Scientific Notation III: Linear vs Exponential

Teacher Lesson Plan


After investigating examples of exponential growth (bacteria colonies, human populations ...) students will:

  1. Have an intuitive feel for the concept of exponential (as opposed to linear) growth of a system.
  2. Be able to interpret data plotted in logarithmic form, and to create logarithmic plots of gathered data.

  1. Basic math skills, including the use of decimal notation for fractions.
  2. Basic graphing skills. See for example the graphing exercises in the ASPIRE gas law lessons.
  3. Familiarity with bacteria and their growth in Petri dishes.
  4. 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.


  1. 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).
  2. Java-capable and enabled browser with the Java Run-Time Environment plug-in installed. For questions/assistance write .
  3. 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 lab data.


  1. 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" grows.
  2. 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).
  3. 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 (108) 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?
  4. 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?

Time unit
Natural Increase

Source: U.S. Census Bureau, Current Population Projections, and  
(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:

  1. 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?

  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?


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:

  1. 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?

  2. 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?

  3. 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 contributing factors?

Back to main Scientific Notation page

Last Updated: 15 August 2000 by MS