## Work to a Conclusion

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.

The amount of work in dropping a whelk to break it open depends on the height of the drop and the number of times a whelk has to be dropped.

*W* = *H* * *N*

If you've been working with a data set, the data should automatically be filled in below. Otherwise, enter your own data or select one of the two other data sets.

**Based on the computed values of work for the individual data points, between what heights is the amount of work the smallest?**

The relationship between the height of the drop and the number of drops can be used to investigate the work. Using the method of transforming the data and
using linear regression, the relationship between *H* and *N* for the
sample peanut data is:

The equation for the amount of work for the sample peanut data is:

### Activity 1

**Use your own graphics calculator to find the height corresponding to the minimum work.**

- What is the height at which the minimum work occurs? How do values for the work near this height compare to the minimum work?
- Compare the location for the minimum work you found using the equation to the value for the minimum work you observed from the data? Which finding do you think should be reported and why?
- What is true about the work for large heights? Give an explanation for your observations.
- What are the asymptotes for the work equation?

Both the expression for the number of drops in terms of the height and the work in terms of the height are rational expressions. Rational expressions are those that can be expressed as the quotient of two polynomials.

Rational expressions can be written in several different forms. Three common ways of writing a rational expression are show below for the same rational expression:

Standard Form:

Factored Form:

Proper Fraction Form:

Each form of a rational expression provides different information and insight into the nature of the function.

### Activity 2

*N*and

*W*in each of these forms. Graph each function and determine features of the graph such as asymptotes, zeros, or the general shape.

- What information can you determine about the function from each form?
- Examine the proper fraction form. What does this form tell you about the amount of work for very large heights? for very small heights? Hint: What is the major contributing factor to the amount of work required in each case?

### Reflection Questions

*N*and

*W*.

- Some biologists hypothesize that the dropping of whelk from a height of 5 meters by the crows is an example of optimal foraging. Does the Large Whelk data provide evidence to support or refute this claim? Give specific evidence.
- Work also depends on the weight of the object. Northwestern crows drop only
large whelk which are fairly comparable in weight of about 8.8 grams.
- How would you alter the equation to include the factor of weight?
- How is the height at which work is a minimum affected if the factor of weight is included?

- How was knowledge about rational functions useful in finding a mathematical model for the work involved?

**References**

- Contemporary Mathematics in Context: A Unified Approach, from the
*Core-Plus Mathematics Project*, Course 4, Unit 5. Coxford, Arthur F., James T. Fey, Christian R. Hirsch, Brian Keller, Harold L. Schoen, Eric W. Hart, and Ann E. Watkins. 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), 475-489.
- Smith, Cynthia. A discourse on discourse: Wrestling with teaching rational equations, Mathematics Teacher 91 (December 1998): 749 - 753.
- Zach, Reto. Selection and dropping of whelks by northwestern crows. Behavior 67 (1978): 134 - 147.
- Zach, Reto. Shell dropping: Decision-making and optimal foraging in northwestern crows. Behavior 68 (1979): 106 - 117.

- Computers with internet connection

### Whelk-Come to Mathematics

Explore sets of names and create bar graphs, pictographs, glyphs, and circle graphs.

### Northwestern Crows

### Conduct an Experiment

### Analyze the Data

### Learning Objectives

Students will:

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

### NCTM Standards and Expectations

- Recognize how linear transformations of univariate data affect shape, center, and spread.

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

- Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.

- Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.

- Evaluate published reports that are based on data by examining the design of the study, the appropriateness of the data analysis, and the validity of conclusions.

- Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.

- Use simulations to construct empirical probability distributions.