**Lesson 2.4**

**Introduction**

**Course Objectives**

This lesson will address the following course outcomes:

· 9. Compare proportional relationships represented in different ways, considering units when doing so.

**Specific Objectives**

Students will understand that

· population density is a ratio of the number of people per unit area.

· population density may be described proportionately to compare populations.

Students will be able to

· calculate population densities.

· calculate population density proportions from density ratios.

· compare and contrast populations via their densities.

**Problem Situation 1: Using Ratios to Measure Population Density**

Earth’s human population has grown from about 1 billion people to nearly 7 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.

**Definition:** 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 “reduced” by division in order to compare them with a standard area measurement.

**Art Stage**

**Example: Performance Art Stage**

Suppose an artist created a performance art event. On one stage that measures 20 feet by 20 feet there are 100 people spaced so that each person stands on a 2 foot by 2 foot square.

The population density of the stage could be expressed as 100 people per 400 square feet, 100 people400 square feet100 people400 square feet .

Or this could be rewritten with a 1 in the numerator to indicate how much space each person has. To do this, divide the numerator and the denominator values by 100, getting 1 person per 4 square feet, or it could be expressed as fractions: 100 people400 square feet=1 person4 square feet100 people400 square feet=1 person4 square feet .

Population density is often expressed as the number of people per square unit. To rewrite the ratio this way, express the ¼ as the decimal 0.25: 1 person4 square feet=0.25 person1 square foot1 person4 square feet=0.25 person1 square foot .

#1 Points possible: 5. Total attempts: 5

In another part of the performance art event, on a nearby patio of size 10 feet by 10 feet there are 25 people, each standing on his or her own 2 foot by 2 foot square. How does this population density compare with that of the stage described above?

· It would be larger

· It would be smaller

· It would be the same

#2 Points possible: 5. Total attempts: 5

Remember the 100 people evenly spaced on the 20 foot by 20 foot stage. Next they move around to form conversation subgroups by clustering close together. How does this affect the population density of the stage?

· It would be larger

· It would be smaller

· It would be the same

**A Classroom**

Imagine a classroom of students.

#3 Points possible: 6. Total attempts: 5

In the picture above, there are 30 students standing in a 18 foot by 20 foot rectangle. Calculate the population density (rounded to 2 decimal places), selecting the appropriate units.

Suppose we wanted to compare the densities of several populations, knowing the number of people in each population and where they live. Using the ideas from the last few problems, we could do this by dividing the number of people by the area in which they live (in square units: square miles, square kilometers, square feet, etc.).

Consider the room to be a “county” containing a city, a suburb, and rural area. In the picture below an area has been designed a “city”, another area has been designated a “suburb”, and the remaining space is the “rural” area. The students have moved so that the city is fairly crowded, the suburb is less crowded, and the rural area has only a few people.

#4 Points possible: 12. Total attempts: 5

In the picture above, 16 students have crowded into the 8ft by 8ft region designated the “city”, while 12 students are in the 12ft by 10ft region designed the “suburb”. Calculate the population density of each region (rounded to 2 decimal places), selecting the appropriate units.

Population density of the “city” region:

Population density of the “suburb” region:

#5 Points possible: 5. Total attempts: 5

Compare the population density of the city and the suburb regions.

The population density of the city is times the population density of the suburb.

#6 Points possible: 5. Total attempts: 5

Did the population density of the “county” (the classroom as a whole) change when people moved to the city and suburb (as compared to when they were scattered throughout the room)?

· No, the population density of the country didn’t change

· Yes, the population density is smaller after the move

· Yes, the population density is larger after the move

Notice a few important things:

· In the last set of problems, the density of the whole classroom was smaller than the density of the “city” region, even though there were more people in the whole classroom. The point is that the density depends on both the number of people *and* the area, and that more total people doesn’t necessarily mean a higher density.

· A ratio such as population density is useful because it helps you compare groups and areas of different sizes. You could compare the density of your classroom to the density of a group in the gymnasium because you are taking the ratio down to a unit rate (number of people per 1 square foot). This standardizes the measurement. This is a common use of ratios.

· A limitation of population density is that it measures an “average” as if the population were spaced out evenly. In a country or state, the population density would be greater in the cities, and smaller in the rural areas. Looking at the average population density for a country ignores this.

#7 Points possible: 6. Total attempts: 5

Imagine a very crowded large city, with each person standing on his or her own 2-foot-by-2-foot square, where the squares are adjacent. Calculate the population density per square mile. (1 mile = 5,280 feet)

Try the problem on your own first. If you are having trouble after 2 tries, we will break it down.

Population density: people per square mile

#8 Points possible: 5. Total attempts: 5

Imagine an average city like Lakewood or Puyallup, where each person could stand on his or her own 100-foot-by-100-foot square. Calculate the population density per square mile, to the nearest whole person.

people per square mile

**China**

**Problem Situation 2: Making Comparisons with Population Density**

How crowded is China, compared to the United States?

#9 Points possible: 5. Total attempts: 5

In 2010, in the United States, approximately 309,975,000 people occupied 3,717,000 square miles of land. In China, approximately 1,339,190,000 people lived on 3,705,000 square miles of land. Use this information to answer the following questions.

Using your estimation skills, is the density of China more or less dense than the United States?

· China is more dense than the United States

· China is about the same density as the United States

· China is less dense than the United States

#10 Points possible: 5. Total attempts: 5

Using estimation and the numbers from the last problem, determine the best *whole number* that completes the sentence below.

China’s population density is approximately times as dense as that of the United States

#11 Points possible: 15. Total attempts: 5

Calculate more precisely, to one decimal place, the densities per square mile of the Chinese and U.S. populations.

Density of U.S.: people per square mile

Density of China: people per square mile

Based on those calculations, complete the sentence below with a value accurate to one decimal place.

China’s population density is times as dense as that of the United States

**HW 2.4**

#1 Points possible: 5. Total attempts: 5

Which of the following was one of the main *mathematical* ideas of the lesson? The questions refer to the following quantities:

50 people/mi2 | 20 ft/5 sec | 34% |

· All three figures are population densities.

· Only the first two figures are ratios.

· All three of the figures are ratios.

· The second figure could be written as 4 ft/sec.

#2 Points possible: 15. Total attempts: 5

Use the picture below to answer the questions.

a. What is the density of the stars in Rectangle B? Round to the nearest hundredth of a star per square foot. stars/ft2

b. What is the density of the stars in Rectangle A (Note: The stars in Rectangle B are also in Rectangle A). Round to the nearest hundredth of a star per square foot. stars/ft2

c. Suppose three new stars were added in the gray part of Rectangle A. Which of the following statements would be correct?

· The density of Rectangle A would increase. The density of Rectangle B would increase.

· The density of Rectangle A would decrease. The density of Rectangle B would decrease.

· The density of Rectangle A would decrease. The density of Rectangle B would stay the same.

· The density of Rectangle A would increase. The density of Rectangle B would stay the same.

· The density of Rectangle A would stay the same. The density of Rectangle B would increase.

#3 Points possible: 10. Total attempts: 5

Wikipedia states the following about Anchorage, Alaska: “The city constitutes more than 40 percent of the state’s total population.1

a. Calculate the population density for Anchorage, based on a 2010 population of 291,826 people living on 1,961.1 square miles. Round to the nearest person per square mile. people/mi2

b. Wikipedia also says that the small Alaskan town of Ketchikan has the densest population in Alaska. Ketchikan had a population of 7,368 in 2010 and an area of 4.1 square miles. Calculate the population density of Ketchikan. Round to the nearest person per square mile. people/mi2

#4 Points possible: 5. Total attempts: 5

The following information comes from the lesson:

Population | Land Area (sq. miles) | |

United States | 309,975,000 | 3,717,000 |

China | 1,339,190,000 | 3,705,000 |

In India, about 1,184,639,000 people live on 1,269,000 square miles of land2. Which of the following statements is false?

· The population density of India is approximately three times that of China.

· The population density of India is approximately 11 times the population density of the United States.

· The population densities of these three countries ranked from smallest to largest are United States, India, and China.

· The population density of the United States is approximately 83.4 people per square mile. That is smaller than the population densities for China and India.

#5 Points possible: 5. Total attempts: 5

Which of the following population densities are equivalent to a density of 20 people/mi2? There may be more than one correct answer.

· 1 person/0.5 mi2

· 1 person/0.05 mi2

· 200 people/2 mi2

· 100 people/5 mi2

· 1 person/0.2 mi2

#6 Points possible: 10. Total attempts: 5

One way to measure a country’s economy is per capita gross domestic product, or per capita GDP. This is the value of all the products and services produced in a country over the course of a year divided by its population.

a. According to the CIA’s The World Factbook, in 2010, the United States had a per capita GDP of $47,400. If the population was about 309 million, which of the following is a reasonable estimate for the GDP of the United States?

· $60 trillion

· $60 billion

· $15 billion

· $15 trillion

· $2.3 trillion

b. Also according to The World Factbook, in 2010, China had a per capita GDP of $7,400 and a population of around 1,337,000,000. Which of the following is a reasonable estimate for the GDP of China?

· $60 trillion

· $60 billion

· $15 billion

· $2.3 trillion

· None of the above

#7 Points possible: 5. Total attempts: 5

The population of Nebraska is 1,826,341, and its population density is 23.8 people per mi2. The population of New Hampshire is 1,316,470, and its population density is 146.8 people per mi2. Which of the following statements is a valid interpretation of this information?

· The population of New Hampshire is approximately 139% of the population of Nebraska.

· Nebraska is approximately six times more densely populated than New Hampshire.

· The population of Nebraska is approximately 72% of the population of New Hampshire.

· New Hampshire is approximately six times more densely populated than Nebraska.