Population Explosion Using an Exponential Function

Project Overview

This primary source RWLO is intended for use in an Intermediate Algebra course or a College Algebra course after students have been introduced to function notation and exponential equations. Instructors may use this RWLO as a project to be done during class or it may be used as a supplement to the course to be done by students outside of class.

The Population Explosion RWLO will require students to access the U.S. Census Bureau website to find estimates for population and population growth in 1999. Using function notation and the U.S. Census estimates, students will develop a constant growth rate model and a continuous growth rate model to calculate U.S. populations in years after 1999. Students will then compare their estimates with the estimates at the U.S. Population Clock website. These models present students with a real world use for function notation and exponential equations.

The documents for this RWLO can be found by navigating the folders in the frame to the left.

Student Learning Objectives

Students should be able to:

·  Interpret data from the U.S. Census Bureau

·  Utilize function notation

·  Solve an exponential function when given a value for the independent variable

·  Calculate populations for several years using a constant growth rate model

·  Calculate populations for several years using a continuous growth rate model

·  Analyze their calculations and compare them to that of the U.S. Population Clock (are the student results reasonable?)

Procedure

Time: Approximately 30 minutes.

Materials: Pencil, paper, scientific calculator, and computers with internet access.

Implementation: This RWLO is recommended as a project that is done outside of class as homework, with a suggested in-class introduction to the project. However, this project could be done entirely as an in-class project. Students may discuss their procedures with other students, but it is recommended that each student goes through each step and does his/her own calculations.

Steps:

  1. Review functions, function notation, and exponential equations. Also, show students how to use the e button on their calculators (if necessary).
  2. Print off copies of the Constant Growth Rate Model and Continuous Change Model worksheets (in the Content Materials folder). Students will need to print off a copy of each of these worksheets for themselves or you may provide copies for them.
  3. Instruct students to read through the directions and examples provided in the Course Materials Folder and print off the Constant Growth Rate Model and Continuous Change Model worksheets.
  4. On the Constant Growth Rate Model and Continuous Change Model worksheets, students should fill in the data, show their calculations, and answer the questions.
  5. One week after assigning the project, collect the Constant Growth Rate Model and Continuous Change Model worksheets. A brief discussion on students’ results may follow.

Content Material

In this project, you will be examining population growth using two different models: Constant Growth Rate and Continuous Change. Print off a copy of the Constant Growth Rate Model worksheet (in the Content Materials folder) and fill it in while you go through Part I: Constant Growth Rate Model. After completing Part I, print off a copy of the Continuous Change Model worksheet (in the Content Materials folder) and fill it in while you go through Part II: Continuous Change Model.

Part I: Constant Growth Rate Model

When looking at population growth over a short period of time, it may appear to follow a linear function.

While a linear function can be used to model population growth that has a constant increase or decrease in the number of people, an exponential function can be used to model population growth that has a constant percentage change in population.

Since the average annual percent change in a population (growth rate) is often relatively constant during a short period of time, it is not uncommon to fit an exponential model to population data.

Exponential Function
Constant Growth Rate Model
f(t) = abt
f(t) = population after t years
a = initial value
b = base or growth factor
t = time in years

This exponential model can be used to predict population during a period when the population growth rate remains constant.

  1. From the U.S. Census Bureau's Historical National Population Estimates, 1900 to 1999, record the estimated national population in 1999 and the estimated average annual percent change (growth rate given in percent) for that year.
  1. Assuming that the growth rate remains constant, give an equation to express the population as a function of t, the number of years after 1999. Remember:

f(t) = Population t years after 1999

a = Population in 1999

b = Growth factor = 1 + growth rate in decimal form

t = number of years after 1999

Example: In July 1964 the population was 191,888,791 and the growth rate was +1.39 % or 0.0139.
The growth factor = 1 + 0.0139 = 1.0139
This means that the population in 1964 was approximately 1.0139 times the population in 1963.
An equation to find the population t years after 1964 would be:
f(t) = (191,888,791)(1.0139)t
  1. Using the equation you determined for 1999 data, predict the population in 2000, the current year, and 2010.
  2. Compare your results to the estimated values given in the International Database (IDB) Summary Demographic Data for the U.S. as well as the U.S. Population Clock. How close were your results? Why might they be different?

Part II: Continuous Change Model

There are other functions that can be used to model exponential growth. If a population grows (or decays) continuously over a period of time such that the rate of change of population is proportional to the total population, then a continuous change model for exponential growth can be used. This model is similar to the model used to determine how much a savings account grows when the interest is compounded continuously.

Exponential Function
Continuous Change Model
A(t) = Pert
A(t) = amount of population after t years
P = initial Population
e = exponential constant
r = annual growth rate
t = time in years

(Click here for more information about e.)

This exponential model can be used to predict population during a period when the growth of a population is continuous.The constant growth rate model does not assume continuous growth.

  1. From the U.S. Census Bureau's Historical National Population Estimates, 1900 to 1999, record the national population for 1999 and the average annual percent change (growth rate given in percent) for that year.
  2. Assuming that the growth rate remains constant, give an equation to express the population as a function of t, the number of years after 1999. Remember:

A(t) = Population t years after 1999

P = Population in 1999

e = 2.718....

r = Annual growth rate in decimal form

t = number of years after 1999

Example: In July 1964 the population was 191,888,791 and the annual growth rate was +1.39 % or 0.0139.
An equation to find the population t years after 1964 would be:
A(t) = (191,888,791) e (0.0139)t
  1. Using the equation you determined for 1999 data, predict the population in 2000, the current year, and 2010.
  2. Compare your results to the estimated values given in the International Database (IDB) Summary Demographic Data for the U.S. as well as the U.S. Population Clock. How close were your results? Why might they be different?
  3. Do you think this continuous change model is any better than the constant growth model for predicting population in the long term? Support your answer by showing what today's predicted population would be if the continuous change model was used for a year in the past, assuming that the growth rate remained at the same value as in that year.

Assessment

The following is a recommended form of assessment:

Allow students one week to complete this project and then collect the Constant Growth Rate Model and Continuous Change Model worksheets. Require students to show their calculations. They may attach a sheet of paper showing their work. Suggested point values are listed in red (10 points per worksheet). If a minor mistake is made in a calculation, 1 out of 2 points may be awarded. Other partial credit may be given at the instructor’s discretion.

Constant Growth Rate Model/Continuous Change Model

Name______

1. Data from Historical National Population Estimates, 1900 to 1999.

Estimated national population in 1999: ______

Estimated average annual percentage change: ______

2. Write your function: ______(2 points)

3. Using your function from step 2, predict the population in 2000, the current year, and 2010. Fill in Your Results in the table below.

2000 / Current Year / 2010
Your Results / (2 points) / (2 points) / (2 points)
International Database
U.S. Population Clock

4. Compare your results to the estimated values given in the International Database (IDB) Summary Demographic Data for the U.S. as well as the U.S. Population Clock. Fill in the rest of the above table with these results.

How close were your results? Why might they be different (explain below)?

(2 points)

Links to Course Competencies

This project is appropriate for an Intermediate Algebra course or a College Algebra course.

Students should be able to:

·  Demonstrate an understanding of and simplify exponential expressions

·  Work on the internet

·  Work collaboratively (if the instructor allows group work)

·  Evaluate functions

·  Use a scientific calculator

·  Assess the reasonableness of their calculations

Supplementary Resources

·  Historical National Population Estimates, 1900 to 1999

·  International Database (IDB) Summary Demographic Data

·  U.S. Population Clock

·  U.S. Census Bureau

·  Ask Dr. Math

·  Info about e (courtesy of Ask Dr. Math)

·  World Population Clock (check it out, just for fun)

Recommendations

Recommendations for Integration:

This RWLO is intended to be used as a supplement to a unit with functions and exponential equations in an Intermediate Algebra or College Algebra course. Instructors may choose to allow students to work in class on this RWLO or may prefer to assign it as a project to be worked on outside of class. While an instructor may choose to have students work individually, working in pairs or small groups may encourage discussion on why students’ calculations differ from those of the U.S. Population Clock before the entire class discusses this topic. Therefore, allowing students to work in groups may be more productive.

Back-up:

Print off copies of the Constant Growth Rate Model and Continuous Change Model worksheets (in the Content Materials folder) so that you may distribute copies to your students if this website is inaccessible. Also, print off copies of all documents and referenced websites in the Content Materials section. If necessary, skip the Population Clock portion of the activity.

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