**capture-recapture method**in the Higher Tier specification:

Source: Edexcel GCSE (9 - 1) Mathematics Teaching Guidance |

Here's the sampling content in more detail:

Source: Edexcel Content Exemplification FAQs |

It's worth noting that both Edexcel and AQA list stratified sampling as a topic that has been

*removed*from GCSE. However, this comes with a caveat - both sampling and proportional reasoning do feature in the 9 - 1 GCSE, so it would be reasonable for exams to include a stratified sampling question even if students haven't been explicitly taught this topic (as long as the question doesn't use the word 'stratified' without explaining its meaning). So my advice - whatever board you're using - is to look at a few stratified sampling questions with your GCSE class, whether in a statistics lesson or a ratio and proportion lesson.

Stratified sampling is a great opportunity to use proportional reasoning, as is the capture-recapture method. If you've taught Edexcel's GCSE Statistics then you'll already be familiar with capture-recapture, but I'll explain it here in case you've not seen it before.

**A simple example**

Try this question... it will only take you a second.

I captured 50 fish from a lake. I marked a big cross on the back of each fish with a permanent marker*...

I put the marked fish back in the lake and they happily swam away to join their friends.

The next day I captured 20 fish from the same lake. 10 of them had a cross on their back.

*no fish were hurt, promise.Can you estimate the total population of fish in the lake?

I'm sure you spotted that the proportion of marked fish in the second sample was 0.5, and we can assume the same proportion of marked fish in the whole population. Given that I marked 50 fish, we can estimate that there are 100 fish in the lake.

**A formula**

If the numbers are less straightforward so the estimation can't be done mentally, it's easy to set up a formula to work out the population. This is certainly not a formula that students will need to memorise - it can be deduced using proportional reasoning.

You can see that the formula on the left simply shows that proportion of marked fish in the population is equal to the proportion of marked fish in the sample. The formula on the right has been rearranged to make N the subject.

Here's an example from the Biology section of BBC Bitesize. It would be better if they had shown the proportional reasoning and rearrangement process rather than just give a final formula.

Source: BBC Bitesize |

The example goes on to list some assumptions - these are certainly worth discussing with your students.

- There is no death, immigration or emigration (ie the population is
*closed*) - The sampling methods used are identical
- The marking has not affected the survival rate of the animals

We also assume that animals do not lose their marks, that marking does not affect the likelihood of recapture, and that sufficient time is left between marking and recapture for all marked individuals to be randomly dispersed throughout the population.

**An exam question**

A scientist wants to estimate the number of fish in a disused canal. He catches a sample of 30 fish from the canal. He marks each fish with a dye and then puts them back in the canal. The next day the scientist catches 20 fish from the canal. He finds that 4 of them are marked with the dye.

(a) Estimate the total number of fish in the canal. (2)

(b) Write down any assumptions you made. (2)

For part b, candidates have to mention two ideas, including something about the population being unchanged, or the idea of randonmess, or that markings remain unchanged.

**Teaching ideas**

I think this will be quite a nice topic to teach. Here are a few useful links:

- Capture and Recapture - nrich
- A Lesson: The Capture-Recapture method - MathsMuggle
- Capture-Recapture lesson and resources - NCTM
- 'Something Fishy' project - PBS
- Capture Recapture: class experiment (page 1 & 2) and practice worksheet (page 3 and 4) - Mario Martinez, Cerritos College
- Capture recapture worksheet - youngscientistsmhs.weebly.com
- Select 'Estimating Populations' on the mathsbot.com GCSE question generator
- James Gurung's video explains how to answer an exam style capture-recapture question
- Mark and recapture garden snails (graphic for practical activity) - The Wildlife Trusts on TES

I expect we'll see some more student resources appear on TES over the coming year.

Finally, here's a nice video to show in your lesson - Johnny Ball estimates the number of black cabs in London.

The Johnny Ball video is lovely, but I don't think I like the way round he did the calculations. I prefer to work with (using his ping pong numbers) 17/100=0.17 or 17%, then question 100=17% of ?. I find this way tends to make more sense conceptually to students (although, granted, I've not taught this topic to huge numbers of children).

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