At a glance. One lesson of about 60 minutes. It assumes the sampling lessons and interpolation. Students build a line from the mean points of two halves, predict inside the data, choose and allocate a sample for a plan, and judge a written conclusion for sample, range and cause.
Calculator: no calculator. The numbers are chosen so the means and gradients are exact by hand, which keeps the attention on the design rather than on arithmetic.
What students will learn
Know
- Sort the data, split into halves, find each half's mean point and join them.
- The line passes through the mean point of all the data when the halves are equal.
- A claim should name its sample, its range and its limits.
Understand
- The line is only as good as the sample and its range.
- Volunteers or convenience samples can give a tidy line that does not describe the population.
- A link is not a cause.
Do
- Build the line from a table of eight pairs and predict inside the data.
- Choose and size a sensible sampling plan.
- Write a critique of a report in three points.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: mean points, gradient, intercept, allocation (questions 1 to 8).
- Criterion B. Investigating patterns: the first figure. Ask which line stays close to the typical score across the whole range.
- Criterion C. Communicating: the worked critique, written as sample, range and cause.
- Criterion D. Applying mathematics in real-life contexts: planning a bivariate project from the sampling design to a limited conclusion.
The one thing to do with the figure
Ask for a verdict, then the evidence. In the first figure, ask which sample students trust more and why, before switching. Then slide to 1 hour on both and compare the gaps.
In the second figure, have students predict each stage before moving the slider: what will the two half-way mean points be, and where will the line cross the overall mean?
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: two good students, one question. Collect predictions for 3 hours. |
| 10 min | First figure and key idea: the chain of an investigation. |
| 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 D, ask which plan could produce a pattern across all ages. |
| 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 | Interpolation; 8; 10. |
|---|---|
| Hook | At 3 hours the spread sample's line gives 51 and the volunteers' line gives 52.8, both near the typical 50, but at 1 hour the volunteers' line is 5.3 above the typical score against 2.5 for the spread sample. The volunteers never saw a student who studied little. |
| Worked examples | y = 2.8x + 9 and 23 at x = 5; the canteen report has a sample, a range and a cause to question. |
| Faded example | Lower mean point (4, 10), upper mean point (12, 22), y = 1.5x + 4, and 19 at x = 10. |
| 1. Choosing a sample for an investigation | B. A is tiny and chosen by convenience; C has a narrow range; D is one convenient place. |
| 2. Gradient through two mean points | 4. (39 − 15) ÷ (8 − 2). |
| 3. Intercept of the line | 2. 20 − 6 × 3. |
| 4. Predicting from the built line | 26. 3 × 5 + 11. |
| 5. Choosing a sampling method for a plan | C. A picks a special group; B has no chance in it; D chooses itself. |
| 6. Allocation for a plan | 11. 220 ÷ 900 × 45. |
| 7. Judging a conclusion | D. A trusts the fit; B blames size only; C asks for an irrelevant condition. |
| 8. A justified conclusion | C. A claims a cause; B claims a rule for every student; D extrapolates far outside the data. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Uses too few cars, a narrow age range, or one convenient street. | Ask: what range of ages does this plan cover, and how were the cars chosen? |
| 24, 0.25 or 6 in question 2 | Gives the rise only, inverts the fraction, or gives the run only. | Write rise over run with the numbers labelled. |
| 38, 18 or 120 in question 3 | Adds the term, gives the gradient term only, or multiplies the wrong values. | Substitute the point into y = 6x + c and write the equation to solve. |
| 15, 16 or 55 in question 4 | Leaves out the intercept, leaves out the gradient, or multiplies the intercept by x. | Read y = 3x + 11 as: multiply x by 3, then add 11. |
| A, B or D in question 5 | Chooses a special group, a fixed block, or volunteers. | Ask: could every student have been chosen, and who decides? |
| 9, 4400 or 9900 in question 6 | Splits equally, multiplies by the scale factor, or forgets to divide by the population. | Find the fraction of the school sampled first. |
| A, B or C in question 7 | Trusts the fit, blames size only, or asks for the origin. | List the three questions: who, how far, and what else. |
| A, B or D in question 8 | Claims a cause, claims a rule for every student, or extrapolates far. | Underline what the sample can support and what it cannot. |
Questions worth asking
- Why are two lines from different samples of the same data both fair? Each is a summary of its own sample, and samples differ.
- Which part of the chain is hardest to repair afterwards? The sample: if it was biased at collection, no later step fixes it.
- Can you ever say that one thing causes another from a scatter graph? Not from the scatter graph alone: a controlled experiment or a clear mechanism is needed.
- What does the line say about someone outside the sample range? Very little: that is extrapolation.
Differentiation
A way in for anyone stuck
- Give a table for the half-way means with labelled boxes for x and y of each half.
- Provide a three-question checklist for a report: who? how far? what else?
- Give the sorted data already split with a bracket.
Stretch
- Show that for eight dots, the line through the two half-way mean points passes through the mean point of all eight.
- Design an investigation of your own with a question, a population, a sampling method, a sample size and a limit on predictions.
Students learning in English as an additional language
“Investigation” means a planned search for an answer. “Chain” here means a series of steps where each depends on the last. “Convenience” means whatever is easiest to reach. “Justified” means supported by evidence.
| Word | Plain meaning | Example |
|---|---|---|
| investigation | a planned piece of work that answers a question using data | a project on sleep and screen time |
| variable | a quantity that can take different values | hours studied each week |
| half-way mean point | the mean point of the lower half or the upper half of data sorted by x | the point found from the four smallest x values |
| convenience sample | a sample chosen because it is convenient to reach, not by chance | the people in the room |
| justified conclusion | a statement the data actually support, with its limits named | the last paragraph of a report, which a reader could check against the data |
Sentence starters for explaining aloud or in writing:
- The lower half has mean point … and the upper half has mean point ….
- The conclusion is not justified because the sample ….
- The data support … for x between … and ….
Two things not to say
“A good line proves the link is real everywhere.” It describes the sample and its range.
“A bigger sample fixes any plan.” The method and the range decide the quality first.
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.
- GradientMean points (1, 10) and (5, 18). Find the gradient.
8 ÷ 4 = 2. - Predicty = 2x + 8. Find y when x = 3.
14.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA line built from data on x = 4 to 20 is y = 1.5x + 10. Predict at x = 12 and say whether to trust it.
28; inside the data, so reasonable. - Earlier work (Lesson 3)A school has 400 students, 120 in the junior section. A stratified sample of 50 is taken. How many juniors?
120 ÷ 400 × 50 = 15.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy does a sample whose x values span a wide range produce a more useful line than one that spans a narrow range, even with the same number of dots?
The line is anchored at both ends, so predictions across the range are better supported. - Working backwardsA line through half-way mean points (3, 12) and (9, 30) is used to predict. What x gives y = 24?
Gradient 3, intercept 3, so 24 = 3x + 3 and x = 7.
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 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 the eight pairs of the faded example on squared paper and have them build the line with a ruler.