.
Estimating the Number of Monarchs in a Roost
Let's Practice

People are dazzled by the beauty of fall "butterfly trees." It's a rare opportunity to see more monarch butterflies than a person can count. If you found a monarch roost, how would you estimate the number there? Scientists often use photographs to help them estimate things that are moving or numerous. Let's practice by using these photos.

 Picture #1 (Click to enlarge.) Courtesy of Tim Mostrom

How many monarchs are in the roosts in these pictures?

1) As a class, take a 10-second look at Picture #1. (A quick glance is often all a scientist may have!) Without discussing, guess and record the number of butterflies you think you just saw.

2) Compare the different guesses the students in the class made, and record.

3) Next, print out Picture #1 for a careful look. Working in groups, students should count out 10 butterflies. Carefully notice how much space the 10 butterflies occupy. Students can train their eyes to see how the clusters are arranged.

As you work, write down the challenges you face. What makes estimating difficult? (The butterflies aren't all visible. They're not evenly distributed. The clusters aren't same size, etc.)

4) Now estimate the total number of butterflies in the picture by extrapolation. (That is, look at the whole picture with your trained eye and count the number of "10 butterfly groupings.")

5) Come back together as a class, and compare and record your estimates.

6) Finally, make an actual count. How did your class do? Was this method better than guessing?

 Picture #2 (Click to enlarge.) Courtesy of Tim Mostrom

Now you're as ready as you'll ever be to move to bigger trees.

7) Take a 30 second look at Picture #2. Without discussing, students should guess and record the number of butterflies they think they just saw.

8) How would you go about counting this entire roost? Can you use (or rework) the method above to come up with an estimate for the roost? Do you think a grid system overlay would work? Go back to student groups and make your estimates. Come back as a group and record the results from each group.

During research, scientists carefully reflect on their own methods. They often learn from mistakes and refine their methods in the second round. A careful critique of methods is a good habit for scientists.

In your journal, record your thoughts after attempting to estimate the number of monarchs on a single branch and in a roost. What seemed to work well? What would you change if you did this again? How would YOU estimate numbers if you were to discover a fall monarch roost?

• "The pine trees bordering our property have been covered with monarchs. Every time I walked under the trees I was greeted by a cloud of monarchs. I have no idea how many are roosting in the trees. I counted 20 on a small branch and while I was counting there were at least 50 flying from tree to tree. Several of my students have reported large groups of monarchs in the trees in their yards as well."
Mr. Fahrendorff, Grade 5, Hillcrest Elementary, Ellsworth, Wisconsin
• "I am totally intimidated by the prospect of trying to estimate numbers in a major roost," said a biologist who was new to studying monarchs.

National Science Education Standards

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

Use appropriate tools and techniques to gather, analyze, and interpret data. (5-8)

National Math Standards

Number and Operations
Compute fluently and make reasonable estimates.

Problem Solving
Solve problems that arise in mathematics and in other contexts.