At a glance. One lesson of about 60 minutes. It assumes the regression line from Lesson 4 and interpolation from Year 11 Standard. Students find residuals, check that they add to 0, read a pattern, and weigh r against the shape of the residuals and the position of x in the data.
Calculator: GDC optional. A GDC can store and plot the residuals after a regression, which is the check this lesson asks for. It also gives a prediction at any x without a warning, which is why the judgement of interpolation and extrapolation stays with the student.
What students will learn
Know
- The regression line and its equation.
- How to substitute into a line.
- The meaning of interpolation and extrapolation.
Understand
- A residual is the dot's y minus the line's y, and the residuals of a regression line add to 0.
- A U or an arch in the residuals shows the line is the wrong shape, whatever r is; a fan shows predictions are less reliable where it is wide.
- A prediction outside the data assumes the pattern continues, and its error can grow.
Do
- Find a residual and a missing residual.
- Read a residual pattern and judge a linear model.
- Quantify the error of an extrapolation as a percentage.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: residuals, their sum and their squares (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Ask which set has the higher r before showing the second.
- Criterion C. Communicating: writing a two-part judgement, shape from residuals and trust from the position of x (the exit ticket and the what-if).
- Criterion D. Applying mathematics in real-life contexts: sales forecasts, growth and choosing between models.
The one thing to do with the figure
Make students predict r for the curved set. Before switching, ask which of the two sets will have the higher r. Most will say the rough band. Switch and let the surprise stand for a moment.
Then move through the eight dots on the curved set and ask what the residual bars are doing. Students should say “positive at both ends and negative in the middle” and name the shape. Finish by asking what to report.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: which set has the higher r? Vote. |
| 10 min | The figure and key idea: residuals, their sum, the pattern. |
| 10 min | Warning card, the two worked examples and the faded example in pairs. |
| 5 min | The check. If many choose A, ask them to write the sizes of the residuals in a row. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 3 above; the total of the squared vertical gaps; x = 12. |
|---|---|
| Hook | The curved set has the higher r (0.98 against 0.93) and the worse straight-line model: its residuals 7, 1, −3, −5, −5, −3, 1, 7 form a U. |
| Worked examples | Residuals 2, −1 and 0; and a U shape with total 0. |
| Faded example | No pattern: a straight line is a fair model. |
| 1. Reading residuals | B. A reads only the signs; C sees a bend that is not there; D reads the zero total as a fit. |
| 2. Finding a residual | 4. 34 − (2.5 × 8 + 10). |
| 3. A missing residual | −3. 0 − (1.5 − 2 + 0.5 + 3). |
| 4. Total of the squared residuals | 13. 3² + (−2)². |
| 5. High r with a pattern | D. A trusts r alone; B and C each assume r rules out a pattern. |
| 6. The cost of extrapolating | 25. (45 − 36) ÷ 36 × 100. |
| 7. Choosing between two analyses | B. A prefers simplicity over fit; C reads the zero total as a fit; D thinks fitted models must agree inside the data. |
| 8. Two checks before predicting | D. A trusts r; B reads the zero total as a fit; C checks the position of x but not the residuals. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Reads only the signs, sees a bend that is not there, or reads the zero total as a fit. | Write the sizes of the residuals in a row. |
| −4, 44 or 24 in question 2 | Subtracts the wrong way round, adds the intercept, or ignores the gradient term. | Find the line's y first, then subtract. |
| 3 or 0.75 in question 3 | Loses the sign or takes a mean. | The five must add to 0, so cancel the four. |
| 1, 5 or 25 in question 4 | Adds the residuals, adds the sizes, or squares the sum. | Square each residual, then add. |
| A, B or C in question 5 | Trusts r alone, or assumes a high r rules out a pattern. | Plot the residuals before deciding. |
| 20 or 9 in question 6 | Uses the prediction as the base, or gives the difference. | Percentage of the measured value: divide by 36. |
| A, C or D in question 7 | Prefers simplicity, reads the zero total as a fit, or thinks fitted models must agree inside the data. | Ask which analyst's line misses in a systematic way. |
| A, B or C in question 8 | Trusts r, reads the zero total as a fit, or checks only the position of x. | Ask for both checks: the residual pattern and where x = 12 lies. |
Questions worth asking
- Why do the residuals of a regression line always add to 0? The line passes through the mean point, so the gaps above and below balance.
- What patterns in the residuals change what you say about a line? A U or an arch: the line is the wrong shape. A spread that grows or shrinks as x changes: predictions are less reliable where the spread is wide.
- Can r be high and the model still be wrong? Yes: a gentle curve gives a high r, and the residuals show the curve.
- What would you write beside a prediction at x = 40 from data with x up to 20? That x = 40 is outside the data, so the prediction assumes the trend continues and may be well off.
Differentiation
A way in for anyone stuck
- Give the residuals as a row of signs and sizes and ask students to colour the positives and negatives.
- Give a number line with the data range marked for the interpolation questions.
- Use a sentence frame: the residuals are ___ at the ends and ___ in the middle, so a straight line is ___.
Stretch
- Show why the residuals of a regression line must add to 0, using the intercept formula a = mean of y − b × mean of x.
- Suggest a change of variable that would make a straight line suitable for the curved data, and test it on the figure's heights. (Plotting y against x² gives the exact line y = x² + 2, with every residual 0.)
Students learning in English as an additional language
“Residual” means what is left over: the part of y that the line does not explain. “Appropriate” means suitable. “Systematic” means following a pattern, not random. “Extrapolate” means go outside the data. “Shapeless” means with no pattern.
| Word | Plain meaning | Example |
|---|---|---|
| residual | a dot's y minus the line's y at the same x | a plant 2.5 cm below the line has residual −2.5 |
| model | a rule or equation used to describe or predict real data | the regression line used to forecast sales |
| appropriate | suitable for the data and the purpose | a straight line is appropriate when the residuals have no pattern |
| systematic | following a pattern rather than being random | residuals that are positive, then negative, then positive |
| extrapolation | predicting for an x outside the range of the data | forecasting sales beyond the months measured |
Sentence starters for explaining aloud or in writing:
- The residuals are ___, so a straight line is ___ appropriate because ___.
- r is ___ but the residuals show ___, so I would ___.
- The prediction is ___ the data, so it is ___ and I would trust it ___.
Two things not to say
“r is high, so the line is a good model.” Look at the residuals as well.
“The residuals add to 0, so the line fits well.” They add to 0 for every regression line.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- ResidualA line is y = 4x − 2. A dot is at (5, 21). Find its residual.
21 − 18 = 3. - MissingFour residuals are 2, −5, 1 and 3. What is the fifth if it is a regression line?
−1.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextResiduals in order of x are +3, +1, −2, −4, −2, +1 and +3. Describe the pattern and say what it suggests.
A U shape (positive at the ends, negative in the middle) with total 0: a curve would suit the data better. - Earlier work (Lesson 4)The mean point is (6, 20) and b = 3. Find a.
20 − 18 = 2.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningGive an example where extrapolating a straight line gives a result that cannot be true.
A population line that goes negative, or a plant height that exceeds any plant. - PercentageA prediction of 66 is measured as 55. Find the error as a percentage of the measured value.
11 ÷ 55 = 20%.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to the slider or the button, then the arrow keys or the space bar) and by touch. Under it, “Show me the steps” gives the same states in words.
No device? Give each pair a printed scatter of the curved set with the line drawn and a ruler: measure the gaps and plot them.