## Northwestern Crows

Sea gulls and crows feed on various types of mollusks by lifting them into the air and dropping them onto a rock to break open their shells. Biologists have observed that northwestern crows consistently drop a type of mollusk called a whelk from a mean height of about 5 meters. The crows appear to be selective; they pick up only large-sized whelks. They are also persistent. For instance, one crow was observed to drop a single whelk 20 times. Scientists have suggested that this behavior is an example of decision-making in optimal foraging.

Why do you think crows consistently fly to a height of about 5 meters before dropping a whelk onto the rocks below?

Click to play a movie showing the carrion crow dropping mussels on rocks.

The figure below left shows the possible flight paths of northwestern crows when they are dropping whelks. The figure below right shows the detail of a whelk, a large marine gastropod (snail) found in temperate waters.

### Think About This Situation

Consider the dropping of large whelks by northwestern crows.

- Which flight path, A or B, do you think the crows use most? Why?
- What factors do you think influence the height at which the crows choose to drop the whelk?
- Do you think there is a minimum or maximum number of drops required to break a whelk?
- Do you think there is a minimum or maximum height at which a whelk can be dropped to break?
- What classroom experiment could model the dropping of whelks to collect and
analyze data?
- What questions would you attempt to answer in your experiment?
- How would the relationship between the number of drops and the height of the drops help you answer your questions?

- Sketch a possible graph of the number of drops required to break a whelk as a function of the height of the drop. How are your answers above evident in your graph?

### References

- Coxford, Arthur F., James T. Fey, Christian R. Hirsch, Brian Keller, Harold L. Schoen, Eric W. Hart, and Ann E. Watkins. Contemporary Mathematics in Context: A Unified Approach, from the Core-Plus Mathematics Project, Course 4, Unit 5. Glencoe/McGraw‑Hill, 2001.
- Boswall, Jeffery. Birds of the Lands of Four Seasons. [Videotape] Churchill Films, 1987.

Keller, B. A., and H. A. Thompson. "Whelk-come to Mathematics." Mathematics Teacher, 92 (6) (September 1999), 475‑89. - Smith, Cynthia. "A Discourse on Discourse: Wrestling with Teaching Rational Equations."
- Mathematics Teacher 91 (December 1998): 749‑53.
- Zach, Reto. "Selection and Dropping of Whelks by Northwestern Crows." Behavior 67 (1978): 134‑47.
- Zach, Reto. "Shell Dropping: Decision-Making and Optimal Foraging in Northwestern Crows." Behavior 68 (1979): 106‑17.

- Computer and Internet connection

### Conduct an Experiment

### Analyze the Data

### Work to a Conclusion

### Learning Objectives

Students will:

- Use rational functions to investigate the behavior of Northwestern Crows.

### NCTM Standards and Expectations

- Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable.

- Understand histograms, parallel box plots, and scatterplots and use them to display data.

- Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.