At a glance. One lesson of about 60 minutes. It assumes the mean and percentage of an amount. Students separate population from sample, use a sample share to estimate a count, apply capture and recapture, and meet sampling variation before any named sampling method.
Calculator: no calculator. Every calculation is a short division or multiplication that is better done by hand, because the reasoning about shares is the content.
What students will learn
Know
- Population and sample, and the sample mean as an estimate.
- Count estimated as sample share times population size.
- Capture and recapture: tagged × second catch ÷ tagged in the second catch.
Understand
- Two samples from one population are expected to differ.
- A larger sample is closer on average, not every time.
- A difference is only a problem when the method leans one way.
Do
- Estimate a population total or mean from a sample.
- Combine two samples into one mean using their sizes.
- Say in a sentence what a sample result does and does not show.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: population, sample, estimate, and the share calculation (questions 1 to 8).
- Criterion B. Investigating patterns: the two figures. Ask for a prediction of the sample mean before each slider move.
- Criterion C. Communicating: writing the estimate as a sentence with the assumption stated (the faded example).
- Criterion D. Applying mathematics in real-life contexts: quality checks, wildlife counts and polls.
The one thing to do with the figure
Ask for a number before each move. In the first figure, hide the readouts with a hand and ask “Will a sample of 20 be closer to the true mean than a sample of 10?” Take a show of hands.
In the second figure, ask what the four means would look like if samples never differed. Then slide through all four to show that the differences are normal.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the grower and the harvest. Collect predictions. |
| 10 min | First figure and key idea: estimate, and sample size. |
| 10 min | Second figure, the warning card and the two worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A or B, redraw population and sample as a large and a small circle. |
| 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 | 8; 32; a little. |
|---|---|
| Hook | Not steadily: the first figure reads 212 g at a sample of 5, then 214.9 g at 10, then closer, with no promise on any single step. The gap is 0 at the end only because every mango is in the sample. |
| Worked examples | 6 ÷ 80 = 0.075 and 0.075 × 2500 = 187.5, so about 188; 4 ÷ 20 = 1 ÷ 5 and 25 × 5 = 125. |
| Faded example | 9 ÷ 150 = 0.06 and 0.06 × 6000 = 360. |
| 1. Population and sample | C. A reverses population and sample; B treats the estimate as exact; D expects every sample to agree. |
| 2. Sample mean | 23. 138 ÷ 6. |
| 3. Estimating a count from a sample | 240. 3 ÷ 50 × 4000. |
| 4. Capture and recapture | 140. 35 × 24 ÷ 6. |
| 5. Combining two samples | 49. (1040 + 1410) ÷ 50. |
| 6. A new value changes the mean | 43. 403 − 360. |
| 7. Estimating a count from a percentage | 14 500. 0.58 × 25 000. |
| 8. Reading two different samples | B. A treats one estimate as exact; C treats sample variation as a mistake; D forgets to scale from 40 to 1200. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Swaps population and sample, treats the estimate as exact, or expects every sample to agree. | Draw a big circle (everyone) and a small one inside it (those measured). |
| 24, 138 or 27.6 in question 2 | Gives the median, leaves the total undivided, or divides by 5. | Count the values on the page before dividing. |
| 120, 12 000 or 80 in question 3 | Uses the wrong share, multiplies by the count, or finds only the scale factor. | Write the sentence: out of 50 bulbs, 3 are faulty. |
| 8.75, 53 or 4 in question 4 | Inverts the formula, adds the counts, or stops at the second catch ratio. | Ask: what fraction of the lake is tagged? Draw the lake as five equal parts. |
| 49.5, 99 or 2450 in question 5 | Averages the two means, adds them, or forgets the final division. | Ask: which sample should count for more, and why? |
| 31, 1 or 403 in question 6 | Gives the new mean, the change in the mean, or the new total. | Write total = mean × count for the 12 values and then for the 13. |
| 116 or 1 450 000 in question 7 | Finds the number in the sample, or uses 58 in place of 0.58. | Say aloud what the answer is a count of: voters in the town. |
| A, C or D in question 8 | Treats an estimate as exact, blames a mistake, or forgets to scale the sample count up to the school. | Ask: is 12.5 a number of students in a sample or in the school? |
Questions worth asking
- Is a sample that disagrees with the population a bad sample? No: some difference is expected. Look at the method before the result.
- Why can a sample of 10 land further away than a sample of 5? A sample is a chance selection. Bigger is closer on average, not every time.
- What do we assume when we estimate a count from a sample share? That the whole population has the same share as the sample.
- Why does capture and recapture need the second catch to be a fair mix? Tagged fish must be spread through the pond for the tagged share to be representative.
Differentiation
A way in for anyone stuck
- Give the share line already written: “out of ___ in the sample, ___ have the feature”.
- Use smaller numbers first (a sample of 10 from 100) and then scale the same sentences.
- Provide a two-circle template (population, sample) for questions 1 and 8.
Stretch
- Why must capture and recapture assume the population has not changed between the two catches? Name two ways it could fail.
- Invent a population of 10 values and list three different samples of 3. Find the three sample means and compare them with the population mean.
Students learning in English as an additional language
“Population” here is every item of interest, not only people. “Sample” is a part taken from the whole, like a taste of soup. “Estimate” means a careful guess based on evidence. “Tagged” means marked so it can be recognized again.
| Word | Plain meaning | Example |
|---|---|---|
| population | every item or person the question is about | all the pupils in a school year |
| sample | the part of the population that is actually measured | 30 pupils chosen from a year group |
| estimate | a value worked out from a sample to stand in for the population value | using a sample mean in place of the population mean |
| sample size | the number of items in a sample | the count of readings taken |
| tagged | marked so that it can be counted again later | a bird with a small ring fixed to its leg |
| variation | the natural difference between one sample and another | the weights of two handfuls of the same flour |
Sentence starters for explaining aloud or in writing:
- The population is … and the sample is ….
- The sample share is …, so I estimate … in the whole population.
- Another sample could give a different answer because ….
Two things not to say
“The sample mean is the mean of the population.” It is an estimate of it.
“A bigger sample is always better.” It is better on average, and only if the method is sound.
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.
- EstimateA sample of 40 books from a library of 2 000 has 5 damaged. Estimate the damaged books.
5 ÷ 40 = 0.125 and 0.125 × 2 000 = 250. - MeanFind the mean of 14, 18, 11, 17.
60 ÷ 4 = 15.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA ranger tags 18 deer. A later count of 24 deer shows 3 tagged. Estimate the herd.
18 × 24 ÷ 3 = 144. - Earlier work (Year 9)A line of best fit passes through the mean point (5, 60). What is y when x = 5?
60.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningWhy is the combined mean of two samples not the plain average of their means unless the samples are the same size or have the same mean?
The larger sample has more values, so it carries more weight. - Working backwardsSample A has 10 values with mean 20. Sample B has 40 values. The combined mean is 26. Find the mean of sample B.
(50 × 26 − 200) ÷ 40 = 27.5.
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, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Give each pair a bag of 40 numbered counters, have them draw 10 four times and compare the means.