Counting
all Butterflies
Estimating Population Size
Background
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 overwintering 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.
Activity
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 studentgroup 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 visitand 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.
Discussion
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. (58)
Mathematics is important in all aspects of scientific inquiry. (58)
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.
Algebra
Represent and analyze mathematical situations and structures using algebraic
symbols.
