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Content Area: Linear Algebra, Linear and Exponential FunctionsLinear and Exponential Regression Models,Percent operations, Rounding

Prerequisites: Linear Algebra; Writing Equations Of a line

Standards met: NM-Num.9-12.1aNM-Num.9-12.2aNM-Num.9-12.3NM-Alg.9-12.1a,e,fNM-Alg.9-12.2cNM-Alg.9-12.3NM-Alg.9-12.4NM-Data.9-12.1NM-Data.9-12.2NM-Prob.Pk-12.1,2,3,4NM-Prob.Comm.Pk-12.1,2,3,4NM-Prob.Conn.Pk-12.1,2,3NM-Prob.Rep.Pk-12.1,3

Time required: 1-2 class periods

Materials: Graphing CalculatorsProtractors

6 Billion and Counting

Teacher Background Information:

No discussion of the future of our planet can go without a consideration of the impact of population. Students will need to look at this issue as a framework for all other data regarding environmental, economic and social equity.

Goals: To apply statistical modeling tools to world population data. The models that students develop will then be used to predict the population of the world in 2050.

Objectives: Students will . . .

Calculate ratios and percentages. Represent data via circle graphs. Use exponential equations to make

calculations. Will use multiple math tools to analyze data.

Model data using trend lines and the regression capabilities of the graphing calculator

Procedure: Ask students to read the article “Population

and the Environment: The Global Challenge”. Lead a brief discussion as to each of these

points and how some of the pressures on the earth might be alleviated if the population was cut in half, a third, etc.

Hand out the Global Population by Continent Analyze Data Using Pie Charts student sheet.

Review and answer any questions. Give students time to complete the sheet. Hand out the World Population: An Explosion or a Hic-up? Student Sheet Review and answer questions. Give students time to complete the worksheet. Have students read the article, “How many people can the earth support?

©2010 Beyond Benign – All Rights Reserved.

Global Population by Continent Analyze Data Using Pie Charts:Student Worksheet

Name: ____________________________ Class Period:______________

When studying population growth and sustainability it is important to be acquainted with how the world’s population is distributed today. In this problem you will;

First: Make a pie chart to illustrate world population distribution in the year 2008.

Second: Use historical growth data to predict the world population for the year 2050.

Third:Make a new pie chart illustrating the projected world population distribution in the year 2050.

Fourth: Compare and contrast the 2005 World Population pie chart with the 2050 World Population pie chart

1. Using continental population data from the U.S. Census International Data Base. A. Calculate the world population total.

B. Calculate the percent of total for each continent.

C. Calculate the size of the pie slice for each continent.

Continent Population 2008 Percent of Total Slice (sector) size

Africa 972,752,366

Asia 4,053,868,153

Europe 729,553,228

Oceania 34,368,042

North America 527,831,687

South America 388,619,456


2. Create your pie charts on the first circle on the following page. Use a protractor to approximate each slice. ‘Fudging’ a little is OK. Be sure to give your graph a title and label each sector.

3. The annual population growth rates (in percent) for each continent during the years 2000 to 2005 were: Africa: 2.2%, Asia: 1.2%, Europe: 0%, Oceania: 1.3%, North America: 1.0%, and South America 1.4%. (Statistical Yearbook 2007; Statistics Norway)

Use the exponential growth formula to find the expected continental population for the year 2050. Record your results in the chart below.

2050 Population = (2008 population)(1 + r)t

r = rate of population growth (2.2% = .022)

t = the number of years from 2008 until 2050

Example: AFRICA 2050 Population = (972,752,366)(1.022)42 = 2,426,274,419 people

Continent Growth rate (r)

Projected Population 2050 (millions)

Percent of Total

Slice (sector) size

Africa 2.2% 2,426,274,419

Asia

Europe

Oceania

North America

South America

TOTALS

After calculating total world population, find the values for columns 4 and 5.


4. Create a second pie chart in the space below. Be sure to give the graphs a title and label each sector.

5. Which continent experienced the greatest increase in population growth?

6. Which continent experienced the greatest percent increase in population?

7. Given a continuation of current growth rates, will Africa’s population ever exceed Asia’s population? Explain?

8. Given a continuation of current growth rates, will North America’s population ever exceed Africa’s population? Explain?


9. Use the table on your graphing calculator to determine how long it will take for North America’s population to exceed the population of Europe.__________________________________________________________________


Global Population by Continent Analyze Data Using Pie ChartsStudent Sheet – Teacher Key







1. Using continental population data from the U.S. Census International Data Base.

A. Calculate the world population total. 6,706,992,932

B. Calculate the percent of total for each continent. Continental Population6,706,992,932

C. Calculate the size of the pie slice for each continent. Answer in B *360



Africa 972,752,366 0.145 = 14.5% 52º

Asia 4,053,868,153 0.604 = 60.4% 218º

Europe 729,553,228 0.109 = 10.9% 39º

Oceania 34,368,042 0.005 = 0.5% 2º

North America 527,831,687 0.079 = 7.9% 28º

South America 388,619,456 0.058 = 5.8% 21º

2. Create your pie charts on the first circle on the following page. Use a protractor to approximate each slice. ‘Fudging’ a little is OK. Be sure to give your graph a title and label each sector.








Continent Growth rate (r)

Projected Population 2050 (millions)

Percent of Total

Slice (sector) size

Africa 2.2% 2,426,274,419 0.213 =21.3%

77º

Asia 1.2% 6,690,385,536 0.587 =58.7%

211º

Europe 0% 729,553,228 0.064 =6.4% 23º

Oceania 1.3% 59,122,323 0.005 =0.5% 2º

North America 1.0% 801,665,432 0.070 =7.0% 25º

South America 1.4% 696,816,858 0.061 =6.1% 22º

TOTALS 11,403,817,798 100% 360º



World Population by Continent

2008

2050



Asia has the greatest increase. Its population increased by 2,636,000,000


Africa experienced the greatest percent increase of almost 60%

7. Given a continuation of current growth rates, will Africa’s population ever exceed Asia’s population? Explain?Yes, because Africa has an exponential growth rate that is greater than Africa’s. Africa’s population will exceed China’s in 146 years at current growth rates.(approximately)


No, North America’s population is growing at a slower exponential rate than Africa’s population. The population of North America will always be less.

9. Use the table on your graphing calculator to determine how long it will take for North America’s population to exceed the population of Europe.

In approximately 33 years North America’s population will exceed Europe’s



World Population: An Explosion or a Hic-up? Student Worksheet


Goal: In this problem you will graph world population over time. Then you will explore which function; linear, quadratic or exponential, best models this data.

1. Round world population to the nearest hundred million.

Year Total World Population Total World Population (rounded)

1950 n= 0 1955 n= 5

2,555,948,654 2,780,907,497

2.6 billion

1960 n = 10 1965 n =

3,041,593,413 3,349,157,095

1970 n = 1975 n =

3,711,800,786 4,089,026,773

1980 n = 30 1985 n =

4,452,766,336 4,852,574,467

1990 n = 1995 n =

5,282,371,928 5,690,982,026

2000 n = 6,084,907,596

2. Graph World Population data below. Be sure to give the graph a title and label each axis.


3. Draw a trend line (also know as a line of best fit) that goes through the middle of the data set. Your best line will meet two characteristics

a. The total vertical distance to the trend line from the data points above the trend line will (as close as possible) equal the total vertical distance from the data points below the trend line to the trend line.

b. The total vertical distance for all points is as low as possible.

4. Select two points on your trend line.

1st point: _________ 2nd point: ________

Write the equation of the line that goes through these two points.

Equation: ________________________________

5. a. What does the slope represent?


b. What does the y-intercept represent? Be specific.

6. Enter this data into your graphing calculator. Find the linear regression equation (LinReg) and its corresponding correlation coefficient. (Hint: touch Catalog: go to turn diagnostic on enter)

Linear: _________________________________________ r = _______

7. Compare the equation of your trend line to the linear regression. Are the two close?

Describe what you could do to make your trend line more like the LinReg line?

8. Now you will examine two other regression models, quadratic and exponential.

Quadratic: _______________________________________ r = _______

Exponential: ______________________________________ r = _______

Answer the following questions:

1. Which curve fits the data best; linear, quadratic or exponential? Explain.

2. Sketch the curve with the highest R2 value onto the graph above. (Note: do not round off)


Predict World Population for the year 2005

3. Enter the regression equations into your graphing calculator. (Note: do not round off.) Go to the table scroll down to n= 55 = 2005. Record the predicted population for 2005.

Method Linear Regression

Quadratic Regression

Exponential Regression

Predicted World Population for 2005

4. The actual population for 2005 was 6,470,340,436. Compare this total with the predicted values from the above table. Which equation was closest?. Any surprises??


5. This time go to the table scroll down to n = 100 = 2050 and record the predicted population for 2050.





6. How old will you be in the year 2050? ______________


7. Which population prediction do you think is most accurate?

8. How much will the world population have increased since 2005?

Population Change: __________________

Percent increase in Population: ___________________

9. Do you feel that the planet can sustain the predicted world population in 2050? Please share your thoughts. Please write in complete sentences.

World Population: An Explosion or a Hic-up?


Student Sheet – Teacher Key Name: _________________________ Class Period:______________

Goal: In this problem you will graph world population over time. Then you will explore which function; linear, quadratic or exponential, best models this data.

1. Round world population to the nearest hundred million.

Year Total World Population Total World Population (rounded)

1950 n= 0 1955 n= 5

2,555,948,654 2,780,907,497

2.6 billion

1960 n = 10 1965 n =

3,041,593,413 3,349,157,095

1970 n = 1975 n =

3,711,800,786 4,089,026,773

1980 n = 30 1985 n =

4,452,766,336 4,852,574,467

1990 n = 1995 n =

5,282,371,928 5,690,982,026

2000 n = 6,084,907,596

2. Graph World Population data below. Be sure to give the graph a title and label each axis.


3. Draw a trend line (also know as a line of best fit) that goes through the middle of the data set. Your best line will meet two characteristics

1. The total vertical distance to the trend line from the data points above the trend line will (as close as possible) equal the total vertical distance from the data points below the trend line to the trend line.

2. The total vertical distance for all points is as low as possible.

4. Select two points on your trend line. 1st point: __(0,2.45)_ 2nd point: __(15,3.5)_ Write the equation of the line that goes through these two points.

Equation: ___ P(t) = 0.07t + 2.45__* Answers will vary____________

5. a. What does the slope represent? b. What does the y-intercept represent? Be specific.

a. The slope 0.07 represents a 0.07 billion increase in the world population per year.

b. 2.45 represents world population in 1950, the starting point of this statistical study.


Time in Years

6. Enter this data into your graphing calculator. Find the linear regression equation (LinReg) and its corresponding Pearson correlation coefficients. (Hint: touch Catalog: go to turn diagnostic on enter) Linear: _P(t) = 0.0727t + 2.3636________________________ r = ___0.9955___

7. Compare the equation of your trend line to the linear regression. Are the two close?

Yes, my rate estimate was a growth of 70 million/yr. LinReg estimates a growth rate of 72million/year.

Describe what you could do to make your trend line more like the LinReg line?

Really, not much. But I might drop the left side somewhat while raising the top.

8. Now you will examine two other regression models, quadratic and exponential.

Quadratic: __P(t) = 0.000429x 2 + 0.05128x + 2.5244_____ r2 = _0.9978_

Exponential: _P(t) = 2.575 • (1.01797)t____________ r2 = _0.9980_

Answer the following questions:

1. Which curve fits the data best; linear, quadratic or exponential? Explain.

The quadratic model is somewhat better than the exponential and linear equations. However all three fit the data well.

2. Sketch that curve onto the graph above.


3. Enter the regression equations into your graphing calculator. (Note: do not round off.) Go to the table scroll down to n= 55 = 2005. Record the predicted population for 2005.






6.36 billion 6.6425 6.8581

4. The actual population for 2005 was 6,470,340,436. Compare this total with the predicted values from the above table. Which equation was closest?. Any surprises??

Yes! The linear model was the closest even though the LinReg model had the weakest correlation of 0.9955 vs. 0.998 for ExpReg and 0.9988 for QuadReg.


5. This time go to the table scroll down to n = 100 = 2050 and record the predicted population for 2050.





9.6356 billion 11.942 billion 15.285 billion

6. How old will you be in the year 2050? ___Age +_______ = _____

7. Which population prediction do you think is most accurate?

9.6356 Billion via LinReg

8. How much will the world population have increased since 2005?

Population Change: ___3.2756 billion_______________

Percent increase in Population: _____51.5 % increase______________

3.2756 = 0.515 6.36


9. Do you feel that the planet can sustain the predicted world population in 2050? Please share your thoughts. Please write in complete sentences.

Answers will vary


Global Population by Continent Analyze Data Using Pie ChartsStudent Sheet – Teacher Key







1. Using continental population data from the U.S. Census International Data Base.

A. Calculate the world population total. 6,706,992,932

B. Calculate the percent of total for each continent. Continental Population6,706,992,932

C. Calculate the size of the pie slice for each continent. Answer in B *360



Africa 972,752,366 0.145 = 14.5% 52º

Asia 4,053,868,153 0.604 = 60.4% 218º

Europe 729,553,228 0.109 = 10.9% 39º

Oceania 34,368,042 0.005 = 0.5% 2º

North America 527,831,687 0.079 = 7.9% 28º

South America 388,619,456 0.058 = 5.8% 21º

2. Create your pie charts on the first circle below. Use a protractor to approximate each slice. ‘Fudging’ a little is OK. Be sure to give your graph a title and label each sector.







Continent Growth Projected Population 2050

Percent of Total

Slice (sector) size


rate (r) (millions)

Africa 2.2% 2,426,274,419 0.213 =21.3%

77º

Asia 1.2% 6,690,385,536 0.587 =58.7%

211º

Europe 0% 729,553,228 0.064 =6.4% 23º

Oceania 1.3% 59,122,323 0.005 =0.5% 2º

North America 1.0% 801,665,432 0.070 =7.0% 25º

South America 1.4% 696,816,858 0.061 =6.1% 22º

TOTALS 11,403,817,798 100% 360º



World Population by Continent

2008

2050



Asia has the greatest increase. Its population increased by 2,636,000,000


Africa experienced the greatest percent increase of almost 60%

7. Given a continuation of current growth rates, will Africa’s population ever exceed

Asia’s population? Explain?

Yes, because Africa has an exponential growth rate that is greater than Africa’s. Africa’s population will exceed China’s in 146 years at current growth rates.(approximately)


No, North America’s population is growing at a slower exponential rate than Africa’s population. The population of North America will always be less.

9. Use the table on your graphing calculator to determine how long it will take for North America’s population to exceed the population of Europe.

In approximately 33 years North America’s population will exceed Europe’s


Spreadsheet Calculations

2008 Population % of total pie slice5yr growth r

972,752,366 0.145035544 52.21279571 1.0224,053,868,153 0.604424098 217.5926753 1.012

729,553,228 0.108775011 39.15900386 134,368,042 0.00512421 1.844715694 1.013

527,831,687 0.078698709 28.33153535 1.01388,619,456 0.057942428 20.85927413 1.014

6,706,992,932

2008 Population r^422050 Population % of Total Pie slice

972,752,366 2.494236461 2426274419 0.212759837 76.593541424,053,868,153 1.650370778 6690385536 0.586679448 211.2046014

729,553,228 1 729553228 0.063974472 23.030810134,368,042 1.720270354 59122323.79 0.005184433 1.866395705

527,831,687 1.518789895 801665432.3 0.070297987 25.30727523388,619,456 1.793057059 696816858.7 0.061103823 21.99737611

114038177986,706,992,932

2050 - 2008 % change1,453,522,053 0.5990757032,636,517,383 0.394075553

0 024,754,282 0.418696022

273,833,745 0.341581081308,197,403 0.442293264


Population and the Environment

The Global ChallengeFrom http://www.infoforhealth.org/pr/m15edsum.shtml

As the century begins, natural resources are under increasing pressure, threatening public health and development. Water shortages, soil exhaustion, loss of forests, air and water pollution, and degradation of coastlines afflict many areas. As the world's population grows, improving living standards without destroying the environment is a global challenge.

Most developed economies currently consume resources much faster than they can regenerate. Most developing countries with rapid population growth face the urgent need to improve living standards. As we humans exploit nature to meet present needs, are we destroying resources needed for the future?

Environment Getting Worse

In the past decade in every environmental sector, conditions have either failed to improve, or they are worsening:

Public health. Unclean water, along with poor sanitation, kills over 12 million people each year, most in developing countries. Air pollution kills nearly 3 million more. Heavy metals and other contaminants also cause widespread health problems.

Food supply. Will there be enough food to go around? In 64 of 105 developing countries studied by the UN Food and Agriculture Organization, the population has been growing faster than food supplies. Population pressures have degraded some 2 billion hectares of arable land—an area the size of Canada and the US.

Freshwater. The supply of freshwater is finite, but demand is soaring as population grows and use per capita rises. By 2025, when world population is projected to be 8 billion,48 countries containing 3 billion people will face shortages.

Coastlines and oceans. Half of all coastal ecosystems are pressured by high population densities and urban development. A tide of pollution is rising in the world's seas. Ocean fisheries are being overexploited, and fish catches are down.

Forests. Nearly half of the world's original forest cover has been lost, and each year another 16 million hectares are cut, bulldozed, or burned. Forests provide over US$400 billion to the world economy annually and are vital to maintaining healthy ecosystems. Yet, current demand for forest products may exceed the limit of sustainable consumption by 25%.


http://www.infoforhealth.org/pr/m15edsum.shtml

Biodiversity. The earth's biological diversity is crucial to the continued vitality of agriculture and medicine—and perhaps even to life on earth itself. Yet human activities are pushing many thousands of plant and animal species into extinction. Two of every three species is estimated to be in decline.

Global climate change. The earth's surface is warming due to greenhouse gas emissions, largely from burning fossil fuels. If the global temperature rises as projected, sea levels would rise by several meters, causing widespread flooding. Global warming also could cause droughts and disrupt agriculture.

Toward a Livable Future

How people preserve or abuse the environment could largely determine whether living standards improve or deteriorate. Growing human numbers, urban expansion, and resource exploitation do not bode well for the future. Without practicing sustainable development, humanity faces a deteriorating environment and may even invite ecological disaster.

Taking action. Many steps toward sustainability can be taken today. These include using energy more efficiently; managing cities better; phasing out subsidies that encourage waste; managing water resources and protecting freshwater sources; harvesting forest products rather than destroying forests; preserving arable land and increasing food production through a second Green Revolution; managing coastal zones and ocean fisheries; protecting biodiversity hotspots; and adopting an international convention on climate change.

Stabilizing population. While population growth has slowed, the absolute number of people continues to increase—by about 1 billion every 13 years. Slowing population growth would help improve living standards and would buy time to protect natural resources. In the long run, to sustain higher living standards, world population size must stabilize


How many people can the Earth support?Tim Radford The Guardian, Thursday November 11 2004

It's an old question. Two hundred years ago, Thomas Malthus said population would race ahead of food supply, but he wasn't the first. The early Christian writer Tertullian said (around AD 200, in De Anima): "We are burdensome to the world, the resources are scarcely adequate for us... Truly, pestilence and hunger and war and flood must be considered as a remedy for nations, like a pruning of the human race becoming excessive in numbers."

That was when the population of the whole planet was maybe 100 million or so. We reached the first billion mark by about 1850. By 1950, it was about 2.5 billion. In less than one short lifetime, this figure doubled. It passed six billion in the late 1990s. Note that: humans took 150,000 years to get to the first billion. The most recent billion arrived in just 12 years.

Nobody knows how many people the planet could hold. The UN predicted this week that fertility would decline and longevity would increase until the global population stabilized at nine billion in 2300. Some optimists have argued that the planet could support 1,000 billion; others look at what is happening right now and wish that it had stayed at ancient Roman levels.

Joel Cohen, the Rockefeller University population biologist, argues in a 1995 book (How Many People can the Earth Support?) that it isn't a question like "How old are you?" which only has one answer at any one time. Cohen argues that you could fit one billion people each a metre apart, into a field 32km square. So everybody in the world would fit easily into Yorkshire. But it takes 900 tonnes of water to grow a tonne of wheat, and there is only so much water, so much land and so much sunshine. Human action has its own "ecological footprint"; there has to be so much land to provide food, clothing, shelter, medicines, building material, fresh air and clean water for any one human. It takes, according to some calculations, 2.1 hectares of land and water to provide for one average human. The important word is: average. The American footprint is about 10 hectares. So if all humans lived at US standards, we'd need another four Earths


Documents

Packaging e-ratio - WordPress.com · Web viewEurope Oceanla North America South America 0.15 0.60 0.11 5.00E-03 0.08 0.06 Sheet3 Sheet2 Sheet1 Chart1 0.21 0.59 0.06 5.00E-03 0.07