Unit 2 WebQuest - Internet Project

Population Explosion

Introduction | Task | Process | Guidance | Conclusion | Questions

Introduction
The world population reached 6 billion in 1999. In addition, the world population has doubled in about 40 years and gained 1 billion people in just 12 years. Assuming middle-range fertility and mortality trends, world population is expected to exceed 9 billion by 2050, with most of the increase in countries that are less economically developed. Did you know that the population of the United States has increased by more than a factor of 10 since 1850? The table below shows how the population of the U.S. has changed over the years.

     In this project, you will use quadratic and polynomial mathematical models that will help you to project future populations.

The Task
Your social studies teacher and your mathematics teacher are collaborating on a project for your class. Each student will prepare a Web page showing an application of mathematics to social studies. These will be posted on your school's Web site. You have decided to focus on presenting population data and making predictions about population. You have met with your teachers about your project proposal and they want you to present data on world population and population for two other areas, which could be countries, states, or cities. They also want you to be sure that your Web page contains the following information:

• population data for the entire world for at least the past 200 years;
• population data for two other areas, which could be countries, states, or cities for a period of at least 50 years;
• graphs and/or tables displaying the population data. The areas that you choose can show either growth or loss of population;
• at least one mathematical model for the population of the world and your two chosen areas. You may propose more than one population model, if you prefer;
• a prediction of the population of the world and the two areas for the year 2050.

You will get some ideas for population models in the exercises in Unit 2 of your textbook.

• Find data on the population of the world for at least the past 200 years. For help, try these Web sites.
  www.statistics.com
  www.census.gov
  www.infoplease.com and search for population data of your choice
  www.gazetteer.de/home.htm
  www.yahoo.com and search for population data of your choice
  www.citypopulation.de/cities.html
• Select the other two areas you wish to include in your project. Find population data for these two areas for at least the past 50 years. For help, try the Web sites listed above.
• Display each population data set in an appropriate table and/or graph.
• Refer to Unit 2 in your textbook for ideas about mathematical models for your population data.
• Be creative. Add some additional data, information, or even pictures to your Web page. Try these Web sites for additional information.
  www.census.gov/main/www/popclock.html
  fisher.lib.virginia.edu/census

Search for the Web site of the country, state, or city for additional information to add to your Web page.

1. How has the median age of the population changed over the last 100 years? What problems could this present in the future?
2. What factors affect population growth?
3. What is the history of the census in the U.S.? How do other countries measure their population?
4. What areas of the world are experiencing a high population growth rate? In what areas, if any, is population decreasing?
5. Compare the population models that you chose for the world and the other two areas. How are the models similar? How are the models different?
6. What factors can affect the accuracy of population estimates made using mathematical models?
7. How has the population density of your chosen areas changed over the last 50 years?

• Present your project to your class or at a family night.
• Write a one–page summary of your project, including what you have learned from researching this topic.

Lesson 5–1
For your project, you find this table of population for the world from 1650 through 2000.

1. Write each population in scientific notation.
2. Will using the values in scientific notation make it easier to graph the data? If not, suggest another way to write the values.
3. Rewrite each year as Years Since 1650. For example 1650 will be 0, 1750 will be 100, and so on. How will this make the data easier to graph?
4. Make a scatter plot of the data using the ordered pairs (years since 1650, population). Describe the shape of the scatter plot.
5. Find a linear equation to model the data. How well does this model fit the population data? Explain.

1. Find a quadratic equation whose graph best fits the data.
2. Graph the equation and the data on the same screen. Do you think the graph of the equation fits the data? Justify your answer.
3. Predict the world population for 2050 using the quadratic model. Do you think your prediction is a good estimate for the population in 2050? Why or why not?

1. Find a cubic polynomial function to model the population data.
2. Graph the equation and the data on the same screen. Do you think the equation models the data fairly well? Explain.
3. Compare the linear (Lesson 5–1), quadratic (Lesson 6–6), and cubic models for the data. Which one do you think best models the data? Explain your reasoning.
4. Use the equation you think best models the data to predict the world population in 2050.

In this project, students will research population. All students will research world population growth and be given accurate data to use in the exercise in Lesson 5-1. They will need to find the population data for two other areas, which can either be countries, states, or cities. They can do two of the same type of area or one city and one state or any other combination. The exercises help students to see what types of equations might model their data. If you prefer, you can encourage students to try other types of functions as models for the data. For example, they can try exponential functions since those are easily found using a graphing calculator or graphing software. Even thought they do not study exponential functions until Unit 3, they can still explore these models for the data since they have probably seen them in previous mathematics courses.

The Guidance section of the WebQuest contains questions that would be good for a whole-class discussion and for providing interdisciplinary connections to social studies. You may want to collaborate with a social studies teacher at your school just like the scenario set for this project.

Several Web sites are included in the project to help students in completing this WebQuest. Encourage students to find additional sites and to share those sites with other students. They will probably need to find their own Web sites if they want pictures or maps of various countries, states, and cities to enhance their Web page.

Students will work on this project in Unit 2.

Lesson	5–1	6–4	7–4
Page	237	332	404

ANSWERS
Lesson 5–1

1. Year	Population
   1650	5.5 • 108
   1750	7.25 • 108
   1850	1.175 • 109
   1900	1.6 • 109
   1950	2.556 • 109
   1980	4.458 • 109
   2000	6.08 • 109

2. Since some values are times 10⁸ and some are times 10⁹, you can’t just use the decimal portion as the value of the graph. However, you could change them all to values times either 10⁸ or 10⁹, so that you could work with smaller values on your graph.
3. The values on the axis for the year will be smaller and will start with 0.
4. Sample answer: Use years since 1650 on the x–axis and population values rewritten as p • 10⁸. If you connect the points, the graph is a curve that gets steeper as time goes by.
   
5. A linear equation is y = 0.1333x – 4.6465. It does not fit the data very well because the data points show a curve and the r² value is only 0.676.

1. Sample using years since 1650 on the x–axis and population values rewritten as p • 10⁸. y = 0.0008x² – 0.1617x + 8.7296.
2. It fits fairly well since a quadratic function is a curve. The r² value is 0.9197.
3. 2050 is 400 years after 1650, so substitute 400 for x. y = 0.0008x² – 0.1617x + 8.7296 = 0.0008(400)² – 0.1617(400) + 8.7296 = 72.0496. Since the model was written with values times 10⁸, the estimate is 7,204,960,000.

1. y = 0.000005x³ – 0.0017x² + 0.17x + 4.9287
2. The cubic function models the data fairly well with an r² value of 0.9826.
3. The cubic fits best since you can see that it passes close to most data points and its value of r² is the highest of the three equations.
4. Choose y = 0.000005x³ – 0.0017x² + 0.17x + 4.9287 = 0.000005(400)³ – 0.0017(400)² + 0.17(400) + 4.9287 = 120.9287. Since the values in the graph were written as values times 10⁸, this would be 12,092,870,000 people.