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Counting all Butterflies
Estimating Population Size

Population estimates provide critical information which biologists need in order to manage a species properly, but these estimate are very hard to make. Imagine trying to count all the monarchs in Mexico!

In this lesson students use two of the same methods biologists use when estimating the number of monarchs at the over-wintering sanctuaries. In the first method, a number of individuals is counted and students extrapolate to make an estimate. The second method is called "Mark, Release, Recapture (MRR)". After trying both methods, students can make an actual count and test the accuracy of each method.

1. Divide the class into work groups. For each group, put 1 cup of rice into a jar, leaving extra room for mixing.

2. Have students imagine that the kernels of rice are monarchs in a wintering sanctuary. Give each student-group 3 pieces of paper, and label:

  • Count and Extrapolate
  • Mark, Release, Recapture (MMR) Estimate
  • Actual count

3. Count and Extrapolate
Have students take out a small number of rice kernels, count them, and put them back into the jar. Encourage them to notice the amount of space the small number of rice kernels occupies in the jar. Now, have them extrapolate in order to estimate the number of rice kernels in the full jar.

On their "Count and Extrapolate" paper have them:

  • record this estimate
  • record the time it took to do the estimate
  • list the pros and cons of using this method with butterflies

4. Mark, Release, Recapture (MRR)
This method involves 2 "visits" to a population. Some monarchs are captured and marked on the 1st visit. On the 2nd visit, the biologist again captures individual monarchs and records the number that are "recaptures" (those caught and marked on the first visit). By comparing the numbers captured and recaptured, the total number of individuals can be estimated using the equation provided below.

First Visit
"Capture" rice kernels and mark them with a colored marker. Put all the rice back into the jar and mix well. Record the number of pieces you captured and marked on your "1st Visit".

Second Visit
Now go back for a "2nd Visit". Record the number of MARKED individuals you recapture on the 2nd visit. Also record the TOTAL NUMBER of individuals you capture. (Do not return the rice to the jar until you have captured the full number.)

Important: The MRR equation will seem difficult at first. However, if students are allowed to experiment capturing, marking and recapturing individuals, they will quickly internalize the reasoning behind the math.

Practice Runs
To practice, set up a small population of 10 individuals. Run 3 different trials, in which students mark 10, 5 and 1 individuals, respectively. For these experimental trials, have them guess how many they will recapture on the return visit--and explain why. The basic reasoning you're looking for would be along these lines:

  • The more you mark on your 1st visit, the more marked individuals you'll capture on the 2nd visit.
  • If you mark them all on your 1st visit, all the recaptures will be marked on the 2nd visit.
  • If you only mark 1 on your 1st visit, it may take 10 tries (1/10 chance) to recapture the one marked.
  • On your 2nd visit, the % of butterflies you recapture should be the same as the % of the total population you marked on your 1st visit.

After the practice runs with the small population, use the full cup of rice and have students "capture" at least 100 pieces of rice.

Make Your Estimate
Now, figure your estimate according to the MRR equation. Remember, you are solving for "Total Population Size", or "b" in the equation below. Again, the basic idea is that the number of individuals marked on the first visit (a) is to the total number in the population (b), as the number of marked individuals captured on the 2nd visit (c) is to the total number of individuals captured on the 2nd visit (d).

MRR Equation: a/b = c/d

a= # Individuals Marked on 1st Visit
b= Total Population Size
c= # Marked Individuals Recaptured on 2nd visit
d= # Individuals Captured on 2nd Visit, in Total

Here's an example:

a=100 Individuals Marked on 1st Visit
b= Unknown (Total Population Size)
c= 20 Marked Individuals Recaptured on 2nd visit
d= 100 Individuals Captured on 2nd Visit, in Total

Actual Count
Divide the rice among the students in the group, and have them count the actual number of rice kernels in the jar.


1. Which method of estimation was more accurate? How did your 2 estimates compare to the actual number of objects?

2. What assumptions are made in the MRR method? List as many as you can. (For example, between the 2 visits there are *no births or deaths, * no arrivals or departures, * there's equal chance of capture and recapture, * monarchs don't learn to avoid being captured, etc.)

3. Do you think MRR is a reliable method?

4. When you count and extrapolate, what happens if your original count is not correct? (Experiment by changing the original count and reviewing your end results.)

5. Does the estimate become more accurate if you mark a greater number of individuals? Why?

National Science Education Standards

Science as Inquiry
Use math in all aspects of scientific inquiry. (5-8)
Mathematics is important in all aspects of scientific inquiry. (5-8)

National Math Standards

Number and Operations
Understand meanings of operations and how they relate to one another.

Compute fluently and make reasonable estimates.

Data Analysis and Probability
Understand and apply basic concepts of probability.

Represent and analyze mathematical situations and structures using algebraic symbols.

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