Illuminations: Whelk-Come to Mathematics

Whelk-Come to Mathematics

Unit Overview

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Conduct an Experiment

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.

Learning Objectives

Students will:

use rational functions to investigate the behavior of Northwestern Crows

Materials

Computer and Internet connection

Instructional Plan

Are the crows minimizing their work by dropping whelks as they do? The amount of work depends upon the height of the drop and the number of times the crow has to fly to this height. To answer the question, the relationship between the height of the drop and the number of drops is needed.

Reto Zach^* conducted the following experiment. He repeatedly dropped a whelk from a fixed height until the whelk broke. He recorded the height and the number of drops required. He repeated this for several different heights. The dropping of whelks can be simulated by dropping peanuts or other objects. Peanuts are a good choice because they are relatively inexpensive and fairly uniform.

If you do not
wish to have students
conduct an experiment
on their own, they can
use the sample data
provided in the format
below.

To model the dropping of whelks, get a meter stick and a cup of whole, blanched peanuts that have been removed from their shells. Start with a height of 15 centimeters. Repeatedly drop a peanut until it breaks into two pieces. Record the number of drops needed for the peanut to break. Repeat this experiment for at least eight peanuts at this same height. Find the mean number of drops required to break open a peanut. Repeat this experiment for heights of 20, 25, 30, 35, 40, 50 and 60 centimeters. You may want to pool your data with data obtained by other groups in the class. Record the data in a table similar to the following.

Whelk Experiment Spreadsheet

Have students examine the patterns in the data that they have gathered. Ask them to compare their findings with the conjectures that they made during the Think About It lesson.

Thoughts for Teachers

Questions for Students

Are your conjectures confirmed or disputed? What changes in your conjectures would you make?

[The horizontal asymptote should be confirmed from the experiment as well as the general shape of the graph.]

At which height should more peanuts be dropped to get more accurate data? How can you decide when enough peanuts have been dropped at a given height?

[There should be greater variation in the number of drops for the smaller heights. Thus more drops will be needed at these heights. Later, as students try to find a model, this fact will arise again. The model will fit the worst for small heights because, in part, of the larger error in these readings. Responses will probably be informal such as noting when the mean seems to have 'settled down'. There are statistical tests or rules of thumb that can be performed to answer this question.]

Do you think there are a minimum number of drops required to break open a peanut?

[Yes, 1 drop is the minimum number of drops needed.]

Do you think there is a minimum height required to break open a peanut?

[Responses will vary. This question is intended to remind students that this issue is still open.]

What type of function do you think could be used to model the data that you've collected?

[Students may focus on the horizontal asymptote alone - indicating hyperbolic, power, or exponential. If consensus has been reached about the existence of the vertical asymptote, then exponential functions should be eliminated from the possibilities.]

NCTM Standards and Expectations

Data Analysis & Probability 9-12

Recognize how linear transformations of univariate data affect shape, center, and spread.
Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.
Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.

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.


		1 period
		NCTM Resources Principles and Standards for School Mathematics