Illuminations: Whelk-Come to Mathematics

Whelk-Come to Mathematics

Analyze the Data

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
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. That is,
Work = Height × Number of Drops
or
w = h × n

To investigate the work solely as a function of height, a relationship between the number of drops and the height is required.

The goal of this lesson is for students to find a good model for the relationship between the height of the drop (h) and the number of drops (n). To find such a model, students can use the Model Analysis worksheet. (Alternatively, rather than having all students access this page, you could also display it for the class to see and refer to it as part of a whole-class discussion.)

Model Analysis Model Analysis Worksheet

In previous lessons, students may have observed that the data points for (h, n) resemble the graph of the hyperbolic function y = 1/x, which has a horizontal asymptote at x = 0 and a vertical asymptote at y = 0. The graph of this function is shown below.

The general equation for a hyperbolic graph is

Reto Zach carried out an experiment of dropping whelks of various sizes from different heights. Since northwestern crows feed only on the largest whelk, the data from his experiment for the large whelk are provided within the Model Analysis worksheet.

Have students open the Model Analysis page. Using the interactive graph, allow them to try different values of a, b and c to find a graph that approximates the data. (For this exercise, they can use their own data, a sample set of data from the peanut drops, or the data that Reto Zach collected regarding whelks.)

Based on their work with the interactive graph, have students answer the following questions:

  1. How do the values of a, b and c change the shape of the graph?

  2. Based on the situation, what is a reasonable conjecture and possible explanation for the most likely value of a?

  3. Using this value of a, what are values for b and c so that the function closely models the data?

  4. Which data points cause the most difficulty regarding getting the model to fit? What explanation can you give for this observation?

  5. Explain why a function of the form:

    is a reasonable model for the whelk data.

Explain to students that experimenting with different values for a, b and c is not a systematic method for producing a model; different people may settle on different values. A good method of finding a model would allow anyone to reproduce the results and would be based on robust procedures.

Linear regression is a widely accepted and reproducible method. If the value of a can be assumed to be 1, then it is possible to transform the question of finding a hyperbolic model into a question of finding a linear model, and the method of linear regression can be used to find a valid equation.

The equation above can be transformed to a linear equation using algebraic methods, as follows:

Ask students, "Why does the last equation show that a linear relationship exists between h and (n – 1)-1?" Students should be able to see that, if y = (n – 1)-1 and if x = h, then this equation can be written as:

which is the equation of a line with slope 1/b and y‑intercept c/b.

Once students understand this translation to linear form, have them click on the Show Graph for the Transformed Data button. Students can adjust the values until they find a line that approximates the data; in addition, they can turn on the plot for Y2 under the Functions tab, and this will cause a regression line to appear on the graph.

Conclude the lesson by having students consider the following questions:

  1. Find an equation of the line for the relationship between h and (n ‑ 1)-1. Rewrite this equation to express the hyperbolic relationship between h and n.
    • Compare the equation derived using this method to the equation you found above.
    • Be sure to write your equations using the variables h and n.

  2. Why do you think it is necessary to assume a value for a to be able to use this method?

  3. What are the horizontal and vertical asymptotes for your equations? Do these values make sense?

Extensions
 
Reto Zach collected data on different sizes of whelk based on weight. The graph of his results and the curves he sketched by hand are shown in the diagram below. Show this diagram to students, and have them think about the following questions:
  1. What do you observe about the vertical and horizontal asymptotes for the different sizes of whelks? Explain why the asymptotes might differ for the different sizes of whelks.

  2. What are possible equations for each size of whelk?

  3. Using the data for the largest whelk, find an equation using the methods of this investigation.

NCTM Standards and Expectations
 
Data Analysis & Probability 9-12
  1. Understand how sample statistics reflect the values of population parameters and use sampling distributions as the basis for informal inference.
  2. 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
  3. Understand the concepts of sample space and probability distribution and construct sample spaces and distributions in simple cases.

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.
  
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NCTM Resources

Principles and Standards for School Mathematics
National Council of Teachers of Mathematics Thinkfinity Verizon Foundation
More and Better Mathematics for All Students
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