# Understanding Exponential Growth in the Context of Population Models

This material is replicated on a number of sites
as part of the
SERC Pedagogic Service Project

#### Summary

This set of three assignments gives students practice with exponential models in the context of the growing human population. These assignments are designed to:

- help students see that growth at a constant percentage rate implies repeated doublings over a specific time interval
- give students practice describing differences (in growth rates and in population sizes) in both absolute and relative terms
- introduce students to real population data, and
- get students to think about the limitations of the simple exponential model and consider the multiple factors that affect population growth.

- calculations
- creation of tables and graphs
- a short comparison and contrast exercise, and
- a critique of the model.

## Learning Goals

**Finding Information**

- Become acquainted with some key data sources on populations.

**Calculating and Modeling**

- Model exponential growth of a population using doubling times and the exponential growth formula.
- Solve for the intersection of two exponential curves.

**Presenting Data**

- Present these values clearly in a table and in a graph.

**Understanding through Writing**

- Understand that growth at a constant percentage rate implies repeated doublings over a specific time interval.
- Recognize that small differences in percentage growth rates can lead to large differences in population values over time.
- Practice with the language of measuring change and differences. In particular, describe differences in percentage rates appropriately (e.g., one scenario's growth rate is 0.35 percentage points larger than the other's); clearly describe changes in population size over time for a given growth rate; and clearly describe differences between the population sizes of two scenarios at a particular point in time.
- Recognize the simplifying assumptions of our model and its limitations.
- Think critically about factors that contribute to population change.
- Think about the ramifications of population growth for resources, infrastructure, etc.

## Context for Use

This set of assignments is designed for a first-year college course for students who need help enhancing their quantitative reasoning skills. Some knowledge of Excel (or another spreadsheet) is required. The assignment is given as a problem set.

## Description and Teaching Materials

The first assignment has students calculate the doubling times for two potential world population growth rates and use those doubling times (and a starting population value) to plot the world's population size over a 200-year period. They also are to use the exponential growth formula to calculate the projected population 200 years hence. Having made these calculations, students are to write a couple of paragraphs describing the projected populations under the two scenarios.

The second assignment has students examine the CIA's World Factbook for data about the population size and growth rates of the world and of each country. Students are to choose two countries: one with a smaller current population but a faster growth rate, and the other with a larger current population and a slower growth rate. Assuming exponential growth, students are to calculate when the first country will surpass the second country in population.

The third assignment contains more open-ended questions and asks students to consider what they have learned from reading UUM and the Web sites about the factors affecting growth rates. They are to explain the limitations of using an exponential growth model and to describe various factors that would be used in a more complex model of population growth. Handout of the problem set (Microsoft Word 26kB May11 08)

The second assignment has students examine the CIA's World Factbook for data about the population size and growth rates of the world and of each country. Students are to choose two countries: one with a smaller current population but a faster growth rate, and the other with a larger current population and a slower growth rate. Assuming exponential growth, students are to calculate when the first country will surpass the second country in population.

The third assignment contains more open-ended questions and asks students to consider what they have learned from reading UUM and the Web sites about the factors affecting growth rates. They are to explain the limitations of using an exponential growth model and to describe various factors that would be used in a more complex model of population growth. Handout of the problem set (Microsoft Word 26kB May11 08)

## Teaching Notes and Tips

Students often have trouble starting the table with the doubling times, so I introduce this exercise during our weekly computer lab. I give students feedback in lab, and once they understand how to set that table up, the rest seems to fall in place fairly easily. Students have a week to finish the problem set.

## Assessment

This Word file provides an answer to the first assignment and a sample answer to the second assignment. Answers to problem set (Microsoft Word 41kB Aug13 08)

## References and Resources

Bennett, Jeffrey O. and William L. Briggs. 2002. Using and Understanding Mathematics: A Quantitative Reasoning Approach. (2nd edition)

Conollay, Eric. Undated. Quantitative Reasoning. Chapter 3: More on Mathematical Modeling. Unpublished manuscript. Wellesley College.

Census Bureau's Population pages:

http://www.census.gov/population/projections/

CIA's World Factbook:

https://www.cia.gov/library/publications/the-world-factbook/index.html

Population (number persons):

https://www.cia.gov/library/publications/the-world-factbook/fields/2119.html

Estimated growth rates:

https://www.cia.gov/library/publications/the-world-factbook/fields/2002.html

Conollay, Eric. Undated. Quantitative Reasoning. Chapter 3: More on Mathematical Modeling. Unpublished manuscript. Wellesley College.

Census Bureau's Population pages:

http://www.census.gov/population/projections/

CIA's World Factbook:

https://www.cia.gov/library/publications/the-world-factbook/index.html

Population (number persons):

https://www.cia.gov/library/publications/the-world-factbook/fields/2119.html

Estimated growth rates:

https://www.cia.gov/library/publications/the-world-factbook/fields/2002.html