2.1: How Crowded Are We?

INTRODUCTION

In this collaboration, you will learn about a way of measuring how crowded the people feel in the place where they live. This measurement is called population density.

To help you prepare, think about visiting a shopping mall on a Tuesday at 11:00 AM in May and on a Saturday at 11:00 AM in December. Discuss in your group which day would be more crowded. Come up with some ways that you might measure how “crowded” a place is. What happens if the rooms or places you are considering are different sizes?

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

• population density is a ratio of the number of people per unit area.
• proportionality can be used to compare population densities.

By the end of this collaboration, you should be able to

• calculate population densities.
• determine if two population density ratios are proportional to each other.
• compare and contrast populations via their densities.

PROBLEM SITUATION 1: USING RATIOS TO MEASURE POPULATION DENSITY

In Collaboration 1.2, you learned Earth’s human population has grown from about 1 billion people to about 7.9 billion in the last two centuries. However, populations in different regions do not always grow uniformly. For example, populations tend to increase in areas where people already live close enough to one another to find mates. On the other hand, crowded populations decrease when deadly diseases, such as smallpox or Ebola virus, sweep through them. In this lesson, you will compare geographic regions by their population densities. The population density of a geographic region is a ratio of the number of people living in that region to the area of the region. Population density ratios are typically reported in a way that makes it easy to compare the ratio to a standard area measurement, such as one square mile or one square kilometer.

Example: Imagine 200 people standing in an open field that measures one mile by one mile (also known as one square mile). The population density of the open field could be thought of as 200 people per one square mile. This can be expressed as a ratio. The expression below shows the population density of the open field as a ratio in fraction form.

\[\dfrac{200\;people}{1\;square\;mile}\nonumber \]

(1) A proportion is the equality of two ratios. Quantities are proportional if they satisfy a proportion. Picture the open field from the example that has a population of 200 people per one square mile. Then, picture a second open field. The length of each side of the second open field is double the length of each side of the first open field (making it 2 miles by 2 miles). The amount of people standing in the second open field is double the amount of people standing in the first open field. Is the population density of the second field equal to the population density of the first field?

(2) An island in the archipelago nation known as the Maldives is threatened by global warming. The island, called the Ari Atoll, is approximately one square mile in size, and has a population of 1,260 people. What is the population density per square mile?

(3) Imagine that global warming caused flooding in Ari Atoll, so the island is now half of its former size. However, the same number of people still live in Ari Atoll.

(a) Before doing any calculations, decide if you think the population density of Ari Atoll increased, decreased, or stayed the same after the flooding. Explain your answer.

(b) Calculate the population density of Ari Atoll after the flood. Try this on your own first before sharing in your group.

(4) Suppose that since everyone is now living closer together, the birth rate on Ari Atoll increases rapidly. Soon the population has quadrupled. What is the population density now?

(5) Picture the open field from the example that contains 200 people and measures one square mile. Now picture another open field that measures five miles by five miles and has exactly the same population density as the first open field.

(a) How many people are standing in this field?

(b) Write a mathematical expression that shows that the population density of the first open field and the population density of the second open field are equal to each other.

(6) Consider a tiny island in the Atlantic Ocean. There are 50 survivors shipwrecked there. When they first arrived on the island, they all set up camp near the waterfall in the middle of the island. About six months later, a professor who was among the survivors claims the island can only support a population density of one person per acre (an acre is 43,560 square feet, which is a little less than the size of an American football field). If the survivors then decide to space their campsites out evenly across the island, how will this affect the overall population density of the island? Discuss with your group.

PROBLEM SITUATION 2: MAKING COMPARISONS WITH POPULATION DENSITY

How crowded is China compared to the United States?

(7) In 2022, in the United States, approximately 333,000,000 people occupied 3,794,000 square miles of land. In China, approximately 1,412,000,000 people lived on 3,705,000 square miles of land. Use this information to answer the following questions.

(a) A student carefully calculates the population densities of China and the United States. He decides that China is less dense than the United States. Using your estimation skills, decide if you think this student’s calculation is correct.

(b) At a lecture, you hear someone claim that, in terms of population, China is more than four times as dense as the United States. Using your estimation skills, decide if you think this statement is correct.

(c) Calculate more precisely the densities (per square mile) of the Chinese and U.S. populations. Based on your calculation, how many times more dense is the more crowded population? Be ready to share your calculations during the class discussion.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.