have sketched a scatter. Once they can produce a line in four keystrokes they stop looking at the data, and every mark on this sub-topic that is worth having comes from looking at the data.
Do not use "strong correlation" and "steep line" in the same breath. It is the exact pairing that produces the question 3 error, and it is very easy to say without noticing.
Three tiers, ramping the way practice should: the method on its own, then the method inside something real, then a challenge. Set the tier the class in front of you needs rather than one undifferentiated sheet. Answers are given so these can go straight onto a board.
The method on its own, with friendly numbers. Set these first and move on quickly once they are secure.
The same skill inside a real situation, where the first job is working out what is being asked.
Reasoning, working backwards, or spotting an error. These are where the top grades are decided.
Works on a phone, though dragging points is easier with a mouse or on a larger screen. The library is served from this site rather than a public CDN, so it works behind a school firewall that blocks third-party scripts. Nothing a student does is saved or sent anywhere.