After investigating examples of exponential growth (bacteria
colonies, human populations ...) students will:
- Have an intuitive feel for the concept of exponential
(as opposed to linear) growth of a system.
- Be able to interpret data plotted in logarithmic form, and
to create logarithmic plots of gathered data.
- Basic math skills, including the use of decimal notation for
- Basic graphing skills. See for example the graphing
exercises in the ASPIRE gas law lessons.
- Familiarity with bacteria and their growth in Petri dishes.
- Familiarity with large numbers, for example as presented in
the ASPIRE "Big Numbers" online lab. Familiarity with
scientific notation, for example as presented in the ASPIRE
"Manipulating Scientific Notation" online lab.
INVITATION TO LEARN
- Computer Lab with Internet connection or a single machine and
suitable projection equipment. We recommend a Pentium class
or Mac OS 8.6 G3 (or later) computer (166 MHz, with at least
32 MB RAM).
- Java-capable and enabled browser with the Java Run-Time
Environment plug-in installed. For questions/assistance write
- Student Lab Packet - This is a printable version
of the lab materials (instructions, frequent questions, graph
formats, and questions/problems) where students can record their
- Teachers may introduce this lab by asking students to
consider the issue of human population growth, which
is an exponential rather than linear phenomenon. Ask
the students how they would explain the increase in
size of the human population. With guidance, they should
arrive at the realization that populations increase not
at a constant rate (e.g. 5 humans/second) but rather
at an INCREASING rate as the size of the "parent pool"
- Setup a metronome clicking 4 times per second to illustrate the
birth rate of people in the world. See the table below.
Can the students guess on their own what the rate of the
metronome represents? For scale, you might set the metronome
ticking at other familiar rates such as the human heart beat
rate (1 per second) or the rate at which people blink (1 blink
per five seconds).
- The growth of bacterial colonies can be effective as a model
of human population growth. A petri dish with suitable nutrients
(e.g. agar) can be "seeded" by something as simple as
having the students cough or sneeze on the dish. Keeping in mind
that - once a bacteria colony becomes visible to the human eye -
the densities of bacteria are of the order of 100 million
per square centimeter, have the students measure and plot the
total number of bacteria per unit time. Do they see an exponentially
increasing rate like they predicted for human population growth?
What eventually limits the growth they observe?
- Reinforce the idea of exponential growth using the
"Doubling Penny" exercise. Suppose that they're doing a
job and they can choose how they'd like to be paid: They
can either be paid $100 per day, or else be paid 1 penny
the first day, two pennies the second day, 4 pennies the
third day, et cetera. Which would they choose? Have
them calculate the total number of pennies received over
time? How many days would pass before they exceeded the
initial $100/day? How many days before they were richer
than Bill Gates? How many days before they were richer than
the entire United States?
Census Bureau, Current Population Projections, and www.npg.org.
(Figures may not add to totals due to rounding)
PRE-ASSESSMENT: Here are several questions of the type you might
use to gauge your student's understanding of the exponential function
and scientific notation:
- If population A increases from 1,000 to 100,000 in one year
and population B increases from 100,000 to 1,000,000 in the
same year, which population showed the greatest increase? Which
population showed the greatest RELATIVE increase, i.e.
relative to its original size? If growth continues in the same
fashion, can you predict what will the populations look like
after year 2?
- Have students plot exponential growth rates (for example the
pennies per day received in the "Doubling Penny" exercise
above) on both "linear" and "logarithmic" graphing
paper. Can they read, interpret, and make accurate graphs
on each type of graph paper? Can they select the appropriate
type of graph paper for different graphical applications?
DIRECTIONS FOR TEACHING THE LAB
This symbol appear when lab partners are expected to discuss
ideas with each other.
Invite students to proceed to the beginning of the Student Lab. They
should first read the introductory statements and other information that
pertains to the lab. As students scroll down the page, they will see the
lab setup. In this lab, the students will record and eventually plot
(with the aid of a log-plot graphing applet) the growth of bacterial
colonies on a virtual "petri dish" by recording the total population
at different times.
Closure and Assessment
This lesson should provide students with an introduction to and
experience in the use of scientific notation to represent the
exponential growth of a system. Collection, recording, and graphically
representing data are emphasized.
Students should be encouraged to make connections from this lesson
to population growth issues impacting their own lives:
- The population curves all plateau at a certain point, regardless
of the starting population or growth parameter. What causes this
plateau? Are there similar forces at work in human populations?
- If the population of a city grows from 10,000 to 20,000 in a
ten-year period, what would be the predicted population after
another ten years? After twenty years? How might this impact
on the actions of city planners and politicians?
- What factors might affect the "growth parameter" for a bacterial
colony? For a human population?
POST ASSESSMENT: Teachers may return to the pre-assessment
questions and use these same questions or compose their own post-assessment
instrument. Hopefully, teachers can also include more difficult, higher-level
questions in their post-assessment.
EXTENSION ACTIVITIES: In addition to the problems and questions given
at the end of the lab activity, students can be encouraged to consider
the extension of the exponential population growth idea to the human
population on the petri dish called "Earth". When do we expect the
human population curve to plateau? What do we expect are the major