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.
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Student Learning Objectives
Students should be able to:
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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?)
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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.
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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.
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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:
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
3. Using the equation you determined for 1999 data, predict the population in 2000,
the current year, and 2010.
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. 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
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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.
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An equation to find the population t years after 1964 would be:
A(t) = (191,888,791) e (0.0139)t
3. Using the equation you determined for 1999 data, predict the population in 2000,
the current year, and 2010.
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. How close were your results? Why might they be different?
5. 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.
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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)
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Links to Course Competencies
This project is appropriate for an Intermediate Algebra course or a College Algebra
course.
Students should be able to:
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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
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Supplementary Resources
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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)
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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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