New Gcse: Capture-Recapture

If your schoolhouse plans to role Edexcel for GCSE from 2017 together with therefore you lot may get got spotted the capture-recapture method inwards the Higher Tier specification:

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

Here's the sampling content inwards to a greater extent than detail:

Source: Edexcel Content Exemplification FAQs

It's worth noting that both Edexcel together with AQA listing stratified sampling every bit a topic that has been removed from GCSE. However, this comes amongst a caveat - both sampling together with proportional reasoning create characteristic inwards the ix - 1 GCSE, therefore it would last reasonable for exams to include a stratified sampling query fifty-fifty if students haven't been explicitly taught this topic (as long every bit the query doesn't role the give-and-take 'stratified' without explaining its meaning). So my advice - whatever board you're using - is to expect at a few stratified sampling questions amongst your GCSE class, whether inwards a statistics lesson or a ratio together with proportion lesson.

Stratified sampling is a slap-up chance to role proportional reasoning, every bit is the capture-recapture method. If you've taught Edexcel's GCSE Statistics together with therefore you'll already last familiar amongst capture-recapture, but I'll explicate it hither inwards instance you've non seen it before.

A unproblematic example
Try this question... it volition solely get got you lot a second.

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

I pose the marked fish dorsum inwards the lake together with they happily swam away to bring together their friends. 

The side past times side solar daytime I captured twenty fish from the same lake. 10 of them had a cross on their back. 

Can you lot approximate the total population of fish inwards the lake?

*no fish were hurt, promise.

I'm sure you lot spotted that the proportion of marked fish inwards the bit sample was 0.5, together with nosotros tin assume the same proportion of marked fish inwards the whole population. Given that I marked 50 fish, nosotros tin approximate that at that spot are 100 fish inwards the lake.

A formula
If the numbers are less straightforward therefore the estimation can't last done mentally, it's slow to ready a formula to operate out the population. This is for certain non a formula that students volition postulate to memorise - it tin last deduced using proportional reasoning.

You tin run into that the formula on the left merely shows that proportion of marked fish inwards the population is equal to the proportion of marked fish inwards the sample. The formula on the correct has been rearranged to brand northward the subject.

Here's an instance from the Biology department of BBC Bitesize. It would last ameliorate if they had shown the proportional reasoning together with rearrangement procedure rather than exactly give a terminal formula.

Source: BBC Bitesize

The instance goes on to listing about assumptions - these are for certain worth discussing amongst your students.

• There is no death, immigration or emigration (ie the population is closed)
• The sampling methods used are identical
• The marking has non affected the survival charge per unit of measurement of the animals 

We too assume that animals create non lose their marks, that marking does non touching the likelihood of recapture, together with that sufficient fourth dimension is left betwixt marking together with recapture for all marked individuals to last randomly dispersed throughout the population.

An essay question

So what mightiness this topic expect similar inwards a GCSE exam? Here's a question from Edexcel's 2014 Higher GCSE Statistics paper.

A scientist wants to approximate the pose out of fish inwards a disused canal. He catches a sample of thirty fish from the canal. He marks each fish amongst a dye together with and therefore puts them dorsum inwards the canal. The side past times side solar daytime the scientist catches twenty fish from the canal. He finds that iv of them are marked amongst the dye. 

(a) Estimate the total pose out of fish inwards the canal. (2)
(b) Write downwardly whatsoever assumptions you lot made. (2)

For business office b, candidates get got to lift 2 ideas, including something nigh the population beingness unchanged, or the thought of randonmess, or that markings rest unchanged.

For to a greater extent than essay questions, see Edexcel's Emporium together with expect under GCSE 1MA1 Practice Papers > Themed Papers.

Teaching ideas

I yell back this volition last quite a overnice topic to teach. Here are a few useful links:

• Capture-Recapture handout together with slides - MathsPad
• Capture together with Recapture - nrich
• Capture-Recapture Slides - pbrucemaths on TES
• A Lesson: The Capture-Recapture method - MathsMuggle
• Capture-Recapture lesson together with resources - NCTM
• 'Something Fishy' project - PBS
• Capture Recapture: shape experiment (page 1 & 2) together with exercise worksheet (page three together with 4) - Mario Martinez, Cerritos College
• data resources library for listings.