At a glance. One lesson of about 60 minutes. It assumes lines of best fit and scatter graphs from Year 9 and the idea of bias from Lesson 4. Students predict from an equation, find an intercept from the mean point, give a range from the largest gap, and judge claims made far outside the data.
Calculator: GDC optional. A GDC finds a regression line from lists and draws the scatter, but it offers a value for any x without comment, so the lesson is about the human judgment around the output.
What students will learn
Know
- Interpolation is inside the range of the data and extrapolation is outside.
- The equation y = mx + c is used to predict y from x.
- A line of best fit passes through the mean point.
Understand
- The further outside the data, the less reliable the prediction.
- Looser dots mean a wider margin at the same x.
- The real situation may have limits the line does not know about.
Do
- Predict from an equation and label it interpolation or extrapolation.
- Find the intercept from the mean point and the gradient.
- Write a sentence judging whether a prediction can be trusted.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: equation, mean point, inside and outside the data (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask for the week at which the line stops being sensible before revealing the later points.
- Criterion C. Communicating: judging a prediction in a sentence that names the range and the reason (the worked examples).
- Criterion D. Applying mathematics in real-life contexts: growth, sales, populations and sport.
The one thing to do with the figure
Make students place the cut-off. In the first figure, ask each pair for the week after which they would not trust the line, and why. Then reveal the later points and compare.
In the second figure, hold x fixed and flip between tight and loose dots. The line and prediction do not move, but the band does. Ask what the band is telling them.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the seedling in week 20. Collect the week at which students stop trusting the line. |
| 10 min | First figure and key idea: interpolation, extrapolation and limits. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A or B, ask which age was the oldest in the data. |
| 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 | 27; 18; no. |
|---|---|
| Hook | There is no single week, but it gets worse beyond week 8: the line gives 43 cm at week 10 against 40 cm measured, and 83 cm at week 20 against 48 cm measured. |
| Worked examples | 190 at 25 °C (interpolation) and 310 at 45 °C (extrapolation, doubtful); the range 37 to 43. |
| Faded example | 25.5 at x = 7 (inside the data); 48 at x = 16 (outside, less reliable). |
| 1. Interpolation or extrapolation | C. A and B are inside the range; D is the end of the range. |
| 2. Predicting from an equation | 53.4. 3.2 × 12 + 15. |
| 3. Intercept from the mean point | 10. 40 − 5 × 6. |
| 4. A negative gradient | 45. 90 − 1.5 × 30. |
| 5. Why a distant prediction is doubtful | B. A blames the sign; C misuses the mean point; D invents a whole-number rule. |
| 6. The range a prediction allows | 66. 62 + 4. |
| 7. Judging a claim | B. A treats the line as exact; C bans lines altogether; D uses an irrelevant fact. |
| 8. Scatter and trust | B. A ignores the scatter; C treats interpolation as a guess; D reads wide scatter as a strength. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Treats an age inside, or at the end of, the data as outside. | Mark the lowest and highest ages on a number line and place each option. |
| 38.4, 18.2 or 180 in question 2 | Leaves out the intercept, adds the coefficients, or multiplies the intercept. | Read y = 3.2x + 15 as: multiply x by 3.2, then add 15. |
| 70, 30 or 200 in question 3 | Adds instead of subtracts, gives the gradient term only, or multiplies the wrong values. | Substitute the point into y = 5x + c and solve for c. |
| 60, 135 or −45 in question 4 | Forgets the multiplier, adds the term that should be subtracted, or omits the intercept. | Write 90 − 1.5 × 30 and work out the product first. |
| A, C or D in question 5 | Blames the sign, misuses the mean point, or invents a whole-number rule. | Ask: where did the data stop, and where is x = 50? |
| 58, 4 or 15.5 in question 6 | Gives the lowest value, the gap itself, or divides. | Draw the prediction on a number line with the gap above and below. |
| A, C or D in question 7 | Treats the line as exact, bans lines, or uses an irrelevant fact. | Ask: what real thing limits a town's population? |
| A, C or D in question 8 | Ignores the scatter, treats interpolation as a guess, or reads wide scatter as a strength. | Draw 50 ± 2 and 50 ± 9 on one number line. |
Questions worth asking
- What does a good fit tell you about where the data stops? Nothing: a good fit applies inside the data only.
- Why can the same line deserve more trust for one data set than another? The scatter around it differs: tight dots mean small gaps.
- Give a situation where a straight line cannot continue. Plant height, a falling price that would go below zero, a population in a limited space.
- Is interpolation always right? No: it is reasonable and has a margin.
Differentiation
A way in for anyone stuck
- Give a number line for each question with the data range already marked.
- Give y = mx + c with the gradient and intercept circled before substituting.
- Use a sentence frame: x = ___ is inside / outside ___ to ___, so ___.
Stretch
- A line y = 4x + 3 was fitted to weeks 1 to 8. Using the later data in the figure, find the week at which the error first exceeds 10 cm, as far as you can tell from the weeks measured.
- Explain why the line through the mean point is not guaranteed to be good outside the data, even if it is an excellent fit inside it.
Students learning in English as an additional language
“Interpolation” means working out a value between known values (inter = between). “Extrapolation” means going beyond known values (extra = outside). “Margin” here means the room for error either side. “Levels off” means stops rising.
| Word | Plain meaning | Example |
|---|---|---|
| interpolation | predicting a value from x inside the range of the data | reading a value between two measured days |
| extrapolation | predicting a value from x outside the range of the data | extending a line past the last measured day |
| line of best fit | a line that follows the trend of a scatter graph and passes through the mean point | the line drawn through the dots |
| prediction | a value worked out from the line for a chosen x | the height the line gives for a certain week |
| margin | the room for error either side of a prediction | the largest gap from the line in the data |
Sentence starters for explaining aloud or in writing:
- The x value is inside / outside the data because the data run from … to ….
- The line gives …, but the real value could be between … and ….
- I would not trust this prediction because ….
Two things not to say
“The line fits well, so the prediction is right.” A good fit applies where the data are, and only there.
“Extrapolation is always wrong.” A small step beyond the data can be sensible: it is a matter of distance and plausibility.
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.
- Predicty = 4x + 9. Find y when x = 6.
4 × 6 + 9 = 33. - ClassifyData run from x = 10 to x = 40. Is x = 25 interpolation or extrapolation?
Interpolation.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA line of best fit for a plant's height is y = 3x + 5 from weeks 2 to 6. Predict at week 4 and at week 15 and comment.
17 (interpolation) and 50 (extrapolation, a guess). - Earlier work (Lesson 4)Give one reason a survey of a million people could still be unreliable.
Bias: who was asked or who replied does not change with size.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningTwo lines have the same equation but one is fitted to tight dots and the other to loose dots. Which prediction is more informative, and why?
The tight dots: the margin is smaller. - Working backwardsA line y = mx + 12 passes through the mean point (5, 37). Find m.
37 = 5m + 12 gives m = 5.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider or a button, then the arrow keys or the space bar) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give each pair a printed graph and a ruler: draw the line, mark where the data end, and read a value on each side.