Exploration of Two Growth Models:
A Unit for the Curriculum Course EMAT 4500/6500
Connections in Secondary Mathematics

Heide G. Wiegel
Mathematics Education
The University of Georgia
Table of Contents

Introduction

As part of the course EMAT 4500/6500 Connections in Secondary Mathematics, groups of pre-service teachers investigate a unit from a contemporary, standards-based high school mathematics textbook. Two textbook series are available to the pre-service teachers: Interactive Mathematics Program [IMP] (Fendel, Resek, Alper & Fraser, 1997), published by Key Curriculum Press, and Contemporary Mathematics in Context [CorePlus], published by the EveryDay Learning Corporation. The requirements for the group projects are twofold: (a) pre-service teachers have to write individual project reports, and (b) each group has to give a presentation of their unit to the class.

Individual reports should include:

  • overview of the unit;
  • write-ups of a representative selection of worked problems in the unit;
  • a summary of mathematical and non-mathematical concepts introduced or reviewed in the unit;
  • a description and evaluation of the group's working arrangements;
  • an affective and conceptual evaluation of the unit (a) with the eyes of the (future) teacher, and (b) with the eyes of the (former) high school student;
  • two questions for the final exam (final exam is designed by the instructor on the basis of the group suggestions; groups are encouraged to share their exam suggestions);
  • material (e.g., overheads) of the group presentation.

The individual reports are evaluated by the instructor.

Group presentations should include:

  1. a brief overview of the textbook for the grade that contains the unit;
  2. a more extensive overview of the unit and the mathematical concepts in the unit;
  3. at least one activity that involves the class and highlights the essence of the unit;
  4. a homework assignment that reinforces concepts of the unit and of the class activity and is comparable to the final exam questions; the homework assignments are graded by the group.

The group presentations are evaluated by all class members, including the presenters and the instructor (see Appendices IA and IB). Groups have access to the teacher's edition of their unit after they have worked through the unit. During the semester, group presentations lasted usually one whole class session (once a week). A different course schedule will require a different organization of the presentations.

Reflecting on the course and on the group presentations and individual reports, I decided to present—in a revised course—one unit myself during the first part of the course. Thus, pre-service teachers will have a model they can use or modify for their own group presentations. I chose the unit Small World, Isn't it? from the third year of the IMP series. I plan to present the unit in one class session and to extend it in a second session.

Small World, Isn't It?

Each IMP unit starts with the unit question that usually is ambiguous and ill defined. The first task for high school students is to interpret the unit question, to define the ill–defined conditions, to think about the issues involved, and to come up with a preliminary list of information, skills, and mathematical tools they need to acquire in order to answer the unit question. As the students work through the unit they collect information and encounter and study the necessary mathematical concepts and skills. Not all classwork or homework investigations are thematically connected to the unit question. Throughout the unit, however, the class returns to the unit question and takes stock of the progress made. The unit concludes with the solution of the unit question and with the compilation of portfolios in which students collect and submit their best work.

The Unit Question

The central question of the Small-World unit is initially presented as

If population growth continues according to its current pattern, how long will it be until people are squashed up against each other? (Teacher's Guide, (see Fendel, Resek, Alper & Fraser, 1999, p. xxvii)).
A table with estimated U.S. population data from 1650 to 1995 and information about the total surface area and the percentage of land (see Overhead 2) are provided. Based on the initial student work and after a definition of squashed up against each other as one square foot per person, the unit question is formulated as
Based on the data in “Small World, Isn't it?” how long will it take until the population reaches 1.6 * 1015 people? (Teacher's Guide,(see Fendel, Resek, Alper & Fraser, 1999, p. 12)).

Outline of Unit (Teacher's Guide, p. xxix)
  • Days 1-2:  Introduction to the unit problem;
  • Days 3-5:  Working with average rates of change;
  • Days 6-10:  Developing the concept of slope and working with linear functions;
  • Days 11-16:  Developing the concept of a derivative from several perspectives;
  • Days 17-18:  Examining situations that are modeled by exponential functions;
  • Days 19-21:  Studying the derivatives of exponential functions, leading to the discovery of the “proportionality property”;
  • Days 22-26:  Working with bases and exponents to see that any base can be used to represent a given exponential function, then searching for the “best” base;
  • Days 27-29:  Developing the connection between compound interest and the special base e;
  • Days 30-32:  Solving the central unit problem, using curve-fitting techniques, and compiling unit portfolios;
  • Days 33-34:  Unit assessments and summing up.
Concepts and Skills (Teacher's Guide, pp. xxx-xxxi)

Rate of change

  • Evaluating average rate of change in terms of the coordinates of points on a graph
  • Understanding the relationship between the rate of change in terms of a function and the appearance of its graph
  • Realizing that, other things being equal, the rate of change in population is proportional to the population

Slope and linear functions

  • Developing an algebraic definition of slope
  • Proving, using similarity, that lines have a constant slope
  • Understanding the significance of a negative slope in terms of the appearance of a graph
  • Seeing that the slope of a line is equal to the coefficient of x in the algebraic representation of the line
  • Using slope to develop equations of straight lines

Derivatives

  • Developing the concept of the derivative of a function of a point
  • Seeing that the derivative of a function at a point is the slope of the tangent at that point
  • Finding numerical estimates for the derivatives of functions at specific points
  • Working with the derivative of a function as a function in itself
  • Realizing that for functions of the form y = bx, the derivative at each point of the graph is proportional to the y-value at that point

Exponential and logarithmic functions

  • Using exponential functions to model real-life situations
  • Strengthening understanding of logarithms
  • Reviewing and applying the principles that a b * a c = a b+c and (a b)c = a b*c
  • Understanding and using the fact that alog ab = b
  • Discovering that any exponential function can be expressed using any positive number other than 1 as a base
  • Learning the meaning of the terms natural logarithm and common logarithm
  • Using an exponential function to fit a curve to numerical data

The number e and compound interest

  • Estimating the value b for which the function y = bx has a derivative at each point on its graph equal to the y-value at that point
  • Developing and using a formula for compound interest
  • Seeing that expressions of the form (1 + 1/n)n have a limiting value, called e
  • Learning that the limiting value e is the same as the special base for exponential functions

Class Organization

The IMP material uses three primary working arrangements: Small–group work in class, homework, and whole–class discussions. The teacher is not lecturing unless there are technical terms to be introduced or reviewed. Most of the conceptual work occurs during the small–group sessions when students form conjectures, look for patterns, or prepare a class presentation. Homework assignments either follow up on group work and class discussions or serve as preparations for the next lesson. The homework assignments consist of a few conceptual problems rather than masses of drill–and–practice exercises.

Whole class discussions are a major part of the lessons. The discussions serve as forums in which students present results from homework and small-group work, clarify their understandings of the concepts involved, and formulate new questions. The teacher serves a moderator rather than as presenter.

Class Activities for Pre-Service Teachers

For my presentation of the unit to pre-service teachers, I am assuming an instructional format in which the course meets for about 3 hours one evening per week. Given the nature and time limits of the class presentation, some lecture (e.g., overview of the unit) will be necessary. However, any kind of lecture needs to be structured by short class activities in order to counteract the late time of the day and the length of the class session. I intend to include two mini-activities and two longer activities, all taken from the student book of the IMP unit.

   Mini-activity 1: Classwork 1 "Small World, Isn't It?" will be attached to the introduction of the unit question (10-15 minutes individual or small group work, according to individual students' choices; followed by a collection of ideas on the overhead; see Overheads 2 and 3).

   Mini-activity 2: Modification of Homework 16, "What's It All About?" (see attached copy of student page, Overhead 5); for the students in a high school class, this homework is a summary of what they have learned about derivatives. For the secondary pre-service teachers, the mini-activity is a review of college work that might have already retreated to the back of their minds. The mini-activity will serve to establish a common basis for the remainder of the presentation.

   Activity 3: Modification of Classwork "Slippery Slopes" (see Overhead 7). In this activity high school students explore the derivative of exponential functions and look for a pattern of how the derivative might relate to the x-value (independent variable), the y-value (dependent variable), or both. I chose this activity for the work with the pre-service teachers because it highlights the experimental approach taken in the IMP materials: students collect data and look for patterns in the data; abstractions are built on a firm ground of experiences. The approach differs fundamentally from the more theoretical approach usually taken in Calculus courses. High school students will most likely use the calculator to find the slope of the secant and the ratio of the slope to the value of the exponential function. Excel (see Appendices IIA, IIB, and IIC) provides an efficient tool for this exploration, and pre-service teachers can use either tool. (See student pages 1 and 2).

   Activity 4: In this activity, the pre-service teachers will use Excel and its many features to explore how the world population might grow in the next 100 or 200 years. I intend to use a second set of data from the UN 1998 world population report (http://www.undp.org/popin/) for comparison. Some differences exist between the two sets of data: (a) The UN data start with a population estimate in Year 0 whereas the IMP data start at 1650. (b) The two sets have different estimates for the year 1750. The UN data show 790 million people, the IMP data show only 694 million. (c) The data for 1990 are closer, but still different (UN—5.27 billion, IMP—5.29 billion). Based on these differences, we should expect different estimates for the time it takes to fill up the earth to standing room only. (See student page 3).

Homework

Problem of the Week [POW] 8: The More, the Merrier? (Essay, see attached copy of student page).

Outline of Session 1
  • Introduction of the unit question and the properties common to IMP unit questions (Teacher presentation [TP]);
  • Mini-Activity 1 (small groups or individual students; whole-class summary);
  • Overview of unit (TP);
  • Mini-Activity 2 (whole class);
  • Overview of Days 17-23: A Model for Growth (TP);
  • Activity 3 (small groups; whole-class summary);
  • Overview of Days 24-29: The Best Base (TP);
  • Activity 4 (computer work; individual or in pairs).

Appendices VIA, VIB, VIC and VII contain some of my own explorations with the data and different approximating functions. Tables and graphs in Appendices VIA, VIB, and VIC are based on UN data; tables and graphs in Appendix VII are based on IMP data.

The Extension: The Logistical Growth Model

IMP's definition of being squashed up against each other as one square foot per person and the assignment The More, the Merrier? invite two questions: "How many are too many?" and "Can population growth go on for ever?" In contrast to the exponential growth model that allows unrestricted growth, the logistic growth model introduces a limiting value K, the carrying capacity. The logistic growth model is also called the Verhulst model, named after the Belgian mathematician P. F. Verhulst who developed and studied the model in the 19th century. Based on the US census data from 1790 to 1840, Verhulst used the logistic model to predict the US population in 1940. His estimate was off less than 1% . For census results after 1940, however, the logistic model does not fit the actual US population growth (see Appendix III and http://www.math.duke.edu/education/ccp/materials/diffeq/logistic/contents.html).

The Logistical Growth Function

The growth rate in the exponential model is proportional to the population, that is, f'(t) = r f(t), or dP/dt = rP. Because the IMP deals with world population data, the proportionality constant r can be interpreted as the birth rate minus the mortality rate, that is, we do not have to consider other variables such as immigration or displacement. In the logistic growth model, the proportionality constant r is replaced by a more complex growth rate:


The function P(t) is given by

It depends on the group of pre-service teachers whether or not the development of the function P(t) is appropriate for the course. However, developing, understanding, and presenting the function might be a worthwhile task for graduate students.

Development of Closed Form

The rate of change is given as


Step 1. Separate variables:


Step 2. Change so it can be integrated:


Step 3. Integrate:


Step 4. Exponentiation gives


Step 5. For t = 0, we get
Therefore,


Step 6. Invert the previous equation:


Step 7. Invert again


Step 8. Finally, we get


The Simulation

Marilyn Stor and William L. Briggs (Stor & Briggs, 1998) introduce the logistic growth model in the context of an investigation of communicable diseases. They present a game-like simulation, a computer simulation, and the development of a recursive rule. Because EMAT 4500/6500 starts with a set of problems that lend themselves to recursive–as well as closed–formula representations, an introduction of the logistic growth model through a recursive approach is appropriate. The simulation game and its analysis will be at the heart of the second session. During Fall 1998, Cathryn Brooks, a master student in Mathematics Education, translated the provided program into PASCAL (see Appendix IV) and developed the computer simulation. Thus, pre-service teachers can experiment with different infection rates. They can also apply the recursive rule in Stor and Briggs (1998) and use Excel for experimentation. The recursive rule is given as

In+1 = In + Nn = In + aIn(T – In)

In and In+1 represent the number of infected persons at stages n and n+1, respectively, Nn is the number of people newly infected at stage n, and a is the proportionality constant. T represents the total population and corresponds to the carrying capacity K in the logistic population function. The recursive rule assumes no births, deaths, or immigration. However, students should discuss how these population changes and other factors such as education, immunization, and recovery might change the infection rate. After the game and its analysis, students will be given local flu data (from Athens Regional Medical Center, not yet available) and several sets of US flu data (see Appendix V and http://www.cdc.gov/ncidod/diseases/flu/fluvirus.htm). Data will be modeled in Excel, and pre-service teachers will experiment with different values of the proportionality constant a in order to find the recursive rule for the flu data. The TI-83 calculator offers logistic regression.

Homework

http://www.imdb.com/Charts/usboxarchive lists earnings of current movies by week. Students will select a movie and investigate the cumulative revenue function of the movie.

Outline of Session 2

  • Presentation of results of Activity 4 and discussion of homework (whole class); transition;
  • The simulation game
    • Introduction
    • Preparation
    • Game
    • Collection of data
    • Analysis: How many are infected?
    • Graph
  • The logistic growth model (TP and class discussion; history, formulas as appropriate);
  • Computer/calculator investigation of flu data;
  • Extension: Investigation of the fit of the logistic growth function to a set of world population data projected to 2150 (logistic regression with the TI-83 or Excel experimentation with different values of r).

Appendices VII and VIII show some of my own explorations with the game and flu data.

Sources
  • Fendel, D. Resek, D. Alper, L. & Fraser, S. (1997). Interactive mathematics program (Year 3, pp. 282–387). Berkeley: Key Curriculum Press.
     
  • Fendel, D. Resek, D. Alper, L. & Fraser, S. (1999). Small world isn't it?, Teacher's guide. Berkeley: Key Curriculum Press.