Illuminations: Whelk-Come to Mathematics: Investigating the Behavior of Northwestern Crows

Whelk-Come to Mathematics: Investigating the Behavior of Northwestern Crows

In this investigation students explore the possible reasons behind the observation that northwestern crows consistently drop a type of mollusk called a whelk from a height of 5 meters. This lesson plan is based on the Whelks i-Math Investigation.

Learning Objectives

Students will

generalize prior experiences with end behavior, asymptotes, and discontinuities
describe and illustrate the shape of a rational function based upon its algebraic representation
understand the different information readily available from different algebraic representations of rational functions
develop basic skills in working with algebraic representations of rational functions
experience statistical analysis in a real-world setting
apply the concept of work in understanding our world

Materials

Each student needs activity sheets and a graphics calculator.
Each group of three to four students needs shelled, whole, blanched peanuts and a meter stick. Different kinds of peanuts will produce different ranges of data.
Important: The instructor must experiment before class to determine an appropriate range over which to drop the peanuts or structure the class to design this component in the experiment process.The data provided in the answers are based on a brand of oily blanched peanuts. Some blanched peanuts broke relatively easily, and a smaller range of 5-30 centimeters was needed. A brand of Spanish peanuts broke less easily and required a range of 0.5 meters to 3 meters. One bag is usually sufficient for every 12 students if the students work in groups of three to four. Some students are highly allergic to peanuts and peanut oil. Check with the students before conducting the experiment.

Instructional Plan

Prerequisites:

Students should have some exposure to functions with asymptotes. This activity may follow a brief introduction of the basic hyperbola 1/x and can lead to a more complete discussion of hyperbolas. This activity can be used to introduce and apply methods of translations including vertical shifts, horizontal shifts, and stretching or compressing of functions.

Most students have some basic notion of work. This notion may need to be clarified or made explicit.

Procedure:

This investigation is broken down into four parts.

Making a Conjecture - Understanding the Context Students should be given an opportunity individual or in groups to formulate initial conjectures and then discussed as a class using either handouts or the internet resources. Part of this work could be done as a pre-assignment.
Conducting an Experiment - Simulating the Dropping of Whelk The experiment can be conducted either in-class or outside of class with students pooling their data for better results.
Analyzing the Results - Finding a Mathematical Model Analyzing the data can be done using any graphics calculator with linear regression. However, students should find the interactive applet for fitting a hyperbolic function initially easier. Also, students can explore the effects of parameters and translation on hyperbolic functions.
Drawing a Conclusion - Optimizing the work done This stage can be done using any graphics calculator with zoom, trace, or minimization features. Handouts should be sufficient. Function graphing applets are provided as an alternative.

The first activity provides the context and the introduction for the investigation. An overhead transparency of this activity may be used in place of distributing this sheet to each student. Prior to answering the questions, students may view a brief film detailing the crows' behavior. The video, Birds of the Lands of Four Seasons by Churchill Films, contains a clip about the crows. This video clip helps to introduce the activity.

The first activity gives students the opportunity to reason about the context and make conjectures to suggest a model. Students' discussion about the crows' flight paths and factors influencing the height leads to their conjecture that the crows minimize the work exerted to break open the whelks. Students should propose how they might conduct an experiment similar to the natural phenomenon. This activity helps to increase students' ownership of the problem and solution. For this reason, distribute subsequent activities after students have had the opportunity to discuss the first activity. In discussing the context, students readily agree that the relationship between the number of drops and the height of the drop is needed. The students' conjecture of this relationship is critical to the analysis following data collection.

Students must reason about the need for horizontal and vertical asymptotes before sketching a graph. Students understand that the minimum number of drops relates to a horizontal asymptote at N = 1. The possible existence of a minimum height and vertical asymptote remains an open issue until the analysis is complete and should be revisited at the end of the activity. After making these conjectures, students typically sketch a graph of a hyperbolic function justifying that a smaller height requires more drops and a larger height requires fewer drops to break open the whelk.

The second activity describes the process of gathering data while students work in groups of three or four. Alternatively, students can use the sample data set given. In this activity peanuts are used in place of whelks and are repeatedly dropped from the same height until breakage occurs. For each of eight heights, eight peanuts should be dropped and the mean number of drops calculated. A second way to proceed is to have the class design this component of the experiment. Students should determine if the patterns found in the heights and averages of the number of drops support or refute their conjectures in question 1. At this time you may want to discuss general steps for determining a model for situations:

Gather the data.
Conjecture a model for the data based on the context and the graph.
Rewrite the model in a linear form.
Transform the data and determine if a linear relationship results.
If the relationship is linear, find the equation of the line and solve for the desired variable.
Verify the model by comparing the final equation with the original data.

The last two activities direct the students in analyzing the data and determining equations for the number of drops and the amount of work. The analysis activity highlights the need for a statistically rigorous method and understanding of data transformation. The final activity explores the optimal value of work and rewriting symbolic expressions.

Questions for Students

Why do northwestern crows consistently fly to a height of 5 meters to drop a whelk?

Are the crows minimizing the amount of work required to break open a whelk? Is this an example of optimal foraging?

What is the relationship between the number of drops and the height of the drop?

Assessment Options

The material is written so that the guiding questions remain open throughout. One possible assessment task is to ask students to use the Large Whelk data provided or read from the graph to prepare a position paper on if the crows are behaving in an optimal fashion.

If the large whelk data and the guiding questions are answered as a class, then students can use an alternative data set such as the Small or Medium Whelk Data from the graph or students own data to repeat the analysis process. In addition, students may write a summary of the mathematics used, the claims and justifications made, and the use of the related skills to other mathematical situations. As students reflect upon the mathematics, the algorithms applied, and the sense-making approach, mathematical insight is deepened.

Students could be asked to compare the work in dropping small, medium and large whelk to investigate the question of why northwestern crows select only large whelk. To answer this question, students must repeat the analysis and minimization process, make comparisons and draw conclusions. Additional information such as the actual weight of the whelk in each category will be needed.

Extensions

Rational functions with oblique asymptotes may be studied as an extension to this activity. The graph of the amount of work has oblique asymptotes. For very large heights, the curve closely resembles the graph of y = H + k. This equation reflects the observation that for large heights the work is essentially the energy needed to drop the whelk once. For heights close to the minimum height, the graph of the work function resembles the graph of y = c*b/(H-c). This equation is very similar to the equation for the number of drops. Thus, for small heights the work is predominantly determined by the number of drops. Students may verify this by graphing the two equations with the graph of the work function. In addition, students may show that the equation for the amount of work is the sum of the two functions. This provides practice in manipulating and making sense of symbolic expressions. This work leads naturally into the division of polynomials.

Modifications for Statistics: This activity could be used in a statistics class as well. After gathering the data, the mean and standard deviation of the number of drops at each height can be calculated. Students discuss, based on the standard deviations, whether the mean appropriately represents the data. If not, the median or mode could be used in place of the mean in the activity sheets. Groups of students could complete the analysis using one of mean, median, or mode and compare which statistic produces the best results. The actual model is not likely to pass through the smaller heights. Examining confidence intervals around each mean provides additional evidence of the appropriateness of the model. Here students can see the importance of the accuracy of a statistic representing the entire data set. The original research reports are definitely accessible to students with some background in statistics.

Additional extensions to this activity may result as students investigate the behaviors of other animals. Students may be inspired to search for other animals whose behavior is an example of optimal foraging. This is a natural connection between the biology or animal ecology curriculum and mathematics.

NCTM Standards and Expectations

Data Analysis & Probability 9-12

Understand histograms, parallel box plots, and scatterplots and use them to display data.
Understand the meaning of measurement data and categorical data, of univariate and bivariate data, and of the term variable.
Identify trends in bivariate data and find functions that model the data or transform the data so that they can be modeled.
Recognize how linear transformations of univariate data affect shape, center, and spread.
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
Use simulations to explore the variability of sample statistics from a known population and to construct sampling distributions.
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.