UNIT 7—Wildlife

Teacher Materials

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TEKS Support
To the Teacher
Overview
	Additional Resources on the Modeling Process
Teacher Notes
	Unit Project—Tracking Wildlife and Other Issues
	Activity 1—What’s the Moose Problem?
	Homework 1—More Moose Questions
	Homework 1—Which Facts Matter?
	Activity 2—Your Moose Model Begins
	Homework 3—Moose Populations: Analytic Representations
	Activity 3—Interim Report to the Commissioner
	Activity 4—A Life–and–Death Situation
	Activity 5—Net-Growth Moose Model
	Homework 5—Simulation of the Moose Model
	Computer Activity—Using the Computer Program
	Activity 6—Final Report to the Commissioner
	Unit Assessment
	Unit Summary
Annotated Student Materials
	Preparation Reading—Who Manages the Wildlife?
	Activity 1—What’s the Moose Problem?
	Homework 1—More Moose Questions
	Homework 2—Which Facts Matter?
	Activity 2—Your Moose Model Begins
	Homework 3—Moose Populations: Analytic Representations (Recursive and Closed Forms)
	Activity 3—Interim Report to the Commissioner
	Homework 4—A New Wildlife Twist
	Activity 4—A Life-and-Death Situation
	Activity 5—Net-Growth Moose Model
	Homework 5—Simulation of the Moose Model
	Activity 6—Final Report to the Commissioner
	Unit Summary
	Mathematical Summary
	Key Concepts
	Unit Project—Tracking Wildlife Issues
	Assessment—Hungry Mantids
Short Modeling Practice
	Solution for Modeling Radioactive Decay
Practice and Review

TEKS Support

• Graphing linear equations.
• Slope of linear equations.
• Graphing nonlinear equations.
• Exponential equations.

• Recursive form of linear equations.
• Recursive form of exponential equations.
• Use of random numbers to model probabilities.

• Graph paper.
• Graphing calculators and simulation software or
• PC and simulation software or
• Transparency set.

To the Teacher

Teaching mathematical modeling, generally, will change your processes within the classroom. You tend to become a storyteller and guide, rather than a lecturer. It is great fun to set the stage and observe how the students decide to play their parts.

In many ways, teaching modeling is similar to the guide of a white-water rapids expedition. You know where the white water is, you know that you want your group to be excited about trying out some of the rapids, yet you steer the canoes away from the dangerous areas. You will be needed to assist the students in deciding which course to take and which to ignore. Through discussion and questioning, you can lead the students to an exciting and rewarding journey along the river of mathematical knowledge. At the same time, you will want to allow them the freedom to make decisions as to the forks to take.

Two of the lessons are listed as optional. In particular, Activity 3, an interim report, may be omitted if time is short for the unit. This lesson is designed to have students consolidate what they have learned in relation to the modeling process as well as the steps taken to analyze the migration model. If this lesson is omitted, you should summarize these ideas in a lecture format, since the ideas will be used in the analysis of the net-growth model and as tools in the final report.

If all lessons are used, the unit should take approximately two to three weeks.

Note: It may help to read the student Unit Summary and Key Concepts to familiarize yourself with the flow of this unit.

Overview

This unit is designed to reinforce the mathematical modeling process, as well as to give students a different point of reference for exponential functions. It is up to you to decide what areas of mathematics you want to emphasize. Throughout these pages, multiple avenues are suggested so that you have enough flexibility to design the unit in the best interest of the students in your class.

There are parts of the unit that should be done by every student or group of students. Other parts may be de-emphasized, depending on the avenue you decide to take. To summarize the necessary elements for students in a concise way, the following list is provided:

• The process of mathematical modeling. If necessary, review the steps of the process.
• The relationship between the recursive and closed form of the exponential should be emphasized.

It is assumed that the students have worked with exponential functions and exponents in previous courses. However, most introductory algebra courses do not formally work with the recursive relationship of the exponential, e.g., the idea of successive multiplication by a value and that it is possible to find the growth rate by finding the slope of the recursive graph.

• The simulations should be done by all students.

The simulation is important for two reasons. First, it gives students a hands-on experience of how a population can grow over time. Second, students realize that many predictions are done using simulation and that variability can be large when there are few trials. When they see predictions stated in papers or on television, they should be inquisitive about how they were done.

• The Unit Project is highly recommended for all students.

This assignment ties the process of modeling to wildlife and environmental issues and any other issues that are relevant in the students’ lives and communities. It is clear that not all students, perhaps even very few, have an interest in the moose population in a New York state park. Yet, it is also clear that the modeling process is used in many other areas to help argue for or against a particular path of action.

• License is given to you to decide whether you want to use the moose example provided in the unit or some other population that could be modeled in the same fashion.

The first time through this unit, you will probably want to use the existing materials to get a clear sense of the process. It will also allow you to have students use the Unit Project to gather more information involving areas you might want to use in the future. However, after you have worked through the unit, you may want to select an example that would be of more interest to your students. With some modification, you could apply the same process to a variety of populations and growth patterns. Remember that the moose example is just one vehicle to emphasize the modeling process and characteristics of relative short-term population growth as an exponential function.

Following is a listing of elements of the unit with suggested ways to use the materials. Please keep in mind that these are just suggestions and that you should feel free to experiment.

Additional Resources on the Modeling Process

Discuss your expectations for the unit. Set up the reasons to study modeling. Discuss the unit, the modeling process, and the idea that various groups may be affected differently by a decision. A helpful resource that discusses the modeling process is A First Course in Mathematical Modeling by Frank R. Giordano and Maurice D. Weir published by Brooks/Cole, Monterey, CA, 1985. See Chapter 2, pages 29–40. Another source is the textbook series Mathematics: Modeling Our World developed by COMAP and distributed by SouthWestern Publishing, Inc. In particular, Course 3, Unit 7, Modeling Your World, is a detailed trip though mathematical modeling.

Teacher Notes

	Unit Project—Tracking Wildlife and Other Issues

Students may decide to select an environmental or wildlife situation related to the unit or any other situation of particular interest related to community or world issues. Guide them to look for coverage of decisions with conflicting points of view or decisions that have been made based on some type of predictions.

When introducing the assignment, make sure the students understand what you want for log entries. You may want to have them report about their particular entries daily, weekly, or at the end of the unit. You may also want to have the entries turned in each week so that you can read them. Whatever you decide, be sure the students are aware of your expectations. If you decide that a unit-ending project is desirable, be sure to let the students know what you want the project to encompass, how you will assess it, and a time line. As an alternative to a long-term assignment, this can also be given as a weeklong or two-day assignment.

This project is designed to encourage students to look outside the classroom and into the world. Opportunities for such vision are abundant. Any time a projection is necessary or there are differing points of view or information is needed, analysis through modeling is appropriate. In today’s world, there is rarely a time when all groups agree on what needs to be done, and, when they do, they rarely agree on the way to get it done.

These are the types of situations to look for in the student papers (or reports). Do they understand that the power of mathematics is in helping all of us deal with such issues?

You might want to have an oral report or a written report from each student, or you may want to use group reports in addition to or instead of individual reports. This decision will depend somewhat on the manner in which you have students report throughout the unit. Several options are described below.

If students maintain their own files of articles and logs, they can report to the class on the articles and events that they find throughout the unit. The teacher, or student, might then lead a discussion relating the work to ideas developed in solving the moose problem.

If you have students turn in the articles regularly during the unit, you can compile and distribute a list of some of the most pertinent topics. Then have the students discuss in their groups how these topics are related to the moose problem and/or the modeling process. Each group will then report to the class on its findings.

You or your students can create a bulletin board of selected items. Students could then pick out an item of interest and write a brief paper about how the chosen topic relates to the moose problem and/or modeling.

For a more directed approach, lead a discussion of topics the students have found that relate either to the moose problem or to the modeling process. Draw attention to the overall problem of making decisions. Ask the students to write summaries of what they have done in the unit and how this might relate to other topics, which affect their lives. The students could read the Unit Summary to assist them in recalling the processes they went through.

	Activity 1—What’s the Moose Problem?

This activity sets the context for the unit. The students have an opportunity to discuss and refine the central question of the unit. If you have decided to use a context other than the moose situation, this is where it should be introduced and examined.

Discussion and Student Exploration

In their groups, ask students to read the situation described and consider the questions they might ask and information needed. Lead a class discussion of their results. Guide the students into focusing on the question of whether it is desirable to add moose to the park. The decision should therefore focus on what the results will be if they do or do not add 100. When this discussion has taken place, look at the Refining the Questions section of the activity. You may want to use the overhead of it (Transparency 2) or discuss it, asking the students to write down the questions.

Allow students time to develop their answer. A class discussion should help the students come to agreement that questions similar to the second and third questions on the transparency are a reasonable first step:

and,

After discussing the need for further refinement, lead the students to the final two questions on the transparency:

and,

Spend as much time as is necessary to develop its ideas. Asking good questions is the key to successful modeling.

	Homework 1—More Moose Questions

This piece is designed to give students a deeper appreciation of the various factors involved in the decision. Question 4 of the assignment asks students to think about others who have an interest in the outcome of the moose decision, since various interest groups may use the information in entirely different ways.

	Homework 1—Which Facts Matter?

Have students read the Moose Fact Sheet. Note that Question 2 of this assignment asks them to revisit the final, refined questions in Activity 1 with these new facts in mind.

	Activity 2—Your Moose Model Begins

This activity is designed to meet two goals. The first is to give the students practice in the techniques of mathematical modeling. When they get to the second part of the unit, they will have tools they can use to explore a different model for population growth. They will examine the techniques of using tables, graphs, and equations to help find patterns for prediction. The second goal is to find relationships between a recursive and a closed-form description of events.

Students set up a first model, using incoming migration as the only factor that affects the moose population. A new modeling diagram is introduced, and the students review the representation of information through recursive and closed-form expressions for linear growth. Students will arrive at an answer to the key question by using graphical, analytical, and numerical representations of the model.

Large- and/or Small-Group Discussion

The first part of this activity emphasizes the questions students will be answering throughout this unit and reviews the modeling process. The discussion develops the assumptions that the students will formulate for the first model.

Moose Migration

Diagram 1 in this section represents the changing population as a flow is introduced to help the students describe their model. This type of diagram will continue to be used.

The materials lead the students through the modeling process using the assumption for the first model—migration only. In the section How Many Moose in 2013? the students are led through the development of a linear model for growth, using the process described. The resulting rate of migration will set the stage for results in subsequent questions. The groups are then asked to project the results to the year 2013.

Setting the Table

The work in this section illustrates another way to arrive at the resulting change and population in the year 2013. The concept of recursive thinking is described and the connection between closed and recursive forms of a linear equation is examined.

	Homework 3—Moose Populations: Analytic Representations

This work asks the students to link the tabular and graphical representations for the results on Case A and Case B (not adding or adding 100 moose). Case B is a good example of a piecewise-defined linear function. Additional work on this type of definition could be used, depending on your students.

This work is designed to reinforce the ideas of rate of change, graphical representations, and piecewise-defined functions. The recursive graphs (Questions 1 and 3) focus on the process of adding moose from year to year, thus lie along the same line since the process does not change. Transparencies 6 and 7 are included for discussion of these questions during the next class period. They assume a migration rate of 2 moose per year. If your class has chosen a different rate, be sure to discuss how that difference would affect each of the graphs.

The assumptions listed on Model One—Migration can be developed through a class discussion of Activity 1’s exploration. If you choose this discussion option, use Model One—Migration as a summary reference.

You may need to explain the modeling diagrams. Discussion might take the following format.

Emphasize that an early step in the modeling process is to simplify the problem. This allows an understanding of the phenomena and an easier development of the initial model. In this particular problem, since the interest is in the moose population over a twenty-year period, assume that the population in the park continues to increase in the same way that it has over the past few years. Namely, migration into the park is the only factor affecting population growth.

In an ideal situation, the students arrive at the following facts and assumptions through group and class discussion. If discussion is not working, these factors should be given to the students, using the student pages for this lesson. Following are the facts known initially.

• Fact 1—Prior to 1980, there were few, if any, moose in the park.
• Fact 2—Between 15 and 20 moose were estimated to be in the park in 1988.
• Fact 3—Between 25 and 30 moose were estimated to be in the park in 1993.

The following assumptions are made to simplify the model.

• Assumption 1—Population does not grow or decrease other than through migration.
• Assumption 2—Net migration in the future will occur at the same rate as in recent years.

Activity 2 may work best using small groups, with help from you as needed. As students work, remind them of the processes they are using and the questions they are trying to answer. They will need to use these same techniques again when they revise their models in later lessons. When everyone is finished, discuss the results. You may want to have each group make a short presentation of its work. Transparencies 4 and 5 are provided to assist in the discussions.

Discuss the idea that the steepest line touching both blocks of data connects the lowest estimation in 1988 to the highest estimation in 1993. Similarly, discuss the fact that the flattest line goes from the lowest estimate for 1993 to a high point in 1988. (Avoid emphasis on the equations of these lines for now.) Use Conclusion #2 to explore the idea of the "x-intercept" and the implication of the fact that "some" moose were seen in the park in 1980.

Review the modeling diagram on Transparency 3 as a representation of inflow. Allow the class to decide on a value to use for the rate of moose migration per year. Have them use their value to answer the remaining questions. Then have the class agree on the year to use as "the initial year"—the last year having zero moose.

Setting the Table

Allow the students to complete the tables and questions. Be sure they reflect the "rate" and "initial year" assumptions made in the first part of this activity. As a class, have the students discuss their results. Although the closed form is traditionally used, the recursive definition is a more natural first description in the context of this problem. It is important that the students review both of these forms, since they will be using them to develop the concept of an exponential model later in this unit.

Note the additional assumptions required in adding the 100 moose:

• The additional 100 do not affect migration rates, and
• The additional 100 do not improve chances of getting a date.

	Activity 3—Interim Report to the Commissioner

This activity gives students an opportunity to summarize their work by preparing brief oral reports in which they discuss the concepts and assumptions of their models. Homework 4 introduces the possibility that population growth is occurring by means other than migration.

Small groups are suggested for this assignment. Students may need help organizing and developing their thoughts. After distributing this assignment, give them a day to work on these reports in their groups. As they are discussing what they should include, go around to the groups and guide them in formulating their work. They should include the factors they considered and assumptions they made to arrive at their model. Have available materials that students could use to display their information. These materials might include chalk, blank transparencies and pens for an overhead projector, graph paper, and blank graph grid transparencies (Transparency 9). Encourage students to discuss not only their model, but also limitations of their model. This is a two-day assignment. It may take a full day, as well as time outside class, for students to prepare their reports. Encourage discussion and thoughtfulness. The reports are not necessarily as important as the process of preparation. Reports should be brief, but as thorough as time allows, so organization is important.

When the presentations have been made, hand out Homework 4 for students to consider. Discuss the modeling process with the students. Due to the new information, they will now have to revise their models, make new assumptions, and consider other factors. This will set up the next iteration of the modeling process.

	Activity 4—A Life-and-Death Situation

In this activity, students realize that the migration model is insufficient. The students will revisit the problem, revise the assumptions that were made, and continue the modeling process. They will be introduced to the idea that the herd’s current population is dependent upon its previous population. They will deal with a number of growth-type models, and will use a simulation game to gather data on the moose population in Adirondack State Park.

Using independent/small-group and large-group discussions, the students begin analyzing the new information and realize the necessity for a new model. Complexity slowly increases to include factors such as deaths, emigration, and births into a net growth model.

	Activity 5—Net-Growth Moose Model

Working in small groups, students will use a simulation game to generate data. They will plot these data to help them begin to understand the nature of the growth in the moose population.

This activity is designed to give the students an idea of what is involved in simulation. It should also allow them to realize that population is quite variable when populations are small, and, more importantly, when there are few females in the herd. The activity will probably take about a day and a half to graph, discuss, and complete. Students may have to do some of the simulation at home.

Students should have fun doing this, when they understand the process. Student pairs are probably best for this activity—one to generate the random numbers and read the survival chart, and one to record. A detailed description of how to run this simulation is given below, and Transparency 15 may be used to guide the class. Reserve some time in class to help students find the rand key on their calculators. It is crucial that each student pair use its own calculator.

Distribute all of Activity 5 , and use Transparency 16 to go through the process for one year’s worth of data with the entire class, to ensure there is a consistent method of recording the data on the sheets. Afterwards display Transparency 17 and have the students begin their own simulations. After completing a year or two of simulations, students in each pair could work individually. By splitting up the work, they have a reasonable chance of completing six to eight years’ worth of simulation.

It might help the process for each pair to post its herd’s population on the board or butcher paper, or on Transparency 17. This would maintain group productivity and let the students see all the possible variations in this simulation.

The "Survival" table showing the assumed birth and survival rates is, in fact, based on our best guesses. You may want to stress that the data in this table were generated by "informed guesswork." In such cases, the modeler is responsible for defending the assumptions as reasonable. It should be stressed that the probabilities listed are for survival to the next year, not the probability that a given moose will survive to that age. For example, you assume that a 10-year-old moose will have a 93% chance to reach age 11, whereas the probability of an infant living until age 11 would be closer to 35% (the product of its annual survival probabilities).

Step-by-Step Directions for the Simulation

Be sure your calculator is in floating-point mode for the following calculations.

1. Select the male in the Year 1 column and age 0–1 row. You want to see if he survives the year. To do so, check the survival chart. Notice that the chance of survival for a male aged 0–1 is 75%. Using your calculator, generate a random number between 0 and 1. If the number generated is less than or equal to 0.75, this moose survives the year—make a mark in the Year 2 column/age 1–2 row (this moose is now a year older) and cross it off in Year 1. If the random number is greater than 0.75, this moose dies—circle the mark in Year 1 that represents the moose to indicate this moose is no longer with us.

2. Next, consider the female(s) aged 0–1. Follow the same procedure as for the males aged 0–1. (Since females do not start birthing until aged 2–3, you do not need to consider births until the age 2–3 row.)

3. Note that there are no moose aged 1–2 in the chart. Go to the age 2–3 row. There is one male moose. Run the simulation for that male. The survival rate is 95% for a male this age. Generate a random number for this moose. If the number is less than or equal to 0.95, the moose survives. Make a mark in the Year 2 column in the age 3–4 row, giving the moose a birthday, then cross it off in Year 1. If the number generated is greater than 0.95, designate that the moose did not survive (circle the mark representing the moose).

4. Consider the age 3–4 row next. The procedure is the same for the three males in this age group. However, the two females could possibly give birth (the birth rate is greater than 0% for this row of females). Accordingly, you have a few more steps to follow.



1. Generate a random number to see if the female lives (less than or equal to 0.95) or dies (greater than 0.95); there is a 0.95 survival rate for this aged female.



1. If the female dies, circle the mark and go on to the next moose.
2. If the female lives, make a mark in the Year 2 column and the age 4–5 row, and cross her off the Year 1 list.



2. If the female lived, you need to see if she gave birth; if she did, find the gender of the calf.



1. In the age 3–4 row, a female has a 90% chance of giving birth. Generate a random number. If it is greater than 90%, she does not have a calf. Go to the next moose. If it less than or equal to 90%, she has a calf. You therefore need to find the gender of the calf.



• If the female gave birth, generate a random number. If it is less than or equal to 50%, assume that it is a female calf. If the number is greater than 50%, assume that it is a male calf.
• If the female gave birth, make a mark in the Male/Female column (as appropriate) under the next year in the age 0–1 row to represent this calf. Then go on to the next moose.


2. Note that you need to consider births for females from 2–3 until 12–13 ages only. The probability of having a male or female calf is equal (50–50).

5. You have now filled in the Year 2 column. Add up the gender totals, and the total number of moose. Fill in these numbers at the bottom of the Year 2 column. Repeat the simulation for eight years to fill in the table.

Have the students plot their individual data on the butcher paper so that all students can see their results. Ask them to record their data in the table as well. With just 8 years of data, it may not be clear of the variability. Ask the students how they might go about using this combined data for predicting the number of moose in the park in the year 2013. One way could certainly be averaging the data at each year. Another might be taking the largest or smallest values at each year, depending on whether it is desirable to over-estimate or under-estimate the total population. This discussion will lead into Homework 4.

	Homework 5—Simulation of the Moose Model

Students will use a computer or calculator program to generate a more complete picture of the growth model.

Distribute the first page of Activity 4. It is designed to start students thinking about how populations change. A number of factors were considered and assumptions were made in Model One that affect the model for the number of moose in the park, including (among other things) flooding, wildfires, new roadways, disease, a change in predators, illegal hunting, and environmental changes in Vermont, New York, and Canada. Certainly, no single model will include all of them. As the students consider these factors, remind them that they are really making their second pass through the modeling process. This is a good time to review the modeling process (see Transparency 1), and the central problem (Transparency 2). The students are reviewing the properties of objects and deciding which properties to model. Students will probably consider the types of change (birth, death, and so on) in a variety of orders. Do not insist on following the order presented in the reading. However, students should realize, with the additional assumption of moose deaths, that the population will reach equilibrium (level off) if no births occur.

After this discussion, hand out the remainder of the activity and asked students to read through it. The section called Summarizing the Net-Growth Model could be used as a shortened reading if you use a large-group discussion of the previous pages of the reading.

Transparencies 10, 11, and 12 illustrate some of the types of change students may have examined in answering the last question. 13 and 14 represent the convention for indicating dependence of one quantity on another using the feedback arrows. However, do not force assumptions. For example, students may want to allow migration to be included in their models. The effect of this decision may then be explored during the student simulation exercise, Using Simulation in Your Model, by having different models implemented by different groups.

The point of the opening discussion is to see the variation in probabilistic outcomes for a small number of trials with small initial herd sizes, in order to motivate the following computer activity. Your students may quickly realize that the number of females is really the driving factor in the growth of their herd’s population. If a herd is unfortunate enough to lose some of the females at first, or if all the newborns happen to be males, the herd could even decline in population at the start. You want the students to understand that simulations are often used not only to predict outcomes, but also to yield some understanding of the range of possible or likely outcomes.

Hand out the first page of Homework 5 and put Transparency 18 on the overhead. Each shows the plot of ten simulations, with each simulation covering 20 years. Each path represents a separate herd. The graph shows two properties that you should discuss with your students.

First, there is a very wide range of growth rates in this graph; in fact, one herd even decreased in population! Your class results will probably vary as well. Ask the group to explain what happened to its herd to cause either a high or a low birth rate.

Second, there is not a smooth growth in the individual herds from year to year. Many of the paths seem to "bounce around." The point is that, if the herd is small enough, it is quite possible for all the offspring to be male (or female) in any given year. That would certainly affect the growth of the herd. If there were a hundred moose in the herd—with maybe 30 offspring—it is much more likely that there would be a more balanced distribution of males and females. The following computer activity assumes that there are 100 additional moose.

Random behavior is a complex topic. Expose your students to these ideas. However, this may be their first exposure to these ideas, so do not insist on mastery or deep understanding at this time.

Computer Activity—Using the Computer Program

Now distribute the other two pages of Homework 5, the breakdown of the moose population and the Tally Sheet. The second part of this homework—the computer simulation—should be fun for the students. With more computers, collection of these data will go faster. If you are running short on time, you can use the data provided in Transparencies 19 and 20, and move directly to the "averaging" task described below. If you select this option, distribute copies of the data from 20. If there is no time for averaging in class, it may be assigned as part of homework. The results are provided on Transparencies 21 and 22.

Run the program "moose127.exe" (or moose simulation if using Windows). The population provided is 127 moose, beginning in the year 1993. If you have Windows, and want to have a little more flexibility, you can install the simulation program by running the "setup" program from Windows. The setup program will install the appropriate software on you hard disk and create an icon that you can click to run the program.

The listings of moose127.txt and moose27.txt are included in the disk, and may be modified to suit the students’ curiosity. In order to modify these, you will need to load the moose.txt file, change the data lines, and/or beginning year, and compile moose.txt to moose.exe. This is possible only if you have QBasic™ (or some compatible software) on your system.

Collect the students’ simulation data on Transparency 17. Then have students help you average the results, year by year, and record the yearly averages in their own notes. Program listings and detailed directions are included in the Appendix to this lesson. If the disk is bad, or you want to alter the program, it is a QuickBasic™ program. You can enter the program and modify it to suit your interests.

The program is a "run-time" version and is provided in this form for both MAC^TM and DOS^TM environments. There is also a version for the TI–82^TM, but it runs much more slowly.

The data collected will be used for the students’ Final Report to the Commissioner in Activity 6.

	Activity 6—Final Report to the Commissioner

It is suggested that the less teacher direction on this activity, the better.

Group work—preferably small groups—is suggested for this activity. The teacher should be available to guide, suggest, and prod the students in the direction of solid mathematical justification.

Various suggestions and a possible sequence of analysis are described below. Students should arrive at a specific destination, but they do not necessarily have to travel the same road to get there.

Resist the temptation to be too directive. However, students should all explore varied simulation data—either those generated by the class or those that are provided—to understand the value of pooling the data by averaging and to plot the growth of "average" herds. Sufficient guidance should be provided so that they investigate representations of the growth, with the goal of writing its mathematical description (equation) in some form. Some students will be able to discover both the recursive and closed-form equations; others will find only the recursive form, and others will not find either on their own. Each student should be prepared to understand the results of all reports, though, and learn from them.

A Possible Approach to the Analysis of Simulation Results

As you know, conservationists proposed augmenting the Adirondack State Park moose herd by an additional 100 moose. The results of 10 simulations are plotted on Transparency 19.

It would be best for the students to have the opportunity to generate such a plot with their own data from Activity 4. You may want to have your students use the sample data from the simulation listed in Transparency 20 in the interests of saving time and ensuring common results. All "answers" in this activity are based on these sample data; results from student simulations should be similar, but not exactly the same. In any case, students should notice that there is much less variability in the possible outcomes, and there is a distinct pattern to the data, as compared to their work with smaller herds. The goal of this activity is to discover and describe this pattern.

Have students plot their data to form a graph like that in Transparency 19. To discuss the simulation results as a single function, students need to combine the results of the simulations into one list. One way to do this is to calculate the average population for each year. Have them do this for their data and make a new graph of average herd size versus year. Results for the data from Transparency 20 are included in Transparencies 21 and 22.

With 100 extra moose added, it becomes very clear from the graph that the growth in population with respect to time is not a linear relationship. On the other hand, it is also very clear that there is some well-defined relationship here, since the growth seems to proceed in a very predictable fashion.

Another relationship to investigate is current population versus previous population. The ordered pairs to be plotted are (P1, P2), (P2, P3), (P3, P4)...(P19, P20).

Plotting these points gives some indication of how populations change from one year to the next. A blank grid is provided as Transparency 23 and in the student pages. The graph in Transparency 24 shows the relationship of current populations with those of the previous year. The relationship shown by the graph is obviously linear. In fact, if one calculates the regression line, one finds that P_current = 1.127 P_previous –1.7.

We strongly suggest that you do not ask your students to find the regression line, because the constant term "–1.7" will unnecessarily confuse the discussion. Instead, ask the students to hold a ruler up to the graph and observe that the y-intercept is very close to zero. This gives the students the right to assert that the form of the current population is the previous population times some number, or P_current = K * P_previous. In other words, a linear relationship exists between current and previous population. The slope of this relationship is the ratio of successive populations. This may therefore be computed either by calculating the slope of a line sketched through the graph (using (0, 0) and a point near the right end of the data) or by calculating the ratios between the successive populations. If time permits, do both. A blank table and a blank graph are provided for this work.

Have the students compute this ratio for all adjacent data points. They should find that the ratio is indeed fairly constant for your data (usually between 1.12 and 1.13). A histogram of these ratios will make the point quite clearly.

Note that the column for the difference between successive populations will likely lead to no useful relationships. It is included here to verify that the growth is not linear.

Unit Assessment

Introduction

The following list of learning outcomes was created for the Wildlife unit. The outcomes are arranged in two different levels: the lower-level skills, describing the more straightforward, technical skills; and the higher-level skills, related to interpretation, critical attitude, and argumentation. (Some of the skills mentioned are not learned explicitly in this unit, but are based on former units.)

Lower-Level Skills

The student should be able to:

• find a formula for a straight line.
• recognize formulations in recursive and in closed form.
• understand the difference between recursive and closed-form formulas.
• calculate values based on formulas in closed and recursive form.
• read data from graphs and put data in a graph.
• read a modeling diagram.
• carry out a simulation based on random numbers.
• recognize a linear and an exponential growth model, in both closed and recursive form.
• handle intuitively the "law of large numbers."
• handle the idea of net growth factor.

Higher-Level Skills

The student should be able to:

• translate information into a modeling diagram.
• design formulas, in both recursive and closed form, for linear and exponential growth.
• interpret data and graphs within the given context.
• transfer knowledge about growth models learned in the context of Wildlife to another context.

Unit Summary

How you use this unit wrap-up will depend on what you decided to do with the Unit Project. You may decide at this point to display and discuss the information the students came up with in their research. The following suggestions are written with that plan in mind.

Any time there is a projection or there are differing points of view or information is needed, analysis through modeling is appropriate. In today’s world, there is rarely a time when all groups agree on what should be done, and, when they do, rarely on the way to get it done. These are the types of situations to look for in the students’ Unit Project papers (or reports). Do they understand that the power of mathematics is in helping all of us deal with such issues?

Final reporting and summary discussion may take place in any of several ways. You might want to have an oral report or a written report from each student, or you may want to use group reports. This decision will depend somewhat on the manner in which you have had students reporting throughout the unit. Several options are described below.

If students have maintained their own files of articles and logs, they can report to the class on the articles and events they have found throughout the unit. The teacher, or student, would then lead a discussion relating the work to ideas developed in solving the moose problem.

If you have had students turn in the articles regularly during the unit, you could hand out a list of some of the most pertinent topics. Then have the students discuss in their groups how these topics are related to the moose problem and/or the modeling process. Each group would then report to the class on its findings.

You or your students could create a bulletin board of selected items. Students could then pick out an item of interest and write a brief paper about how the chosen topic relates to the moose problem and/or modeling.

For a more directed approach, lead a discussion of topics the students have found that relate either to the moose problem or to the modeling process. Draw attention to the overall problem of making decisions. Ask the students to write a summary of what they have done in the unit and how this might relate to other topics that affect their lives.

The students could use the Unit Summary to assist them in recalling the processes they went through. The Unit Assessment materials deal with modeling and modeling diagrams, exponential functions, randomness, and other topics from the mathematics of this unit. You may want to extend the topics to address the societal issues that are important for students to recognize and how mathematics can help answer some of the unanswered questions when it comes to decisions.

Annotated Student Materials

Preparation Reading—Who Manages the Wildlife?

During this unit, you will be doing some of the work of a wildlife manager. Wildlife managers try to understand the natural world so they can help environmental systems to survive. To do that, they have to observe and record population data for a variety of species. As they track changes in wildlife populations over time, they use the data they collect to try to make predictions about future populations. In some cases, they must decide if control of populations by humans is wise and necessary.

Your task will be to use mathematical modeling to solve problems related to predicting the population of moose in a park in New York. You will be using what you learn to advise the commissioner of the New York Environmental Conservation Department. On to the moose!

	Activity 1—What’s the Moose Problem?

Adirondack State Park (ASP) is a six-million-acre wilderness area in upstate New York. Prior to 1980, the last moose recorded in the park was shot in 1861. After 1980, however, some moose were spotted there. In 1988, it was estimated that 15–20 moose were in the park. In 1993, it was estimated that there were 25–30 moose.

The New York State Environmental Conservation Department (ECD) conducted a public opinion survey. A majority of the people surveyed favored a "gradual increase in the moose population as the animals migrate from nearby New England states and Canada and expand their numbers through natural reproduction." Conservationists suggested moving 100 moose into the park over a three-year period. The ECD determined that this would cost $1.3 million, and the commissioner of the EDC had the task of making a recommendation to the governor about this situation.

Imagine that you are going to advise the EDC commissioner. Before you can do that, you need some specific questions answered. Pose one such question about the data above.

What additional information will the commissioner need?

Refining the Questions

Here is one problem you may have posed in this activity:

You will use the mathematical modeling process to simulate what would happen in each case. This will help the commissioner arrive at a decision.

The first step in the modeling process is to state a problem to be solved. The problem we have defined can be restated more specifically:

These two questions might even be too broad. You may also need to specify a reasonable time period.

The conservationists who suggested moving 100 additional moose into the park used a 20-year time frame for their projections. Since the life span of a moose is 15–23 years, it is reasonable to see what happens to the population during one generation. Therefore, we shall further refine the problem by asking the following questions:

Note that the conservationists suggested moving 100 moose to the park over a three-year period. A simplifying assumption can be made that this move occurs all at once, at the beginning of the 20-year period. You may want to return to this assumption and modify it in the future.

	Homework 1—More Moose Questions

The moose scenario is based on a short article in The New York Times from April 7, 1993. Below are some of the questions the article did not answer. As you answer them, think about whether mathematical modeling can help you to answer these types of questions.

1. Remember that the ECD is a state agency trying to make decisions about spending $1.3 million in state funds. Whom might the ECD survey?




Answers will vary—they may have surveyed conservation groups, groups interested in tourism, lumber industry groups, farmers, and homeowners in the areas that may be affected.

2. What questions might they have asked on the survey?




Answers will vary—"Do you favor adding moose for $1.3 million?" "Do you see additional moose to be important to your industry?"

3. Why might conservationists favor moving 100 moose into the park?




Answers will vary—more moose improve the possibility that the population will continue to grow; larger population of wildlife; it’s nice to be able to go into the woods and see moose.

4. What other groups might have an interest in the number of moose in the park?




Answers will vary—hunters, loggers, farmers in the area, and wildlife biologists, among others.

5. Can the questions above be answered by using a simple mathematical model?




No. Although the answers to the questions above are important, the questions generally cannot be answered by a simple mathematical model.

	Homework 2—Which Facts Matter?

The second part of the modeling process is to make assumptions that assist you in solving the problem. Having additional facts about the situation may help. Here are some facts about moose:

Moose Fact Sheet

• Moose are the largest living members of the deer (Cervid) family.
• At birth, a moose calf weighs 24–35 pounds.
• Moose can survive in snow up to 3 1/2 feet deep.
• A Canadian bull moose may tower seven feet tall and weigh up to 1,800 pounds.
• Young bulls are more likely to travel far from their birthplace, and are sometimes driven away by stronger bulls
• The only predator of the moose in ASP is the black bear.
• Many deer carry a parasite benign to deer, but usually fatal to moose. The deer pass this parasite larvae to foliage that is eaten by snails. Moose may consume the snails while eating plants.
• Well-developed moose antlers may weigh 55–75 pounds.
• The average life span of a moose in the wild is 15–23 years.
• A moose can gallop up to 35 mph for short distances.
• Moose do not tolerate prolonged temperature above 75° F.
• An adult moose consumes 35–60 pounds of plant materials daily.

1. Which of the above facts might be most relevant in answering questions about the moose population?



2. Revisit the final, refined questions at the end of Activity 1. Are any of the above facts of major importance in relation to those questions?

Since so many factors are involved in this problem, it may be necessary to make additional simplifying assumptions later in the modeling process.

	Activity 2—Your Moose Model Begins

You are now ready to develop a model to assist you in solving the problem. Recall that the two questions we chose to focus on are

For simplicity, we shall refer to the two parts of the problem as Case A (move no moose) and Case B (move 100 moose).

Mathematical modeling can yield different solutions to the same problem, depending on the assumptions you make and the factors you consider. It is helpful to start with a simple model; that is, make assumptions that allow you to arrive at a solution and understand the problem. This may not be the best solution; however, it affords you some understanding of the problem that you can use in a more refined model. This process of making assumptions and refining the model continues until you arrive at a solution with which you are satisfied.

Review the facts from Activity 2. List assumptions that you think would be helpful (and reasonable) in answering the questions in Cases A and B above.

Moose Migration

In 1988, there were 15–20 moose in the park. By 1993, there were 25–30 moose. If there are only 25–30 moose, with 6 million acres of land in the park, the chance that a moose can get a date on a Saturday night may be slim. The moose may never get together to mate! Therefore, assume that no moose births will occur for Case A. For simplification purposes, in your initial model you can also assume that there are no deaths in the 20-year period.

The modeling diagram may help you picture your initial model. The rectangle represents the moose currently in the park. Refer to this as the "stock" of moose. You might think of this stock as similar to a tank or barrel that can fill up and/or drain out. The arrow represents the migration into the park. Refer to this as the "flow" of moose. You might think of this as similar to flow through a pipeline. The ring is like a "valve" that allows you to adjust the flow of moose into the park. In Case A, you would twist the ring so that the flow would remain open, and migration would occur at a constant rate.

Your goal is to find out how many moose are in the stock 20 years after the New York Times article—that is, in the year 2013. To do this, you need to know the number of moose migrating each year. You know that moose were spotted in the park after 1980. Figure 2 illustrates the estimated information. The bar-like parts represent the range of estimated values: 1988, 15–20; and 1993, 25–30.

Keep in mind that you assume that the population grows only by means of migration and that the number of moose coming into the park is the same each year. This type of growth (constant number each year) may be represented by a line. In this case, the slope of the line represents the rate of growth of population with respect to time. The steeper the line, the faster the growth.

How Many Moose in 2013?

 = 1,6

1. Position a straightedge so that it touches both sets of data. Draw the flattest and steepest lines possible for these data.

2. How many moose are migrating per year if you look at the slowest growth, represented by the flattest line?




One moose per year from 1988 (20) to 1993 (25).

3. How many moose per year are migrating if you look at the fastest growth, represented by the steepest line?




Three moose per year from 1988 (15) to 1993 (30).

4. Based on the two growth rates above, what is your estimate of the actual number of moose migrating into the park per year?




Answers should vary from 1 to 3.

Assumption

Moose are migrating in at a constant rate of _____ moose per year.

With this migration rate as your slope, use your straightedge to draw a line that extends through the two sets of data. Now use this line to estimate the moose population in the year 2013. Explain your method clearly.

Model One Conclusions

If no moose are added, the number of moose in the park in 2013 will be: ______

Extend the line to estimate (to the nearest year), the last year that the population in the park was 0. The initial year: _____.

Setting the Table

Another way to describe this migration is by using a table. Table 1 is designed to pair the current year’s population with the previous year’s population. Verify the results of your graph by filling in the table of values to represent the number of moose in the park each year. Use your conclusions from Activity 3 to assist you in filling out the chart. Circle your initial year.

1. The method you used to fill in the table is called a "recursive" representation. In your own words, or using a mathematical expression, describe this method.




Answers should vary—to find the population for the first year, add the moose migrating into the park to the population in the initial year; to find the population for the next (second) year, add the moose migrating into the park to the first year’s population; and so on.

2. Relabel the last year that you estimate there were moose in the park (i.e., the "Initial Year") as year 0. The next year would be year 1. Then the year 1993 may be represented by the number _____, and the year 2013 represented by the number _____. Using the variable t for the number of years since your initial year, express the number of moose in the park in terms of t. This is called a "closed-form" representation.




Answer: The number of years since their initial year. Population = (migration rate) × (number of years since the initial year).

3. Compare the recursive expression in Question 1 above with the closed-form expression in Question 2. How are they different? How are they alike?




Answers should vary—something like, the recursive is in terms of the year before, and the closed is in terms of the total years since the initial year. Both use the number of migrations per year.

4. For what purposes might the recursive form be preferred? The closed form?




Recursive: writing the "next few years’" populations. Closed: predicting a population for a distant year.

Model One Conclusion—Case A

5. Based on all your work thus far, your answer to the question in Case A (How many moose will be in the park at the end of 20 years (2013) if no moose are moved there?), is _____. Explain what information you used in working this problem.




Answers should vary from around 45 (using one moose per year) to 90 moose (using three moose per year).

	Homework—Moose Populations: Analytic Representations (Recursive and Closed Forms)

Previously, you used a recursive expression to represent the current population, based on the previous population. This yearly change can be thought of as displacement of moose, and may be recursively represented as follows:

A recursive representation requires an initial value. Why?

In your model, the initial value is 0 moose. You can restate the recursive representation as:

A closed representation may be used to represent the population at any time, in terms of the migration rate (moose per year), the elapsed time (years), and the initial population. The closed form equation is:

P_initial is still 0 in this closed-form expression:

P(t) = (migration rate) t + P_initial

Fill in the recursive chart in Table 2 to represent the results of moving 100 moose into the park in 1993.

Model 1 Conclusion—Case B

 = 8, 12, 15, 16

1. There will be _____ moose in the park in the year 2013 if 100 moose are added in 1993.




Answers should vary from around 145 (using one moose per year) to 190 moose (using three moose per year).

2. Sketch a graph of the current moose population in terms of the previous year’s population for Case A (no moose added). Use the table that you filled in for the numeric representation (Table 1) to assist you. What is the slope of the line connecting these points? What is the vertical axis-intercept? What do these values tell you about the growth of the population?




Sample graph:



This graph is recursive and results from using a migration of two moose per year. Student answers will vary, but the pattern should be similar: The slope should be 1 in all cases, and tells you nothing additional; the intercept gives you the starting value, thus, the migration rate. Note that this is a different interpretation from a time series graph, in that the slope is not the rate of change in population. Rather, it represents the rate of change in successive years. Since the quotient Current/Previous is always 2/2 (or whatever the student uses as the migration rate), this quotient is one.

3. Sketch a graph of the moose population over time for Case B. Show the effect of moving 100 moose into the park in 1993.




Sample graph:



The graph above represents an assumption of growth of two moose and an initial year of 1982.

	Activity 3—Interim Report to the Commissioner

You have scheduled a meeting with the commissioner to present your findings to date. You plan to include (as all modelers should) an explanation of the methods you used to arrive at your results, the model that you used, and the factors that you did not include in your model. The commissioner has decided to see you, but has limited time.

Prepare an oral presentation (using whatever presentation materials you have available) for this meeting.

	Homework 4—A New Wildlife Twist

You have finished modeling the problem—or so you think—when some new twists occur. The conservationist group, which has been advocating that 100 moose be moved to the park, has just issued a report. According to their projections, the addition of 100 moose to the park will result in a moose population of around 1,300 in the year 2013. Your model does not come close to the conservationists’ number, but it was the best model you could construct given the limited information you originally had. Also, the ECD receives information that some baby moose have been sighted in the park.

1. What new factor(s) must be considered?

2. Modify a modeling diagram to reflect how you think the new factor(s) affect the moose population (stock).

	Activity 4—A Life-and-Death Situation

You have convinced the commissioner to give your department additional funds to study the problem more deeply. Reconsider your initial assumptions, which led to the migration model. In a short-term situation, an assumption of no moose deaths makes sense. However, your model represents 20 years. The expected life span of a moose is 15–23 years. Some moose migrating into the park at the beginning of the 20-year period may already be aged. Accordingly, it is reasonable to consider deaths as a factor in your model, along with migration. Still assuming there are no births, a revised modeling diagram reflecting this would be:

Death is represented by a flow out of the moose stock. This model may be represented recursively as:

This model assumes that every moose that migrates into the park will stay there until it dies. Repeat your work from Activity 2, Table 1, and Homework 2 using this new model. Write a short paragraph summarizing your findings. However, some moose may migrate into the park (immigration), but later leave the park (emigration). Redrawing the above modeling diagram to reflect immigration, emigration, and deaths as flows into or out of the stock of moose, it might look like:

Write a recursive representation of your modeling diagram.

Modify your model to reflect any other considerations you believe are important.

Adding Births to the Model

In Model 1, you assumed that the moose population grows only through migration. It seemed reasonable to ignore births as a factor, given the large size of the park and the relatively small number of moose. However, the sighting of some baby moose in the park in 1993 indicates that moose might have given birth in the park. You therefore need to introduce births into your model.

The modeling diagram reflecting this situation could be as follows:

Assume that the moose population in the park remains sparse enough that the moose do not feel crowded. Migration therefore remains constant. However, with more moose in the park, they are more likely to meet and mate, so births will probably occur. Similarly, with more moose, some are likely to be old or ill, and deaths will probably occur. Therefore, the numbers of births and deaths depend upon the number of moose in the park. You can indicate this relationship with a thin, curved-arrow connector to form a feedback loop.

Figure 6

The recursive representation of the modeling diagram as refined is:

To simplify this model, call all the changes to the population the "net growth," and rewrite the recursive representation:

net growth = births + immigration – (deaths + emigration).

Therefore, the model in recursive form is:

For purposes of this new model, assume that the immigration and emigration are roughly the same number of moose. Net growth would include only births and deaths. Both factors may be expressed in terms of the previous year’s population. That is, both deaths and births are based on how many moose are in the park.

Combining births and deaths into net growth results in this final diagram:

One further refinement to the model is necessary, however. Since 1993 was the first year that moose calves were detected in the park, this seems to be a good place to start the growth model. Since it was estimated that between 25 and 30 moose were in the park, use 27 in the model. Therefore, set the initial year to 1993 and the initial population to 27 moose.

Summarizing the Net-Growth Model

There has been a major change in the park: the introduction of females, and the observation of births to those females.

In your first model, you assumed that deaths simply did not occur. It seemed like a reasonable assumption at the time, since there are no natural predators in the park and you had data suggesting that most of the moose wandering into the park were very young. On the other hand, you have seen that linear growth would take such a long time to increase the population that deaths by old age would have to be considered in the relevant time period.

Since you have concluded that migrations (producing linear growth) are insufficient for large increases in population, your model must include another factor. You must include births. (There have been two sightings of moose cows with calves in the park, so it is reasonable to assume that births will continue to occur within its boundaries.)

Notice that, in a simpler model (which ignored both births and deaths), the age and gender of the moose were irrelevant. This is no longer the case. Therefore, age and gender information should be included in your model.

Although the migrations were a necessary condition for establishing the herd, you have calculated that the migrations happen at a rate of approximately 2–3 per year. In searching for the effect that might cause the herd to grow to 1,300 in 20 years, 2 or 3 migrations per year will play a very small part. For convenience, your next model will ignore the effects of continuing immigrations/ emigrations into and out of the area.

The consolidated modeling diagram looks as shown in Diagram 8. Remember, the feedback arrow represents the fact that the amount of growth in the population depends, in some way, on the current population. The "two-way" arrow on the flow indicates that net growth can be either positive or negative.

	Activity 5—Net-Growth Moose Model

Using Simulation in your Model

You will now begin a simulation of the moose population in Adirondack State Park. Keep in mind what you are trying to accomplish. You are running the simulation to collect data in order to see if there is a pattern to the growth in the population. You know the approximate initial population.

• The population distribution in the table below incorporates the following known data:
• The herd is estimated at 25–30 in number.
• Moose that migrate from their original territory are usually male, and are usually young (3–5 years old).
• Females were not observed in the park prior to 1988.
• Two mothers with calves were observed in the park in 1993.

Assume that there are 22 males and five females in the population. You can change this assumption and try a different make-up later. Using Table 3, together with the Simulation Checklist, Table 4, and the worksheets, run the simulation for an eight-year period.

Simulation Checklist

1. Identify a particular moose from Table 3 that has not been selected and check its age, gender, and probability of survival. The probability that a moose will survive to the next year can be represented by a decimal between 0 and 1, or by the percentage (between 0 and 100) of moose that will survive to the next year.



1. Generate a random number. Is it less than or equal to the probability of survival? If yes, go to step 1(b). If no, circle the mark to represent that the moose did not survive, and go back to step 1.
2. Is the moose a female? If yes, cross the moose off the current year and place a mark in the box for the next year and the next age for females, and go on to step 2. If no, cross the moose off the current year, place a mark in the "male" box for the next year and the next age, and go back to step 1.

2. Use the age to check the percentage of females that give birth. If the birth percentage > 0%:
1. Generate a new random number. Is it less than or equal to the percentage that give birth? If yes, go to step 2(b). If no, go to step 1.
2. Generate another random number. Is it less than 0.5? If yes, add a female moose to the population 0–1 for the next year, and go to step 1. If no, add a male moose to the population 0–1 for the next year, and go to step 1.

The assumption of 27 moose—5 females and 22 males—was used to fill in year 1.

Homework 5—Simulation of the Moose Model

If you were to complete 20-year simulations of the moose population, the data would probably be similar to the data that are plotted below. The different paths represent different herds.

Notice the wide variety of values for the total population in the year 2013. You are trying to find a pattern in the data, but there is too much variability to allow you to see a clear pattern.

You could try running many more trials, using the 27 moose, to see if a clearer pattern develops. Explain what this would give you.

As suggested by the conservationists, you could also add 100 moose into the model to see if that would allow a more precise picture of the growth. The latter method is described in the computer simulation on the following pages.

Assume that 100 moose are added to the herd. The herd could now be made up as shown in the table to the left (you can modify this later if you want).

Averaging the results from a number of trials might be a good first method for predicting the results within uncertain data. However, to use this method, you need a reasonably large number of trials. As you have probably realized by now, running the simulation using your calculator and a paper tally sheet can be both tedious and time-consuming.

This is the type of problem for which computers were developed.

Using the computer program provided, have each person in your group run five simulations. Using the table, keep track of the total population each year; then give it to your teacher so that the results of everyone’s data may be averaged. Remember, you are trying to find the pattern in the growth of the moose population that will allow you to describe the nature of the "net growth" in Model 2.

	Activity 6—Final Report to the Commissioner

In two days, you are scheduled to report to the commissioner of the New York State Environmental Conservation Department on your new model and the prediction of how many moose will be in Adirondack State Park in the year 2013. The Commissioner also wants to know what the population is expected to be in the year 2033.

Analyze your simulation data (or data your teacher provides), then prepare a written report using all information and data you gathered. You may want to carry out other types of analysis in addition to those suggested by the available worksheets. Describe your methods and justify your conclusions. Give a brief oral description of how you developed your model and the conclusions you reached.

Assessment—Hungry Mantids

The mantid is a creeping insect that looks a little like a cockroach. Ordinarily, it barely moves. However, the hungrier it is, the farther it creeps to get food. You can see this in the graph in Figure 1.

Researchers gathered data about the degree of satiation (measured in grams, the weight of food that the mantid has in its stomach) and its maximal distance of reaction (in centimeters). Call the satiation S and the maximal distance of reaction R.

The researchers have drawn a best-fitting line through the data.

1. Read from the graph how large the maximal distance of reaction is if the mantid is as hungry as it can be.

About 7.2 cm.

2. Suppose a mantid is half-satiated. It discovers food at a distance of 2 cm. Will the mantid go for it?

No, because a half-satisfied mantid's maximal distance of reaction is about 1.5 cm; certainly not 2 cm!

3. Describe in words the behavior of the mantid as its satiation grows. Explain the meaning of "hunger threshold" (marked with the arrow in Figure 1).

As its satiation grows, the maximal distance of reaction decreases in a linear way; this goes on until the mantid isn't hungry anymore (at the "kink," the hunger threshold) at a satiation of about 0.62 gram: from then on R stays the same and remains about 0.2 cm.

4. Find the equations that describe the relationship between S and R. Take S as the independent variable.

R = 7.2 – 11.5S   for 0 < S < 0.62
R = 0.2   for 0.62 < S < I

5. Suppose the mantid shown in Figure 2 is only half-satiated and that the dotted oval around its head shows its maximal distance of reaction at that moment. Use this information to estimate the mantid’s length.

Its maximum action-radius is 1.4 cm in the picture; when the mantid is half-satiated, its R is 1.5 cm, so the mantid is just 1.5/1.4 times bigger than in the picture. If you measure the mantid to be 2 cm in the picture, it will be about 2.1 cm in reality.

Researchers found satiation to be an interesting subject. How fast would the mantid’s digestion work? When would it be hungry again after complete satiation? Again, data were gathered on the degree of satiation related to the time that had passed since the mantid had last eaten its fill. These data were also graphed and a best-fitting curve was drawn (Figure 3). S represents satiation, and T represents time deprived of food.

Figure 3. Time-series graphs of satiation and food

6. The data in Figure 3 are represented by dots with a little vertical line above and below each dot. What do you think these lines mean?

These little lines say something about the variation in satiation at a certain T.

7. Explain why S decreases as T increases.

The longer the mantid doesn't eat, the hungrier it will be, so the less its satiation will be.

8. In the graph in Figure 3, there is a scale on the right side as well. If you look carefully, you can see that there is something strange about this scale. What is the strange thing about this scale? Explain why this feature actually makes sense.

The scale is "upside down" (it starts counting from above). If you read carefully what is written on the axis, you can see that W is the weight of food eaten when the animal is fed again to full satiation. Because "full satiation" is 1.0 gram, when the satiation is 0.4 g, the weight of food the mantid can eat to full satiation is 0.6g.

9. Draw a graph that represents the relation between W and T (this time with W presented in a normal way, on the left axis).





10. The data in Figure 4 are read from the graph in Figure 3, in which you can see that the relationship between T and S is not linear. How can you see from the table that the relationship is not linear?

The first eight hours, S decreases 0.28 g; the next eight hours, S decreases 0.20 g, and the third eight hours S decreases 0.14 g. If the relationship was linear, the decrements should have been equal.

11. You might suppose the relationship to be exponential. Find the relative rates of growth for each 8-hour period and decide whether the data seem exponential. If so, find the growth factor for each 8-hour period and write a closed-form equation. If not, explain how you know.

For n = 0, 1, 2, 3, 4, , which implies an exponential relationship. The growth factor k = 0.7 and the closed form is:

S_n = S₀kⁿ = (0.94)(.7)ⁿ

12. Compare the values of S at T = 72, reading from the graph in Figure 3 and using the formula from Item 11. Do you think the exponential formula fits the graph reasonably well?

At T = 72, this means that t = 9, so

S = (0.94)(0.7)⁹

13. You have seen two graphs about mantid behavior: Figure 1 and Figure 3. In both graphs, satiation is involved. The data from both graphs can be linked.



1. Imagine a mantid that has been without food for 40 hours after eating its fill. What is its maximal distance of reaction then?

T = 40; S = 15; R = 5.5 cm.

2. Imagine another mantid, deprived of food for 8 hours. In this case, what is its maximal distance of reaction?

T = 8; S = 0.66; R = 0.2 cm.

3. Which of the graphs in Figures 5–7 correctly shows the relation between T and R? Explain.

Graph B must be the right one: the horizontal part, i.e., the part where the mantid is satiated enough, so it won't react to food that is more than 0.2 cm away, is in the beginning. The later it gets, the more hungry the mantid will become.



Figure 5. Graph A




Figure 6. Graph B




Figure 7. Graph C

Unit Summary

The model is finally complete. Or is it? Maybe it is as complete as possible at this time. Examine the results from your calculation of the exponential growth model, using a 40-year time frame. What if you use a 100-year time frame? Is it realistic to think that there will be around three million moose in the park after 100 years? That would be one moose for every two acres of ground.

Obviously, the population must be controlled. Population models can be developed that consider these other factors. When you examined the migration model that included deaths, you found that the population reached equilibrium. Similarly, there could be limiting values in the growth model, but, for the time being, you will have to be content with the model you submitted to the commissioner.

Recall how the unit began. You looked at some modeling examples and decided which questions to ask and what factors were important. You also discussed some of the groups that might be interested in the addition of moose to Adirondack State Park. If you were a member of one of these groups, which model would you use to support your concerns?

You continued through the modeling process by making assumptions and "mathematizing" the problem. Developing your first model, you expressed it in terms of the previous year’s population (recursive form) and also in terms of the total number of years since your initial year (closed form).

During your initial report to the commissioner, you became aware of new information about births occurring in the park, and that estimates had been made of up to 1,300 moose in the park.

You then returned to the modeling process to revise your assumptions. Since little information was actually known, you needed to use simulation to approximate the size of the herd. Realizing that you were trying to find the net growth rate, you learned that, in order to reduce the variability of the results, you needed to use a larger population (moving in 100 additional moose) and average the data from many trials.

Using the time-series data created by the simulation, you found a new, nonlinear relationship. You looked at its recursive representation to arrive at the equations describing this model. The resulting exponential function is one that can be used to model a number of real-world situations.

The modeling process serves to simplify complex problems so that solutions can be achieved.

Situations with a wide array of factors can be analyzed through mathematical modeling techniques. In this way, mathematics is a powerful tool that contributes to the understanding of issues in our world.

Mathematical Summary

As you developed your models, you also used many mathematical ideas. Looking at your initial model (migration only) you observed linear growth. You described this growth in several ways. First, using a recursive representation, you described it as the addition of a constant to the previous year’s population:

(1) P_current = P_previous + (new moose per year)

This equation was also represented as a "flow diagram."

Together with the initial population information, this made a very "common sense" description of the population growth.

In addition, you described linear growth as a function of elapsed time using a closed-form expression.

In addition, you described linear growth as a function of elapsed time using a closed-form expression. The closed form is developed from the recursive as shown here.

Given:
An initial population, P₀ at time t = 0; and an annual immigration rate of a.

(2)	P0 = c P1 = a + P0 P2 = P1 + a      = (a + P0) + a P2 = 2a + P0 P3 = P2 + a P3 = (2a + P0) + a P3 = 3a + P0

In general:

P_t = at + P₀

where t = number of years

In the specific case of the moose population, the initial population was 0. Note that "new moose per year" appears directly in each of these two representations. New moose per year is the rate of change of population with respect to time and describes "how" the moose population grows.

Each of these representations has a recognizable graph. The recursive relation is graphed as Current Population Versus Previous Population, and consists of points along a line with a slope of 1 and a "y-intercept" equal to new moose per year. (Of course, that’s exactly what equation (1) says!) The closed-form graph is plotted as Current Population Versus Year. Its form is also a line, but its slope is equal to new moose per year and its y-intercept is the population in "year 0." The fact that the closed-form graph is a line is the reason this kind of growth is called linear.

As you developed your second model, you found the need to gather data to assist you in your predictions. Since you didn’t have enough data to model the problem, you used a simulation to gather more data. The first exercise in this process used a very small herd and only a few simulations. Data obtained in this manner are frequently highly variable (scattered). Increasing the size of the herd and increasing the number of simulations led to decreased variability. The mean populations for each year served to describe the pattern of the data. The idea that the more tests you run, the more the average results will express the underlying pattern, is an important part of probability.

You analyzed the simulation data using the same modeling tools that you used in the migration model. The recursive form of these data was again linear. Estimating the equation of the line led to the discovery that:

(3) P_current = k * P_previous.

That is, population grew by multiplying, not by adding. A flow diagram for this relationship is shown below.

Note the connector from Population to Net Growth, indicating the need to know the population in order to calculate the amount of growth.

The multiplier, k, is the base of the exponential growth. The closed-form equation of this new function is:

(4) P_current = P_initial * k (years since initial year)

As with linear growth, the important constant, k, appears directly in both the recursive and closed-form descriptions of the growth. In addition, as you saw in the linear case, the closed form is the reason for the name—the independent variable appears in the exponent.

The recursive exponential form has an equivalent closed form as the linear form. Start with an initial population, P₀, and an exponential growth base, k, and observe the expression for the population for the first three years.

Pt = P0k P2 = P1k P2 = k(P1k)	Population after 1 year, or 1 time period
P2 = P0k2 P3 = P2k	Population after 2 years, or 2 time periods
P3 = P0k3	Population after 3 years, or 3 time periods

We can make the generalization that the population after n years (or n periods if a different interval and different k are used) can be found from the closed exponential form:

P(t) = P0kn

where P0 is an initial quantity, k is the exponential growth base, and n is the number of elapsed time periods defined by

In this model k is the growth factor for one year, so n is just the number of years.

Throughout this unit, you used the ideas of mathematical modeling to help you answer the questions involved in predicting the moose population. This modeling process can be used in many mathematical problems. It can also be used in assisting you in making other decisions.

Key Concepts

Linear growth (recursive): Add a constant number in each time period.

P_c = P_p + A.

Linear growth (recursive): A line with a slope of 1 is the "graphical signature" of linear growth; the y-intercept is the rate of change per time period.

Linear growth (closed form): The graph is also a line. The slope is the rate of change with respect to time, and the y-intercept is the population in year 0.

P_c = P₀ + A_t

Exponential growth (recursive): Multiply by a constant number in each time period.

P_c = P_p * k

Exponential growth (recursive): A line through (0, 0) is the "graphical signature" of exponential growth; the slope is the base of the exponential function.

Exponential growth (closed form): The graph is a curve.

P_c = P₀ kⁿ.

Exponential growth with base k has a growth rate of (k – 1).

Simulation is the process of "acting out" the assumptions you have made.

The probability that a number randomly selected from between 0 and 1 is also less than "p" (some number you chose first) is p. For example, the probability that such a randomly chosen number is less than 0.3 is 0.3 (or 30%).

The average of many simulations is usually more reliable than a result obtained from very few simulations.

Unit Project—Tracking Wildlife Issues

1. Keep a log of all articles in the newspaper, television or radio coverage, or other sources of information concerning decisions about wildlife or environmental issues in general that must be reached from conflicting points of view or that must make predictions of some type.
2. Note where and when you read or heard about these items.
3. Note the concerned parties and a few of their concerns.
4. Note which concerns are conflicting, and which are common concerns.
5. Is a mathematical model suggested as a solution—or used as a prediction—for this conflict?
6. What, if anything, is the suggested solution?

Short Modeling Practice

Solution for Modeling Radioactive Decay

Students should be led to remember that the key to exponential relations is based on rate of change being proportional to the quantity present.

The rate of change in column D is relatively constant. This supports the guess that the decay is exponential.

The next step is to determine values for k and an expression for n to use in the recursive exponential model.

A = A0(k)n

1. You should choose k to be 1/2 and define n = . Where T is "half-life." You must find an estimated value for T from the data supplied.
2. Then use the data to determine how much time is required for the initial 10023 counts to decrease by a factor of 1/2. This is the time required for the count to drop from 10023 to 5007. The data chart tells us that the count will be ~5000 somewhere between 3 and 4 min. A more accurate estimate can be made by graphing the original data.



One-half the maximum count is ~5000. The time when the cpm is 5000 is estimated to be 3.47 minutes. Using k = 1/2, n = , and initial cpm = 10023, the exponential model is



The student results may vary from these values.

A table and graph produced using this model for time up to 15 seconds will show that cpm drops to 500 at approximately 14.8 minutes. Answers between 14.5 and 15.5 minutes are reasonable.

1. What do you observe about the Rate of Change column? Does the rate increase? Decrease? Stay constant?




The rate of change is nearly constant.

2. Why does this justify using an exponential model?




A constant rate of change with respect to the quantity present is an indicator that the change is exponential.

Practice and Review

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–5

–

0

Yes, it is a linear equation. To get the equation into standard form, multiply the number in parentheses by its coefficient.

C = 5/9 F – 5/9 (32)
C = 0.56 F – 17.78

Here the fractions have been converted to decimals and rounded to two decimal places. The variables are C and F, the coefficient is 0.56, and the constant is –17.78.

1. The students’ graphs should appear generally as shown below.

2. Yes, the relationship is linear.
3. Since the equation is given

Distance = Speed × Time

the student must isolate Time by dividing both sides of the equation by Speed.

=

Time =

The units used for the data in this problem are not consistant with each other. There are two different length units, feet and miles. Change all units to the fundamental units of feet and seconds. Use these equivalent relationships:

1 mile = 5280 feet
1 hour = 3600 seconds

Speed (in fps) = Speed (in mph) × ×

So, for example, using the speed of 50 mph,

Speed (in fps) = 50 mph × × Speed (in fps) = 73.3 fps (rounded)

Then the reaction time can be determined from the equation solved for Time above. Using the data Distance = 51 feet at a Speed = 73.3 feet per second...

Time = Time = 0.70 sec (rounded)

1. Yes, the graph does seem to show a fairly linear relationship. One characteristic of a linear expression is that it graphs as a straight line. This lowering of blood flow does seem to follow a fairly straight line, particularly after age 35.



2. Substitute a value for the age, such as A = 42.



F = –1.18A + 141
F = –1.18(42) + 141
F = 91

For A = 65. . .

F = –1.18(65) + 141
F = 64

Both of these seem to agree well with the graph of the data shown in the text.

Exercise 10

1. The correct choice is equation 4:




T = 0.4D + 15

2. Substitute D = 5.

T = 0.4(5) + 15 T = 17, or 17 min

Substitute D = 10.

T = 0.4(10) + 15 T = 19, or 19 min





First, subtract the constant "15" from both sides.

T – 15 = 0.4D + 15 – 15 T – 15 = 0.4D

then divide both sides by the coefficient 0.4 …

= D =

3. To find the number of days when the tanning exposure will reach 23 minutes, substitute T = 23.



D = D = 20

Thus, on the 20th day following the initial treatment, the maximum exposure time will be reached.

Exercise 11

1. No, this relationship is not a linear one, as can easily be shown by a graph of the data: they do not follow a straight-line relationship (as shown below with the best fit superimposed—the fit discussed in Part b of this exercise).



2. Check the equation with a few values from the table.

For H = 8,

S = –2.817 (8) + 108.9 S = 86.4 (rounded), compared to table value of 90.

For H = 3,

S = –2.817 (3) + 108.9 S = 100.4 (rounded), compared to table value of 97.

For H = 1,

S = –2.817 (1) + 108.9 S = 106.1 (rounded), compared to table value of 105.

From these checks, one can see that the generated values are close, but not accurate. This confirms the conclusion from Part a, that the data cannot be accurately represented by a linear relationship. Computers can easily provide the parameters for a line that is the "best fit" of a group of data. But the user should always examine a plot of the data to interpret results like this from a computer.

Exercise 12

1. From the text of the exercise, the voltage (V) equals the current (I) times the resistance (R). As an equation,

V = I R
where	V is the voltage in volts, I is the current in amperes, and R is the resistance in ohms.

This equation has only variables V, I, and R. It is not a linear equation, since it has variables multiplied by each other.

2. Substitute R = 3300 for 3300 W.

V = 3300 I + 0

This is a linear equation, and it is shown above in standard form. The variables are V and I, the coefficient is 3300, and the constant is 0.

3. Substitute I = 0.0175 for 0.0175 amp.

V = 3300 (0.0175) V = 57.75

So, the voltage is 57.75 volts.

Exercise 13

1. Substitute the known values: L = 28 in. and d = 3 in., and simplify to get an equation in standard form.

B = 2 (28) + 1.625 ( D + 3 ) B = 56 + 1.625 D + 1.625 (3) B = 1.625 D + 60.875

where B and D are the variables, 1.625 is the coefficient, and 60.875 is the constant. Since measurements in inches have been substituted, values for D and B must now be in inches.

2. For a diameter of 6 inches, substitute D = 6.

B = 1.625 (6) + 60.875 B = 70.625, or 70.6 in. (rounded)

Checking the result with the original equation, substitute the values.

B = 2 L + 1.625 ( D + d ) 70.6 2 (28) + 1.625 (6 + 3) 70.6 = 70.625, or 70.6 (rounded)

Yes, it checks.

For a diameter of 10 inches, substitute D = 10 in.

B = 1.625 (10) + 60.875 B = 77.125, or 77.1 in. (rounded)

Checking the result with the original equation, substitute the values.

77.1 2 (28) + 1.625 (10 + 3)

Yes, it checks.

Exercise 14

1. The number of spaces is the number of holes minus 1. The spacing is equally divided into the length of the bar minus 2 3/4 (2 3/4 is the sum of the 1 3/4 on either end that must be maintained).

S =
where	S is the spacing between the holes’ centers in inches, L is the length of bar stock to be drilled in inches, and N is the number of holes to be equally spaced.

2. Substitute N = 9, and put in standard form.

S = S = 1/8 L – 2/8, or S = 0.125 L – 0.25

3. Substitute L = 120 (that is, 10 ft × 12 in per ft = 120 in).

S = 0.125 (120) – 0.25 S = 14.75

So, the spacing would be 14.75 inches.

4. If the spacing between centers was less than or equal to the hole diameter, the holes would be overlapping—probably not the desired result. To find out for what value of L this would occur, substitute S = 0.75 into the equation, and solve for L.

0.75 = 0.125 L – 0.25

Add 0.25 to both sides of the equation, and then divide both sides by 0.125.

0.75 + 0.25 = 0.125 L – 0.25 + 0.25 = L = 8

So, the length is 8 inches.

Exercise 15

1. Contamination = Initial contamination level × 0.60ⁿ



2. 8 day represents 4 periods of 20 days (80 ÷ 20 = 4)...

Contamination = Initial contamination level × 0.60ⁿ
Contamination = 64,000,000 units per m³ × 0.60⁴
Contamination = 8,294,400 units per m³ (rounded)

200 days represents 10 periods of 20 days...

Contamination = 64,000,000 units per m³ × 0.60¹⁰
Contamination = 386,984 units per m³ (rounded)

360 days represents 18 periods of 20 days...

Contamination = 64,000,000 units per m³ × 0.60¹⁸
Contamination = 6500 units per m³

3. You could draw a graph of the formula, and observe how many periods were required before the line dropped to a value of 40,000 units per m³, or you could continue a "trial and error" approach of trying different values for n.

Exercise 16

1. If each of the seven positions can be either a letter or a digit (except for the letter O), there is a total of 35 choices for each position (that is, 10 digits and 25 letters). Thus the number of combinations 35 × 35 × 35 × 35 × 35 × 35 × 35, or 35⁷, which is 6.434 × 10¹⁰, or about 64 trillion!



2. This question should be evaluated in two steps. First determine the number of combinations with three letters followed by three digits. Second, determine the number of combinations with three digits followed by three letters (the same as the first calculation), and add the two numbers to find the total.



Again, there are 25 possible letter choices, and 10 possible digits. Thus the number of possible combinations for the first arrangement (letters followed by digits) is

25 × 25 × 25	×	10 × 10 × 10
Letters		Digits

Or, 15,625,000 combinations. The second arrangement (digits followed by letters) is simply the reverse, and equal to the same number.

10 × 10 × 10	×	25 × 25 × 25
Digits		Letters





Thus, there is a total of 31,250,000 combinations (that is, 15,625,000 + 15,625,000 available for this government.

3. Since only the last three characters, which must be digits, are in question, there are 10 × 10 × 10, or 1000 combinations.



4. This part of the exercise requires some thought by the students. If the local community (from which you expect the suspect car to come0 represents about 2% of the government's population, the number of possible cars is reduced to 2% of 1000, or 20 cars. Furthermore, if only 40% of the government's cars are four-door vehicles, this number is further reducled to 40% of 20, or about 8 vehicles. Thus only about eight cars would need to be investigated, assuming that these eight license numbers were all active license numbers.

Exercise 17

1. The probability of selecting a flawed glove is simply equal to the percentage of flaws in the sample, 1.5% (or 0.015) assuming that the probability of selecting any glove is equally likely.

2. It doesn't matter how many gloves are in the box, the probability remains the same—1.5%.

3. The probability of an examination glove NOT being flawed is 0.975 (that is, 1 – 0.025). Since each glove quality is independent of the others, the probability of 100 gloves not being flawed is the product of their individual probabilities.

Prob {No flawed gloves out of 100 gloves} = Prob {1^st glove now flawed} × Prob {2^nd glove not flawed} × . . . × Prob {100^th glove not flawed}
Prob {No flawed gloves out of 100 gloves} = 0.975 × 0.975 × . . . × 0.975
Prob {No flawed gloves out of 100 gloves} = (0.975)100
Prob {No flawed gloves out of 100 gloves} = 0.080 or 8.0% (rounded)

4. The students should use the formula given in the exercise, substituting N = 100, and the probabilities for flawed and nonflawed examination gloves.

Prob {One flawed glove out of N gloves} = N × Prob {Flawed glove} × Prob {Glove not flawed}^{N – 1}
Prob {One flawed glove out of 100 gloves} = 100 × 0.0255 × (0.975)⁹⁹
Prob {One flawed glove out of 100 gloves} = (0.975)100
Prob {No flawed gloves out of 100 gloves} = 0.204 or 20.4% (rounded)

5. The probability of having no more than one flawed glove out of 100 is the sum of the probabilities determined in Parts b and c above.

Prob {NO more than one flawed glove out of 100 gloves} = Prob {No flawed gloves OR Just one flawed glove}
Prob {NO more than one flawed glove out of 100 gloves} = Prob {No flawed gloves} + Prob (Just one flawed glove}
Prob {NO more than one flawed glove out of 100 gloves} = 0.080 + 0.204
Prob {NO more than one flawed glove out of 100 gloves} = 0.284

Exercise 18

1. Since there are five different colors, and the bin has a uniform mixture of each color, each color must represent 20% (that is 1/5) of all the candies in the bin. Hence the probability that the first candy into the bag is red is 0.20, or 20%.



2. Since the probability that the candy IS RED and the probability that it is NOT RED must sume to 1. . .

Prob {Candy NOT red} = 1 – Prob {Candy IS red}
Prob {Candy NOT red} = 1 – 0.20
Prob {Candy NOT red} = 0.80

3. The second candy dropped into the bag is independent of the first (due to the assumption that the mixture in the very large bin remains uniform). Thus, the probability of not being red remains the same—0.80.



4. Again, since the first and second drops are independent, the probability of both events occurring is the product of their individual probabilities.

Prob {1^st not red AND 2^nd not red} = Prob {1^st not red} × Prob {2^nd not red}
Prob (1^st not red AND 2^nd not red} = 0.80 × 0.80
Prob {1^st red AND 2^nd not red} = 0.64, or 64%

5. If the student cannot perceive the extension of this question from Part d above to create a formula, a great deal of labor (or confusion) can be expected. The simplest way to answer this question is to recognize that every piece of candy into the bag has a probability of 0.80 of NOT being red. The probability that the first two pieces are not red is 0.80 × 0.80, or 0.80². Similarly, the probability that the first three pieces are not red is 0.80³. So, for the nth piece into the bag, the probability that none of them is red is 0.80ⁿ. When the 45^th piece of candy is in the bag, n = 45, and the probability that none of them is red is 0.80⁴⁵, or 0.0000436—extremely unlikely!