The shape of the lesson
By the end of the lesson, learners can
- Plotting and correlation: In the lab, pick a data set and drag any cross.
- The line of best fit: In the line of best fit tab, drag the two blue ends to draw your own line by eye.
- Plot small crosses; never join them.
- Describe the correlation (positive, negative, zero; strong or weak) and say what it means in context.
- Draw one ruled line across the data, near the mean point, with points evenly on each side.
- Estimate by reading up and across from the line.
- Judge: estimates outside the data are unreliable; correlation is not cause.
One picture every idea returns to
Positive correlation: as one variable increases, the other tends to increase. Negative: as one increases, the other tends to decrease. A line of best fit is ONE ruled line across the whole data set with roughly equal numbers of points on each side; use it to estimate only within the data.
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Two measurements, one pattern.. A scatter diagram plots two measurements for each item to show whether they are related. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 17 min | Teach and check: Plotting and correlation | In the lab, pick a data set and drag any cross. Project tab 1 of the lab (Scatter diagrams lab, correlation) and let a student drive it while the class predicts. Model the worked example Describe and interpret on the board, one line at a time. | Whole class, then pairs |
| 17 to 30 min | Teach and check: The line of best fit | In the line of best fit tab, drag the two blue ends to draw your own line by eye. Project tab 2 of the lab (Scatter diagrams lab, line of best fit) and let a student drive it while the class predicts. Model the worked example Use the line to estimate on the board, one line at a time. Your turn: A line of best fit passes through (2, 38) and (8, 68). Estimate \(y\) when \(x = 5\). | Mix-Pair-Share |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 17 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in The line of best fit, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Mix-Pair-Share
Students mix around the room until you call pair; they pair with the nearest classmate, discuss the prompt, then mix again. Three prompts of rising difficulty, three different partners.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| Correlation does not prove cause. | Ice cream sales and sunburn cases have positive correlation, but ice cream does not cause sunburn: both rise in hot, sunny weather. |
| Do not join the points dot-to-dot, and do not force the line through the origin. | Examiners also reject a line that stops short of the data or has most points on one side. |
Scatter diagrams and correlation: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
State the type of correlation you would expect between
(a) the outside temperature and the cost of heating a house (b) the length of a person's foot and their height (c) a person's house number and their mass.
Hours of revision \(x\) and test mark \(y\) for 8 students:
\(x\): 1, 2, 3, 4, 5, 6, 7, 8 \(y\): 31, 40, 41, 50, 52, 61, 63, 70
(a) Calculate the mean point. (b) Describe the correlation.
For the data in the previous question, a line of best fit passes through (2, 38) and (8, 68).
(a) Use the line to estimate the mark for 5 hours of revision. (b) Estimate the hours of revision for a mark of 60.
A student uses the same line to predict the mark for 15 hours of revision. Give two reasons why this prediction is unreliable.
The heights (cm) and masses (kg) of 7 students are (150, 45), (155, 49), (160, 52), (165, 30), (170, 61), (175, 64), (180, 70). Identify the outlier and say whether it should be used when drawing the line of best fit.
A town council finds strong positive correlation between the number of fire engines sent to a fire and the cost of the damage. A councillor says: 'Sending fewer fire engines would reduce the damage.'
(a) What does the positive correlation mean in context? (b) Explain why the councillor is wrong.
Answer key and marking notes
- (a) negative (warmer days, less heating)
- (b) positive
- (c) zero (no correlation)
- (a) \(\bar{x} = \dfrac{36}{8} = 4.5\) and \(\bar{y} = \dfrac{408}{8} = 51\), so the mean point is (4.5, 51).
- (b) Strong positive correlation: students who revised for longer tended to score higher marks.
- The line rises \(68 - 38 = 30\) marks over 6 hours, so 5 marks per hour.
- (a) \(38 + 3 \times 5 = 53\) marks.
- (b) 60 is \(60 - 38 = 22\) marks above (2, 38), which is \(22 \div 5 = 4.4\) hours further, so about 6.4 hours.
- 15 hours is outside the range of the data (1 to 8 hours), so the trend may not continue.
- The line gives \(38 + 13 \times 5 = 103\) marks, which is more than a test out of 100 allows.
- The other points rise by about 4 kg every 5 cm, so about 57 kg is expected at 165 cm. (165, 30) is far below the pattern: it is the outlier.
- It may be a recording error, so ignore it when drawing the line of best fit.
- (a) Fires that had more fire engines sent tended to cause more damage.
- (b) Correlation does not show cause.
- A third factor, the size of the fire, explains both: big fires need more engines AND cause more damage.
- Sending fewer engines would probably make the damage worse, not better.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
What type of correlation would you expect between the number of hours of sunshine in a day and the number of umbrellas sold that day?
- ANegative correlation
- BPositive correlation
- CNo correlation
- DDirect proportion
The scatter diagram shows the temperature and the number of ice creams sold at a kiosk on 11 days. Which point is an outlier?
- A(20, 15)
- B(28, 62)
- C(12, 23)
- D(22, 45)
The line of best fit on the scatter diagram passes through (12, 22) and (28, 62). Use the line to estimate the number of ice creams sold when the temperature is 22°C.
- A47
- B55
- C45
- D48.7
The kiosk owner uses the line of best fit on the scatter diagram to predict the ice cream sales on a day when the temperature is 40°C. Which comment is correct?
- AUnreliable, because 40°C is outside the range of the data.
- BReliable, because the correlation is strong and positive.
- CReliable, because the line gives an exact answer of 92.
- DUnreliable, because the line does not pass through the origin.
The table shows the hours of practice \(x\) and the score \(y\) of five students.
| \(x\) | 2 | 4 | 6 | 8 | 10 |
|---|---|---|---|---|---|
| \(y\) | 11 | 18 | 20 | 27 | 34 |
- A(6, 22)
- B(6, 20)
- C(30, 110)
- D(22, 6)
The line of best fit on the scatter diagram passes through (12, 22) and (28, 62). Use it to estimate the temperature on a day when 50 ice creams are sold.
- A23.2°C
- B20°C
- C16.8°C
- D117°C
Quiz key
On sunnier days fewer people buy umbrellas, so as one variable increases the other tends to decrease: negative correlation. Positive correlation would mean more sunshine goes with MORE umbrellas sold. There is clearly a link, so 'no correlation' is wrong, and direct proportion is an exact rule \(y = kx\) with a positive link, not a description of a scatter of points.
The other points rise steadily from (12, 23) to (28, 62), so at 20°C about 42 ice creams would be expected. The point (20, 15) is far below this pattern, so it is the outlier (perhaps the kiosk closed early that day). The tempting (28, 62) is the largest value, but it fits the pattern well. (12, 23) is the smallest value and (22, 45) lies just below the line; both follow the trend.
From \(x = 12\) to \(x = 28\) the line rises \(62 - 22 = 40\) over 16°C, which is 2.5 per °C. At 22°C: \(22 + 2.5 \times (22 - 12) = 22 + 25 = 47\). The tempting 45 is the plotted point at 22°C, but an estimate is read from the LINE, not from a single point. 55 is \(2.5 \times 22\), treating the line as if it passed through the origin, and 48.7 scales \(62 \times \dfrac{22}{28}\), which also assumes the line goes through (0, 0).
The data only go from 12°C to 28°C. Extending the line to 40°C is extrapolation, and the pattern may not continue (the line would give \(2.5 \times 40 - 8 = 92\), but on very hot days people may stay indoors). The tempting 'strong correlation' answer is wrong: strong correlation inside the data does not make a prediction outside it safe. A line of best fit does not need to pass through the origin.
\(\bar{x} = \dfrac{2 + 4 + 6 + 8 + 10}{5} = \dfrac{30}{5} = 6\) and \(\bar{y} = \dfrac{11 + 18 + 20 + 27 + 34}{5} = \dfrac{110}{5} = 22\), so the mean point is (6, 22). The tempting (6, 20) uses the middle \(y\) value, which is the median, not the mean. (30, 110) gives the totals without dividing, and (22, 6) writes the coordinates the wrong way round: the \(x\) value always comes first.
The gradient is \(\dfrac{62 - 22}{28 - 12} = 2.5\), so the line is \(y = 2.5x - 8\). Put \(y = 50\): \(2.5x = 58\), so \(x = 23.2\)°C (inside the data, so the estimate is reasonable). The tempting 20°C is \(50 \div 2.5\), which forgets the line does not pass through the origin. 16.8°C subtracts the 8 instead of adding it (\(\dfrac{50 - 8}{2.5}\)), and 117 puts 50 in as the temperature (\(2.5 \times 50 - 8\)).
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.