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## Calculating Exponential Growth

• 6-8 9-12
• Length: 60 Minutes

### Overview

Students read an essay, "It's All in the Numbers," about the rapid spread of HIV particles in the body, and learn how to calculate exponential growth using pennies to model HIV particles.

The essay portion of the activity contains stark facts that may be difficult to absorb. Depending upon students’ grade and maturity levels, the essay may be used as teacher background information instead of student reading material. The activity is most appropriate for use with students in grades 6-12.

This activity is from The Science of HIV/AIDS Teacher's Guide. The guide also is available in print format.

This work was developed in partnership with the Baylor-UT Houston Center for AIDS Research, an NIH-funded program.

### Teacher Background

Under favorable conditions and with sufficient time and resources, populations of all organisms, including infectious agents like viruses, have the potential to increase dramatically over time. Even slow-growing organisms can reach astounding population sizes if reproduction is unchecked. Charles Darwin used elephants, which breed very slowly, as a hypothetical example. Beginning with two elephants, which generally produce only six offspring during a reproductive span of 60 years, an elephant population would number only 54 individuals after 200 years. However, after 1,000 years, the population would have grown to 86,000,000 elephants!

Now consider another example, in which a parent cell divides into two daughter cells every 10 minutes. After 10 minutes, there would be two cells; after 20 minutes, four cells; after 30 minutes, eight cells, and so on. After three hours, there would be close to one million cells. When quantity increases by a fixed percentage at regular time intervals, we have what is referred to as exponential growth. On a graph, exponential growth is represented by an upward curve, not a straight line. In addition to the example of cell division, exponential growth can be observed in the accumulation of compound interest, and in the increasing levels of CO2 in the atmosphere. Untreated, HIV also is capable of exponential growth once it begins to replicate and spread within the human body.

Depending upon students' grade and maturity levels, the essay, "It's All In the Numbers," may be used as teacher background information or as a student reading assignment. It is especially effective when read aloud.

• National Institute of Allergy and Infectious Diseases, National Institutes of Health (NIH), offers resources on understanding HIV/AIDS: niaid.nih.gov/topics/hivaids/ and aidsinfo.nih.gov.

• National Institute on Drug Abuse, NIH, offers facts about drug abuse and the link between it and HIV/AIDS: hiv.drugabuse.gov.

• The Centers for Disease Control and Prevention provides up-to-date information on HIV/AIDS prevention: cdc.gov/hiv/topics.

### Objectives and Standards

#### Life Science

• Disease is a breakdown in structures or functions of an organism. Some diseases are the result of damage by infection by other organisms.

• Reproduction is a characteristic of all living systems.

• Cells use and store information to guide their functions. The genetic information stored in DNA is used to direct the synthesis of the thousands of proteins that each cell requires.

• Changes in DNA (mutations) occur spontaneously at low rates. Some of these mutations make no difference to the organism, whereas others can change cells and organisms.

• Living organisms have the capacity to produce populations of infinite size, but environments and resources are finite.

#### Science in Personal and Social Perspectives

• The severity of disease symptoms is dependent on many factors, such as human resistance and the virulence of the disease-producing organism.

• Populations can increase through linear or exponential growth.

### Materials and Setup

#### Teacher Materials (see Setup)

• LCD or document projector, “smartboard” or overhead projector Slides or transparencies of student sheet

#### Materials per Student Group

• Calculator or computer access

• Spreadsheet software, if using a computer

• Copies of "Dollars or Cents” student sheet (one per student; see Lesson pdf)

• Copy of essay (if age appropriate; see Lesson pdf)

#### Setup

1. If not using a document projector, prepare a slide or transparency of the spreadsheets. Also prepare slides or transparencies of the salary graph to show the difference between linear and exponential growth.

2. Have students conduct this activity in groups of 2– 4.

### Procedure and Extensions

1. Depending upon students' grade and maturity levels, have students read the essay, "It's All In the Numbers." Then, lead a class discussion about the meaning of exponential growth, as it relates to HIV. Due to exponential growth, the greater the number of HIV particles present, the faster they will increase in number. Use the following example.

If an HIV particle reproduces itself every minute, at the end of one minute, there will be two particles. After two minutes, there will be four particles; and after 10 minutes, the number will have grown to 1,024. In 20 minutes, there will be more than one million particles, and after 30 minutes, the population will have increased to more than one billion. This is “exponential” growth.

2. Tell students that there are many examples of exponential growth. Pose the following scenario to the class.

Imagine you have applied for a job. Your future employer offers a temporary position lasting just 30 days. Then, something amazing happens: you’re asked to decide if you’d rather be paid in dollars or pennies.

If you choose to be paid in dollars, you will earn \$1,000 on the first day of work, \$2,000 on your second day, \$3,000 on the third, and so on. For each of your 30 days of employment, your salary will be increased by \$1,000.

If you choose to be paid in pennies, you will earn one cent on the first day of work, two cents on your second day, four cents on the third day, and so on. Each day, your will salary will be exactly double the salary you earned the day before. Which payment plan will you select?

3. Give each student group the “Dollars or Cents” page, which includes the challenge just described. Allow time for students to discuss the options and select one of the job’s two possible “pay schedules.” Have students calculate their daily salaries, total income earned so far at the end of each day, and the amount of money they will earn for the full 30-day period.

4. Compare the final balances accrued by each salary schedule. If required for clarification, share the following information with students (also see the answer sheet at the end of this activity).

Being paid in dollars certainly seems like the smart choice. In just five days, you will earn \$15,000. By the end of the next five days, your salary will reach \$55,000. Adding \$1,000 to your salary each day quickly builds up to a 30-day grand total of \$465,000! Not bad for a temporary job.

On the other hand, it takes a lot of discipline (and quick calculations!) to choose to be paid in pennies. Initially, the pay will be dismal. By day 10, you will have only earned a total of only \$10.23. It takes three weeks before your salary begins to pick up. On day 20, you will have earned \$10.485.75. And from that point on, salary growth becomes spectacular. Just five days later, your salary will pass \$335,000. By day 30, you will have earned \$10,737,417.61!

5. Revisit your previous discussion of HIV replication. Ask students to explain how the salary analogy applies to virus multiplication within cells in the body. Or, ask each group of students to summarize what they learned about exponential growth by writing a paragraph in their science notebooks or as a homework assignment.

#### Extension

Ask students, What would happen to the two salaries if the employer retained the employee for one extra day? [The “linear pay” employee’s total salary would increase to \$930,000, while the “exponential pay” employee’s salary would jump to a total of \$21,474,847.22. In six additional days, the exponential salary would climb to more than \$1 billion.]

• ### HIV/AIDS

Teacher Guide

Students read essays, conduct activities, and use actual data from the CDC and other sources to learn about HIV/AIDS and the spread of disease. (5 activities, 5 essays)

• ### X-Times: Career Options

Student magazine: Special issue featuring healthcare professionals who discuss why each chose his or her career, educational requirements needed to obtain the job, and day-to-day responsibilities.