Exploring Rate In The Context of Population
In this lesson, students use an interactive map to compare the crowdedness of three countries of their choice. They learn that to compare countries with different areas and populations, they need to calculate population density—a rate that compares the population of a region to its area.
A ratio is a comparison of two quantities by division. It can be expressed in the forms a to b, a:b, or ab, where b ≠ 0. The value of a ratio is found by dividing the two quantities. A ratio provides a relative comparison of two quantities. A rate is a ratio that compares two quantities measured in different units. Population density is a rate that compares the population of a region to its area. The value is given in number of people per unit of area.
ELL: Identifying key words are crucial for students. Spend some time discussing the key vocabulary in this unit.
Goals and Learning Objectives
- Explore rate in the context of population density.
- Compare three countries to see which is most crowded—that is, which has the greatest population density.
Explore the World
Give students a few minutes to select different countries on the World Map interactive and read the data displayed for each country. Then discuss the question “What two quantities does the map provide for each country?”
Students should notice that for each country, the map provides:
- The area of the country in square miles
- The number of people in the country
ELL: To help students access the mathematics in this problem, review key information before beginning the problem. It would be helpful to walk students through the problem and make sure that the diagram shows all of the given information.
Explore the World
Open the World Map interactive and experiment with some of the countries on the map.
- What two quantities does the map provide?
INTERACTIVE: World Map
Discuss the Math Mission. Students will use the population and area data on the map to find the population density of a country.
Use area and population data to find the population density of a country.
Explore Populations and Areas
Give students a few minutes to select three different countries on the map and write down their population and areas for each country chosen. Then have students answer the questions.
Students will be working with very large numbers, so you may want to allow them to use calculators and to round their answers to the nearest whole answer or an appropriate place value.
SWD: If students are struggling to the point of frustration it may be necessary to help students to move toward particular approaches. The CCSS include student understanding of various ways to solve problems, so while it may be necessary to promote or suggest a particular approach, it is important to clarify for students the various ways to solve the problems.
Student has difficulty getting started.
- What two quantities do you know for each country? What are you trying to find?
- How can you find the number of people in one square mile?
Student does not know how to work with the large numbers in this task.
- The size of the numbers does not change the way they behave. You can use the same calculation methods as you would with smaller numbers.
- Review the calculation methods you used in prior lessons. How can you apply them to this situation?
Student has an incorrect solution.
- When comparing population density in two different countries, can you compare only the populations of the countries? Can you compare only their areas? Why or why not?
- Have you checked your calculations?
- Does your answer make sense?
Student thinks you cannot divide a smaller number by a larger number.
- Think about fractions—how do you change a fraction to a decimal?
Student has a solution.
- Explain how you solved the problem.
- Where do you see each country's population in your solution?
- Where do you see each country's area in your solution?
- What do the numbers in your solution mean?
- What quantities are you comparing in your solution?
Mathematical Practice 1: Make sense of problems and persevere in solving them.
- As students work, identify students who understand how to use the data to find and compare rates of population density. These students are making sense of the problem and understanding the relationships between the quantities.
Mathematical Practice 5: Use appropriate tools strategically.
- Identify students who use a calculator to compute the population and area data, and who detect possible errors by estimating.
Mathematical Practice 6: Attend to precision.
- Watch for students who calculate accurately and who use the appropriate unit in their answers: people per square mile for population density.
- Answers will vary based on the three countries that each student chooses. If you chose the United States, Mexico, and Canada, then Canada has the lowest population density (9 people per square mile) and Mexico has the highest population density (145 people per square mile).
Explore Populations and Areas
- Select three countries using the World Map interactive and write down their populations and areas.
- Calculate the population density for each country you selected. Show your work.
- Which country has the highest population density?
- Which country has the lowest population density?
- Do any of the three countries have an approximately equal population density?
INTERACTIVE: World Map
- Population density is a rate that compares the population of a country to the area of the country. It describes how crowded a country is.
- The countries all have different areas and different populations. Compare the countries by finding the number of people in one square mile of each country.
Prepare a Presentation
Students will prepare a presentation based on their findings from the previous task.
Preparing for Ways of Thinking
Look for students who use division to find the population density of a country. In particular, identify students who divide in two different ways: population by area and area by population.
- In the first case, the result is population per square mile, which is population density.
- In the second case, the result is the number of square miles per person, the inverse of population density.
- Answers will vary.
Prepare a Presentation
Summarize your findings about population density. Use your work to support your summary.
- Calculate the population density of your state and the population density of your city or town.
- Which has a higher population density: your state, city, or town?
Select a variety of student work to bring out common mistakes or misconceptions. In particular, focus on having students present the different methods they used to solve the problem. Encourage the class to critique each method presented. During the class discussion, make sure that students think about how population density indicates more or less crowding.
By the end of the discussion, all students should understand each other's methods, as well as the common mistakes and misconceptions outlined in the Mathematics section.
Help students distinguish between the two rates: population per square mile and square miles per person. Tell them that population per square mile is referred to as population density.
Discuss common errors that students might make:
- Students use only one measure to determine population density (looking only at the population or only at the area).
- Students divide area by number of people and think that the resulting rate means people per unit of area.
Facilitate the discussion to help students understand each other's methods for determining the population density of the countries they selected. Compare students' methods for determining population density. How did they make sense of the problem? Whose method is clearer? Why?
Discuss the concept of averages. For example, ask the students if the population density is the same throughout a given country. Are there always 100 people per square mile in every square mile, or is this figure just an average for the country as a whole?
Ways of Thinking: Make Connections
- Listen to your classmates and take notes on what you learned about population density.
As your classmates present, ask questions such as:
- What does a high population density number indicate?
- What does a low population density number indicate?
- Can a country with a large area have a high population density? How?
- Can a country with a small population have a high population density? How?
- Will the country with the largest population size always have the greatest population density?
- Why can’t you just compare areas (or populations) when determining crowdedness?
- When you divide the area by the number of people, what unit should you use? Why?
Population Density Is a Rate
A Possible Summary
Population density is a rate that compares the number of people in a region to the area of the region. The value of the rate indicates how many people there are per square mile, but since the rate is an average, parts of the area might have a higher or lower number of people per square mile than other parts.
SWD: Students with disabilities often have difficulty with written expressions. Consider providing students with a variety of options to use to express their understanding of the key ideas learned in the lesson (audio recording a verbal summary, creating annotated graphics/drawings that explain the key ideas, and so on).
Additional Discussion Points
- Population density is a rate that expresses the crowdedness of a region.
- When you think about the population density of a region, you need to compare two different quantities that describe the region by dividing the population of the region by its area.
- The rate is the population density. It tells how many people, on average, are in each unit of area.
- A country with a large population can have a low population density if it has a large area.
- A country with a small area can have a high population density if it has a large population.
- Population density is an average rate. Some parts of a city, state, or country may have a higher density and other areas a lower density.
Summary of the Math: Population Density Is a Rate
Summarize the concept of population density—what it is, how to find it, and how it is connected with rates.
- Think about the similarities and differences among unit price, fuel efficiency, and population density.
Check your summary
- Does your summary explain what population density means and how to find it?
- Does your summary include the term rate ?
- Does your summary include the term average ?
Reflect On Your Work
Have each student write a brief reflection before the end of class. Review students' reflections.
If you find some reflections interesting enough to pursue later, you can save them and share them with the class when appropriate.
ELL: When writing the reflection, allow some additional time for ELLs to discuss with a partner before writing to help them organize their thoughts. Allow ELLs who share the same native language to discuss in their language of choice, and to use a dictionary (or dictionaries).
Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.
One thing that still confuses me about rate is …