Reflection: Developing a Conceptual Understanding Mark and Recapture: Population Sampling  Section 2: Getting Started
Because this simulation activity takes place in just one class period and the focus is really on using a formula rather than how the formula works, I ask students to use the simplified formula N = 200/k when calculating their population estimates.
Some more advanced students may be curious about the math behind that formula, and whether this sampling method always uses that specific simplification. The quick answer is no. The simplified formula is based on the particulars of this simulation (taking 20 kernels on the first visit and 10 kernels on each subsequent visit). Different sample sizes would be calculated differently, so for any groups that are curious about how the actual formula works, I cover the following with them:
The formula used to estimate the population size is the LincolnPetersen Method: N = Kn/k
N = Number of animals in the population
K = Number of animals marked on the first visit
n = Number of animals captured on the second visit
k = Number of recaptured animals that were marked
In this simulation, we are going to mark 20 animals on the ”first visit”, so K=20.
We will capture 10 animals for each “return visit”, so n=10.
Therefore (because 20 x 10=200), we will estimate the population (N) with the simplified formula: N = 200/k
Example: If you pull out 10 kernels and 4 are marked, then k = 4. Therefore, N = 200/4.
N would then equal 50 and that is the estimate of the population.
Admittedly, I only ended up going over this with one group that asked how the formula really worked, but I think it helped them see how this formula could be applied realistically in the field.
One thing you might try if you want to really focus on the math and go over this formula's applicability in other situations would be to posit a scenario with different parameters. For example, in a scenario where 15 animals were captured initially but only 5 were captured subsequently, the simplified formula would be N = 75/k.
Mark and Recapture: Population Sampling
Lesson 1 of 10
Objective: Students will be able to estimate the size of a population by using calculations based on the mark and recapture method used by ecologists.
This lesson is used as an activity to introduce the unit on population ecology. In my case, it’s the first lesson taught after a long break, so I feel that an simulation lab is a better way to begin the class rather than diving straight into heavy content. Even if this wasn’t being taught following a break, I think it’s good to introduce units with an activity or simulation to better capture student interest and offer a preview of what’s to come.
1 bag of unpopped popcorn kernels
1 measuring cup or small beaker
Brown paper lunch bags (1 per group)
Permanent markers (1 per group)
In this lab, students will be simulating the mark and recapture method to estimate the size of a population of wild animals. In actual field work, scientists would set traps or otherwise catch a certain number of animals (in our simulation it will be 20) and mark them, usually with paint, a collar, a leg band, etc (we will mark our “animals” with a permanent marker). Then, once the animals have been given sufficient time to mix back in with the general population, a scientist would return to the site and capture another group of animals from the same population (in our case we capture 10 animals on each return visit). By assuming that the proportion of marked to unmarked animals in this return visit is equal to the proportion of animals in the initial sample to the whole population, the size of the whole population can be reasonably estimated.
I really like this lab for three reasons:

It’s very easy to set up and the materials are readily available and inexpensive.

It gives students an opportunity to use quantitative reasoning and apply math skills to solve problems.

It’s fun for students and fairly easy to explain, so they get to dive right in.
Connection to Standard
In this lesson, students use mathematical reasoning to solve a problem, recognize patterns and evaluate the effectiveness of an equation. Students also use a model to simulate field methods of environmental scientists.
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Getting Started
Introducing the Problem of Estimating Unknown Populations:
I begin this lab by asking students to recall the definition of a population: a group of organisms of the same species living in one place at one time. I then hold up a beaker and add one popcorn kernel to it. I ask students how many kernels are in the beaker and they reply something like, “duh… one!” Without asking how they figured that out, I pour several kernels into the beaker and ask them to tell me how many are in there. Some students will invariably offer their wild guesses, but I bring them back to my desire for an accurate count. I then ask students why it was easy to count the single kernel and hard to count the many kernels. Student answers will vary, but I hope they arrive to the idea that a large number is too difficult to make a simple “guesstimate” that’s anywhere close to accurate. I ask how we would get an accurate number and wait for a student to say we would need to pour out the kernels and actually count them one by one.
Then I bring the conversation back to environmental science and ask, “If a scientist was trying to find the size of a population of wild animals (I use cardinals in a forest as an example), which would it be more like, the single kernel or the group of kernels?” Students should realize that a population consists of more than one individual, and that wild animal populations (especially cardinals in a forest) can be quite large.
I then ask why counting the number of cardinals in a forest would be more difficult than counting kernels on a table, even though they both involve counting a large number of things. Students may offer many reasons, but I would hope that they could at least mention:

some may die or new ones could be born

some may migrate into or out of the forest

some may be hiding and hard to count

some might be scared of you and fly away before you see them

most cardinals look the same, you may end up counting the same bird more than once
After students offer a few reasons why it would be difficult to count every single cardinal in a forest, I ask if counting is a practical approach to the problem of knowing the population size.
Students will likely offer that it is not practical, and I ask if it would be practical if there was a sound mathematical way to estimate the population.
I then ask for a volunteer to distribute one lab worksheet to each group. Once all groups have the worksheet, I introduce the mark and recapture method described in the introduction section and on the lab handout,
In this lab activity you will simulate the mark and recapture method of population estimation. With this technique, it is possible to estimate the size of an entire population by first capturing and marking a small sample of the population. Then, on return visits to the habitat, you capture another sample and count the number of marked individuals. By assuming that the proportion of marked to unmarked individuals in the recaptured sample is equal to the proportion of the initial sample to the whole population, we can estimate the size of the population.
Explaining the Lab Procedure to Students:
I then quickly go over the simulation procedure with the students, explaining that they will:
1.Obtain materials from the teacher
2.Shake the bag of kernels and then remove 20 individuals, one at a time.
3.Use a marker to mark the 20 individuals in your initial sample.
4.Return your sample to the bag.
Then, for each “return visit”:
1.Shake the bag and remove 10 individuals, one at a time.
2.Record the number of marked individuals in the table below.
3.Use the formula (population estimate = 200/marked #) to make a population estimate. (Please see the reflection Got Math? for a more detailed breakdown of how this formula works)
4.Return your sample to the bag and repeat steps 14 until you have made 10 “return visits”
Once I explain the basic procedure, I ask each group to send a representative to pick up the materials for the activity. I then give each representative,

a brown paper lunch bag

a permanent marker (if they don’t already have one)

an unknown quantity of kernels poured into the bag (I use the beaker to measure out approximately 20 cm3 of kernels, you may want to experiment ahead of time depending on the size of the kernels so that students aren’t receiving too many kernels… it should be between 100200 kernels).
At this point, I remind students to give the marked kernels a few minutes to dry before returning them to the bag and then allow students to start the simulation.
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Running the Simulation
During the simulation, I move around the room, checking with each group to make sure they are filling in their data table correctly. One issue that comes up fairly often is that they may have zero marked individuals on a return visit. This can leave them confused about whether to divide 200 by 0. Instead of having that math teacher conversation with them, I instead ask them what estimate would result from finding 1 marked kernel (200/1 = 200). Then ask them what finding 2 would result in (200/2 = 100). I then ask, “if the number is lower, what happens to the estimate?”
They should understand that the population estimate is higher when fewer marked kernels are found in the “return visit” samples (suggesting the initial marked sample was a very tiny proportion of the actual population. Once they understand this, I tell them to keep the zero in the # of Marked Individuals in Sample column in the data sheet and write >200 on the population estimate.
Once groups have made 10 return visits, I suggest that instead of making their average estimate be the average of estimates (because some of the estimates may be “>200” and not a specific number), to average the # of Marked Individuals in Sample for the whole exercise, and then enter that into the equation. (for example: If a group’s 10 return visits yielded 1,0,1,3,0,2,1,1,0, and 1 marked kernels, that would add up to 11. The average would then be 11/10 = 1.1. Enter that into the equation, and you get a population estimate of 200/1.1 = 181.8 (which I would remind students to round down, as there are no 80% individuals).
Another problem students may encounter is that they may not be returning samples back to the “wild population” (the bag). This is fairly easy to fix, and even if they had been doing that from the beginning, the simulation doesn’t take too much time, so they can begin again.
The fun part of the activity comes in having the students finally count all the kernels in their bag to acquire the actual population and compare it to their individual estimates and their average estimate. Some groups’ estimates will be very close to the actual population, and other will not. These issues will be addressed in the debrief section.
Debrief
Once students in each group have finished the simulation, I ask them to answer the followup questions at the end of the lab worksheet. During this time, I walk from group to group to see if they are having any difficulties with any question. I save about 15 minutes for a final debrief group discussion and remind students when we have about 10 minutes left for them to complete the worksheet questions.
During this time, I collect the following information from each group,

their actual population

their average estimate
I then find the average of the actual populations for all groups. I then also find the average of the average estimates for all groups. These calculations may offer more comparative data for the whole class to consider about the effectiveness of the method should their particular group’s simulation have yielded a less accurate estimation.
We then have a short debrief discussion where I expect all groups to participate. Although I don’t worry about actually grading their participation in this activity (since it’s a relatively short discussion), I have the same expectations I have when I use this rubric.
For the debrief discussion, we go over the questions from the lab worksheet. Most of these questions really depend on the particular results that each group got from the simulation, so the intent is to show the whole class the range of results. For the final question regarding the limitations of the method. I would make sure students understood that 0 recaptures would "break" the formula because you can't divide by zero. More practically, differential success of recapture arising from unskilled trappers, animals learning to avoid traps, variation in animal behavior (i.e., maybe the initial and recaptures are the only members of a large population that could be captured at all and would skew the estimate towards a smaller population), and migrations based on seasonal or other environmental changes could all skew the results and make this method less effective. At the end of the day, the point is that this method is a way to estimate population sizes... it's imperfect but useful.
Once all questions have been gone over, I ask students what they think about the method and share the data I collected averaging the whole class' info. This additional data may vindicate the method if certain group's estimates were too far off from their actual populations.
Similar Lessons
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Environment: Rural
 LESSON 1: Mark and Recapture: Population Sampling
 LESSON 2: Describing Populations
 LESSON 3: Population Growth
 LESSON 4: Human Geography
 LESSON 5: Predicting Human Population Change Part 1: Age Structure
 LESSON 6: Predicting Human Population Change Part 2: Population Predictions
 LESSON 7: Predicting Human Population Change Part 3: Environmental Impact
 LESSON 8: Critical Editorial Analysis: Is Overpopulation a Problem?
 LESSON 9: What's Your Ecological Footprint?
 LESSON 10: The Demographic Transition