Northwestern crow investigation

Algebra 2 Activity

Introduction

Visit this site:

http://illuminations.nctm.org/java/Whelk/student/crows.html

Background

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?

Think About the Situation…

Consider the dropping of large whelks by northwestern crows. Which flight path, A or B, do you think the crows use most? Why? (Place a post-it note on the board under your prediction.)

Think About the Situation…

Answer the following questions on a sheet of paper and be prepared to respond as a whole group:

• What factors do you think influence the height at which the crows choose to drop the whelk?

• What classroom experiment could model the dropping of whelks to collect and analyze data?– What questions could 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?

Think About the Situation…

Sketch a possible graph of the number of drops required to break a whelk as a function of the height of the drop. (Use the graph paper provided. When you finish, post your prediction on the wall and examine the graphs of your classmates.)

DO NOT MOVE ON UNTIL YOUR TEACHER INSTRUCTS YOU TO.

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 this question, we need to know the relationship between the height of the drop and the number of drops needed to break a whelk.

Gathering Data

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. We will be using sample data gathered to determine this relationship.

Gathering Data

Visit the website: (Note: Heights are measured in centimeters)

http://illuminations.nctm.org/java/Whelk/Whelk-PeanutSpreadsheet.html

Examine the patterns in the data that you have gathered. Now compare your findings with the conjectures you made.

Analyzing Data

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.

Analyzing Data

Visit the site:http://illuminations.nctm.org/java/whelk/analyze.html

Examine the graph and explain why a function

of the form

is a good model for this data?

Using this equation, find a model to fit the data.

DO NOT MOVE ON UNTIL YOUR TEACHER INSTRUCTS YOU TO.

Making Conclusions

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.

Work = Height * Number of Drops W = H * N

Making ConclusionsUse 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?