Implications of Exponential Growth Bill Aten Mathematics Consultant Charlevoix-Emmet ISD Charlevoix, MI With collaborative support provided by US, Inc.

Implications of Exponential Growth Bill Aten Mathematics Consultant Charlevoix-Emmet ISD Charlevoix, MI With collaborative support provided by US, Inc. Petoskey, MI

Presenter Mathematics Teacher – 25 years Mathematics Teacher – 25 years Battle Creek Central High School Battle Creek Central High School Boyne Falls Public School Boyne Falls Public School East Jordan Math, Science and Technology Center East Jordan Math, Science and Technology Center K-12 Principal – 6 years K-12 Principal – 6 years MCTM Regional Director 1997 – 2005 MCTM Regional Director 1997 – 2005 Currently a MMLA Team Member Currently a MMLA Team Member Math Consultant for CharEm ISD and SEE North Math & Science Center Math Consultant for CharEm ISD and SEE North Math & Science Center

"The greatest shortcoming of the human race is our inability to understand the exponential function." Dr. Albert Bartlett Professor Emeritus, University of Colorado

Rationale for Presentation There are over twenty Grade Level and High School Content Expectations (GLCE and HSCE) that speak to students’ understanding of exponents and applications of exponential growth. There are over twenty Grade Level and High School Content Expectations (GLCE and HSCE) that speak to students’ understanding of exponents and applications of exponential growth. There are many real world examples and applications of exponential growth that are current, topical, and potentially motivational. There are many real world examples and applications of exponential growth that are current, topical, and potentially motivational. Research has shown that there is a significant relationship between the 'depth of understanding' of a given concept, and the ability to apply or connect that concept to real world situations. Research has shown that there is a significant relationship between the 'depth of understanding' of a given concept, and the ability to apply or connect that concept to real world situations.

Research Also Suggests Students “get it” and “keep it” longer when they engage with: 1. content at multiple levels (Wahlstrom and Webb) 2. content in multiple exposures (Rogers) 3. content through multiple styles (Gardner)

Bloom’s Taxonomy (partial list) Bloom’s Taxonomy (partial list) Knowledge Knowledge Comprehension Comprehension Application Application Wahlstrom’s Levels of Thinking (partial list) Wahlstrom’s Levels of Thinking (partial list) Recall Recall Relate Relate Connect Connect Webb’s Depth of Knowledge (partial list) Webb’s Depth of Knowledge (partial list) Recall Recall Skills & Concepts Skills & Concepts Strategic Thinking Strategic Thinking Brain Research and Depth of Understanding Demonstrates Increased Level of Understanding

Then….. Then….. if we are to engage our students in research driven instruction, we need to provide them with learning experiences that allow them to: if we are to engage our students in research driven instruction, we need to provide them with learning experiences that allow them to: Apply the knowledge they have learned, Apply the knowledge they have learned, Connect that knowledge to other subjects, Connect that knowledge to other subjects, and, allow them to Strategically Plan the activities that will find solutions to problems. and, allow them to Strategically Plan the activities that will find solutions to problems.

Michigan Content Expectations That Speak to the Understanding of Exponents and Exponential Growth Grade Level Content Expectations (GLEC’s) N.MR.08.07 N.MR.08.09 N.MR.08.07 N.MR.08.09 N.MR.08.08 N.MR.08.11 N.MR.08.08 N.MR.08.11 High School Content Expectations (HSCE’s) L2.1.3 L2.1.3 A1.1.2 A1.1.2 A2.1 A2.1 A2.1.1 A2.1.1 A2.1.2 A2.1.2 A2.1.3 A2.1.3 A2.1. A2.1. A2.1.5 A2.1.5 A2.5.1A2.5.A2.5.3A2.5.4A2.5.5A3.1A3.1.1A3.1.2A3.1.3

GLEC Assessment Samples (N.MR.08.07) The population of a city is about 350,000. Over the next year, the population increased 13%. What is the new population of the city? (N.MR.08.07) The population of a city is about 350,000. Over the next year, the population increased 13%. What is the new population of the city? (N.MR.08.08) In 1938 the minimum wage was $0.25 and by 1997 it had increased to $5.15 per hour. Find the percent of increase for the minimum wage from 1938 to 1997. (N.MR.08.08) In 1938 the minimum wage was $0.25 and by 1997 it had increased to $5.15 per hour. Find the percent of increase for the minimum wage from 1938 to 1997.

A Few HSCE Examples A2.5 Exponential and Logarithmic Functions A2.5 Exponential and Logarithmic Functions A2.5.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay. A2.5.4 Understand and use the fact that the base of an exponential function determines whether the function increases or decreases and understand how the base affects the rate of growth or decay. A2.5.5 Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time. A2.5.5 Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time. A3.1 Models of Real-world Situations Using A3.1 Models of Real-world Situations Using Families of Functions. Families of Functions. A3.1.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P 0 = 300 and a = 1.02 to obtain P = 300(1.02) t. A3.1.2 Adapt the general symbolic form of a function to one that fits the specifications of a given situation by using the information to replace arbitrary constants with numbers. In the example above, substitute the given values P 0 = 300 and a = 1.02 to obtain P = 300(1.02) t.

HSCE Assessment Samples (A2.5.4) When the equation y = 200(g) x models exponential growth, g can be? (A2.5.4) When the equation y = 200(g) x models exponential growth, g can be? a) any whole number, a) any whole number, b) any real number >1, b) any real number >1, c) any real number between 0 and 1, or c) any real number between 0 and 1, or d) any real number d) any real number (A2.5.5) You have deposited $200 into an account that pays 5% interest, compounded continuously. How long will it take for your money to double? (A2.5.5) You have deposited $200 into an account that pays 5% interest, compounded continuously. How long will it take for your money to double? (A3.1.2) A college campus has a population of 4,500 students. The population is growing at 3% per year. What will be the population in 10 years? (A3.1.2) A college campus has a population of 4,500 students. The population is growing at 3% per year. What will be the population in 10 years?

MDE Prototype ‘End-of-Year’ Assessments Samples HSCE: A3.1.2 HSCE: A3.1.1 HSCE: A2.5.1 HSCE: A2.5.5 HSCE: A2.5.4

Classroom Instruction NOTE: The PowerPoint presentation from this point forward can be used as: Introduction to an understanding of exponents Introduction to an understanding of exponents Instructional materials that provide greater depth of understanding of exponential growth Instructional materials that provide greater depth of understanding of exponential growth

Exponents and Exponential Growth

The Origins of Exponents The term ‘exponent’ was introduced by Michael Stifel (1487-1567) in 1544 in Arithmetica integra. He wrote, "Est autem 3 exponens ipsius octonarij, & 5 est exponens 32 & 8 est exponens numeri 256" The term ‘exponent’ was introduced by Michael Stifel (1487-1567) in 1544 in Arithmetica integra. He wrote, "Est autem 3 exponens ipsius octonarij, & 5 est exponens 32 & 8 est exponens numeri 256" Historians credit Rene Decartes, with the creation of exponents as we know them in modern mathematics, in his book Geometrie, circa 1637. Historians credit Rene Decartes, with the creation of exponents as we know them in modern mathematics, in his book Geometrie, circa 1637.

Exponential vs Constant Growth

Rice on a Chessboard (one of the first known applications of exponential growth) There is a story that is told of an ancient Indian mathematician who, according to the fable, invented the game of chess. The emperor of India was so pleased with the game that he tells the mathematician he may have anything in his kingdom he wishes. The mathematician replies that he only asks for a meek amount of rice be placed on the squares of his chessboard. He asks the emperor to place one grain of rice on the first square the first day, two on the second square the second day, four on third square on the third day, et cetera, until each successive square has grains of rice that double the number of rice on the prior square until all 64 squares of the chessboard have had their said amounts. There is a story that is told of an ancient Indian mathematician who, according to the fable, invented the game of chess. The emperor of India was so pleased with the game that he tells the mathematician he may have anything in his kingdom he wishes. The mathematician replies that he only asks for a meek amount of rice be placed on the squares of his chessboard. He asks the emperor to place one grain of rice on the first square the first day, two on the second square the second day, four on third square on the third day, et cetera, until each successive square has grains of rice that double the number of rice on the prior square until all 64 squares of the chessboard have had their said amounts. HSCE’s - A2.5.1 and A2.5.5

1 st Half of the Chessboard Tons SquareGrains of RiceWt (g)kgMetric Tonne(1000 lbs) 11 0.025 0.0000250.0000000250.0001 220.050.000050.000000050.0001 340.10.00010.00000010.0002 480.20.00020.00000020.0004 5160.40.00040.00000040.0009 6320.80.00080.00000080.0018.. 311,073,741,82426843545.626843.545626.843545659,179.8712 322,147,483,64853687091.253687.091253.6870912118,359.7424

2 nd Half of the Chessboard Estimated Biomass of Earth = 1,877.29 x 10 9 tonnes * Mass of Rice (64 th square) 230,584,300,921 tonne Biomass of Earth 1,877,290,000,000 tonne =.1228 Or, approximately 12.3% of the total biomass of earth on one square!! * Source: Biomass, http://en.wikipedia.org/wiki/Biomass

Another Story That Demonstrates Exponential Growth

The Water Lily Story There is a famous story that French children are told in which they are asked to imagine having a pond with one water lily leaf floating on the surface. They are told that the lily population will double in size every day if left unchecked. They are also told that the water lilies will smother the pond in 30 days, effectively killing all other living things in the water. The children are then asked at what point should they concern themselves with cutting back the lily plants? When the pond is 1/8th covered, one-fourth covered, or one-half covered? Many of the children answer that they feel they can wait until the pond is 1/4 th -covered before they need to begin cutting back the lily pads. The teacher then asks the children, “if you wait until the pond is 1/4 th -covered, how many days will you have to do your work? “ HSCE’s - A2.5.1 and A2.5.5

. Lily Pond – Day 1. Lily Count: 1

. Lily Pond – Day 11. Lily Count: 1,024

. Lily Pond – Day 19 Lily Count: 262,144.05 % covered

Lily Pond – Day 20 Lily Count: 524,288.10 % covered

Lily Pond – Day 21 Lily Count: 1,048,576.20 % covered

Lily Pond – Day 24 Lily Count: 8,388,608 1.56 % covered

Lily Pond – Day 25 Lily Count: 16,777,216 3.13 % covered

Lily Pond – Day 26 Lily Count: 33,554,432 Pond is 1/16 th covered

Lily Pond – Day 29 Lily Count: 268,435,456 Pond is 1/2 covered

Lily Pond – Day 30 Lily Count: 536,870,912 Pond is completely covered

When Would You Have Noticed There Was A Problem? If you waited until the pond is a 1/16 th covered, you would have only have FOUR days to correct the problem. If you waited until the pond is a 1/16 th covered, you would have only have FOUR days to correct the problem. If you waited until what the children thought would be acceptable(1/4 th covered), you would only have two days to correct the problem. If you waited until what the children thought would be acceptable(1/4 th covered), you would only have two days to correct the problem.

Real World Implications Response to a problem always involves inevitable delays Delays that include: Delays that include: Coming to an agreement that there is a problem Coming to an agreement that there is a problem Coming to an agreement about a solution Coming to an agreement about a solution Time required to implement a potential solution Time required to implement a potential solution

Questions for Student Response If the Lily Pond were to double in size magically overnight and the given growth rate of the lily pads were to continue, how many more days would you have to respond to the problem before the pond would be completely covered? If the Lily Pond were to double in size magically overnight and the given growth rate of the lily pads were to continue, how many more days would you have to respond to the problem before the pond would be completely covered? Can you think of any current real world situations that might be examples of exponential growth? Can you think of any current real world situations that might be examples of exponential growth? Relating your example to the Lily Pond story, what day do you think we might we be at in your example? Relating your example to the Lily Pond story, what day do you think we might we be at in your example?

In a more modern story, Mike Myers in Wayne’s W  rld 2, tries to explain the implications of exponential growth…… In a more modern story, Mike Myers in Wayne’s W  rld 2, tries to explain the implications of exponential growth…… Reproduced without “expressed written permission“ of Paramount Reproduced without “expressed written permission“ of Paramount Note: You need QuickTime 7.1 to view movie clip. It can be downloaded for free at http://quicktime-downloads.com/ Note: You need QuickTime 7.1 to view movie clip. It can be downloaded for free at http://quicktime-downloads.com/http://quicktime-downloads.com/

The “Doubling” Effect When a quantity grows exponentially, each succeeding value is more than the sum of ALL the preceding values. When a quantity grows exponentially, each succeeding value is more than the sum of ALL the preceding values. Example: y = 2 x

Example: How long does it take 100 to double at 7% annual growth? Example: How long does it take 100 to double at 7% annual growth? 100 (1.07) t = 200 (1.07) t = 2 (1.07) t = 2 log (1.07) t = log 2 log (1.07) t = log 2 t log (1.07) = log 2 t = log 2 / log 1.07 t = log 2 / log 1.07 t = 10.24 years t = 10.24 years Calculating “Doubling” Time HSCE’s - A2.5.1, A2.5.3, A2.5.5 and A3.1.1&2 P = P o (a) t – HSCE A3.1.1 P = P o (a) t – HSCE A3.1.1 log x y = y log x log x y = y log x

Calculating “Doubling” Time This would indicate that any “doubling” time could be calculated by using the following formula. t = log 2 / log 1+ (annual % rate) Example: How long would it take $620 to double at 6% annual growth? t = log 2 / log 1.06 t ≈ 11.896 Check: 620 ( 1.06) 11.896 = $1240.02 HSCE’s - A2.5.5 and A3.1.1

A Quicker Mental Calculation “Doubling” time for a given percent of annual growth can be easily estimated by dividing 70 by the percent of growth. Examples:

An Educational Challenge “If we could get everyone to habitually evaluate news stories, newspaper headlines, or comments made by political leaders that present us with statistics of annual growth rates, and quickly make mental calculations of the doubling times, we might begin to seriously consider the implications of those growth rates and what their impact on our quality of life and the consumption of finite resources means.” “If we could get everyone to habitually evaluate news stories, newspaper headlines, or comments made by political leaders that present us with statistics of annual growth rates, and quickly make mental calculations of the doubling times, we might begin to seriously consider the implications of those growth rates and what their impact on our quality of life and the consumption of finite resources means.” Dr. Albert Bartlett

Potential Application (or Implication) of Doubling Effect “Over the past 30 years nearly half the energy used in the history of the industrial revolution has been consumed.” “Over the past 30 years nearly half the energy used in the history of the industrial revolution has been consumed.” Relocalization: A Strategic Response to Peak Oil and Climate Change, Jason Bradford, PhD Biology, May 23, 2007 “Peak Oil is the theory that predicts that future world oil production will soon reach a peak and then decline.”

Hubbert Peak Oil Theory Hubbert assumed that after fossil fuel reserves (oil reserves, coal reserves, and natural gas reserves) are discovered, production at first increases approximately exponentially, as more extraction commences and more efficient facilities are installed. At some point, a peak output is reached, and production begins declining until it approximates an exponential decline. US Crude Oil Production

“The earth’s endowment of oil is finite and demand for oil continues to increase with time. Accordingly, geologists know that at some future date, conventional oil supply will no longer be capable of satisfying world demand. At that point world conventional oil production will have peaked and begin to decline.” “The earth’s endowment of oil is finite and demand for oil continues to increase with time. Accordingly, geologists know that at some future date, conventional oil supply will no longer be capable of satisfying world demand. At that point world conventional oil production will have peaked and begin to decline.” Relocalization: A Strategic Response to Peak Oil and Climate Change, Jason Bradford, PhD Biology, May 23, 2007

Remaining Oil Reserves If we were to assume that we have used ½ of the world’s supply of oil, and that our consumption rate were to remain the same, how long will it be before the next doubling uses the remaining ½ of the world oil reserves? If we were to assume that we have used ½ of the world’s supply of oil, and that our consumption rate were to remain the same, how long will it be before the next doubling uses the remaining ½ of the world oil reserves? World Energy Consumption Rose 2.4 Percent in 2006 Fox News.com, TUESDAY, JUNE 12, 2007 (3.2% in 2005) Year Growth RateDoubling Time Oil Reserves Depleted 2006 2.4% ( 70 ÷ 2.4 ) = 29.2 yrs 2036 2005 3.2% ( 70 ÷ 3.2 ) = 22 yrs 2029 Using a recent report from Fox News: HSCE - A2.5.5

Questions for Classroom Discussion In your students’ lifetime they will see the peak of oil production. What will life look like when the inevitable occurs, and oil production either plateaus or declines? (will demand plateau?) What will life look like when the inevitable occurs, and oil production either plateaus or declines? (will demand plateau?) How will population growth effect world wide demand for oil? How will population growth effect world wide demand for oil? Using the Lily Pond analogy, what day is it? Using the Lily Pond analogy, what day is it? Are there possible corrective options? Are there possible corrective options?

Financial Applications of Exponential Growth

Annual Percentage Rates Any quantity that grows a given percent per year (or other unit of time), increases in value exponentially. Any quantity that grows a given percent per year (or other unit of time), increases in value exponentially. Annual Interest Rates or Compound Interest Rates, are common examples of exponential growth. Annual Interest Rates or Compound Interest Rates, are common examples of exponential growth. GLCE’s - N.MR.08.07, N.MR.08.08 and N.MR.08.09 P = P o (a) t - HSCE A3.1.1 P = P o (a) t - HSCE A3.1.1

Chart Indicates when the original value has doubled

Graph of Compound Interest “Doubling”

Other “Real World” Applications

US Immigration Which decade saw the greatest increase in immigration of foreign born population? Which decade saw the greatest increase in immigration of foreign born population? What immigrant population is not represented in these statistics? What immigrant population is not represented in these statistics?

% of Increase % of Increase 1.557%.98% 1.806%

Foreign Immigrant Population That Do Not Have US Citizenship Foreign Immigration (Not US Citizens) is an almost perfect example of exponential growth. Foreign Immigration (Not US Citizens) is an almost perfect example of exponential growth.

Immigrant Population Projections Using the US Census Bureau’s statistics of Total US Population of Foreign-Born Immigrants from 1989 and 2000, what was the annual growth rate? Using the US Census Bureau’s statistics of Total US Population of Foreign-Born Immigrants from 1989 and 2000, what was the annual growth rate? If the immigrant population were to continue growing at this annual growth rate, how long would it take the 2000 population to double? If the immigrant population were to continue growing at this annual growth rate, how long would it take the 2000 population to double? 1.041% annual growth rate 70 ÷ 4.1 ≈ 17.1 years

World Population Applications

Growth Rates Population Growth Rate (PGR) (Births + Immigration)-(Deaths + Emigration) Population Population Zero Population Growth Rate (ZPG) (Births + Immigration) = (Deaths + Emigration) PGR = GLCE - N.FL.07.03

Current Growth Estimates The World Factbook states that the world human population currently increases by 203,800 every day. The World Factbook states that the world human population currently increases by 203,800 every day. The 2007 CIA Factbook increased this to 211,090 people every day. The 2007 CIA Factbook increased this to 211,090 people every day. Which is between 8492~8795 each hour Which is between 8492~8795 each hour Or, 142~147 each minute Or, 142~147 each minute Or, 2.4~2.5 every second Or, 2.4~2.5 every second

The world population will have increased by 7,350 people during the course of this 50 minute presentation. This means…. 8,700 if it takes an hour…… 8,700 if it takes an hour……

“Doubling” Time Question: Given that: Data provided by: Central Intelligence Agency – The World Fact Book https://www.cia.gov/library/publications/the-world-factbook/fields/2002.html Assuming that all the land mass is habitable, and the growth rate stays consistent, how many years will it take to reduce the Area per Person to half of its current size? HSCE - A2.5.5

Answer: HSCE - A2.5.5 and GLCE - N.FL.08.11 Equivalent to 27.3% of a HS football field

World Population From Wikipedia, World Population Estimates http://en.wikipedia.org/wiki/World_population_estimates#_note-Census

World Population with Trendline From Wikipedia, World Population Estimates http://en.wikipedia.org/wiki/World_population_estimates#_note-Census

Options That Increase or Decrease Population Growth Rates Increase Population Procreation Procreation Motherhood Motherhood Large Families Large Families Immigration Immigration Medicine Medicine Public Health Public Health Sanitation Sanitation Peace Peace Law and Order Law and Order Scientific Agriculture Scientific Agriculture Accident Prevention Accident Prevention Clean Air Clean Air Ignorance of the Problem Ignorance of the Problem Decease Population Abstention Abstention Contraceptives Contraceptives Abortion Abortion Small Families Small Families Stop Immigration Stop Immigration Disease Disease War War Murder and Violence Murder and Violence Famine Famine Accidents Accidents Pollution Pollution Smoking Smoking From “Arithmetic, Population and Energy” Dr. Albert Bartlett

The Human Dilemma Everything that we regard as “good” increase the human population Everything that we regard as “good” increase the human population Everything that we regard as “bad” decreases human population Everything that we regard as “bad” decreases human population From “Arithmetic, Population and Energy” Dr. Albert Bartlett

Population Overshoot “ The biological phenomenon of ‘population overshoot’ is used by ecologists to describe a species whose numbers exceed the ecological carrying capacity of the place where it lives. “ “ The biological phenomenon of ‘population overshoot’ is used by ecologists to describe a species whose numbers exceed the ecological carrying capacity of the place where it lives. “ "Overshoot is the inevitable and irreversible consequence of continued drawdown; when the use of resources in an ecosystem exceeds its carrying capacity and there is no way to recover or replace what was lost.” "Overshoot is the inevitable and irreversible consequence of continued drawdown; when the use of resources in an ecosystem exceeds its carrying capacity and there is no way to recover or replace what was lost.” Human Population: Why set One Billion as the Upper limit? by Ted Mosquin January, 2006 AN ECOLOATE VIEW OF THE HUMAN PREDICAMENT, G. Hardin; in McRostie, ed. GLOBAL RESOURCES: Perspectives and Alternatives Baltimore: University Park Press, 1980.

Crash “Once a population has exceeded the capacity of its environment, in one life- giving respect or another, there is no recourse, nothing to be done until that population is reduced to the level at which the resources can recover and are once again adequate to sustain it.” “Once a population has exceeded the capacity of its environment, in one life- giving respect or another, there is no recourse, nothing to be done until that population is reduced to the level at which the resources can recover and are once again adequate to sustain it.” AN ECOLOATE VIEW OF THE HUMAN PREDICAMENT, G. Hardin; in McRostie, ed. GLOBAL RESOURCES: Perspectives and Alternatives Baltimore: University Park Press, 1980.

First Law of Sustainability Population growth and/or growth in rates of consumption of resources CANNOT BE SUSTAINED! Which is further complicated by the problem that when a species reaches ‘overshoot’ and begins to ‘crash’, they lose most of their ability to make choices on how to correct the situation, and nature starts making choices for them.

“Democracy cannot survive overpopulation. Human dignity cannot survive overpopulation. Convenience and decency cannot survive overpopulation. As we put more and more people into the world, the value of life not only declines, it disappears. Soon it doesn’t matter if someone dies. The more people there are, the less one individual matters.” “Democracy cannot survive overpopulation. Human dignity cannot survive overpopulation. Convenience and decency cannot survive overpopulation. As we put more and more people into the world, the value of life not only declines, it disappears. Soon it doesn’t matter if someone dies. The more people there are, the less one individual matters.” Isaac Asimov

If we don’t stop population growth, resources will be consumed at a greater and greater rate, and life as we know it will not be able to be sustained. A Seemingly Logical Conclusion is:

What Corrective Choice Do We Make? If we agree with the list of options there are to Decrease the Human Population, then it appears that we, as well as current and future generations, must make a choice from that list, or If we agree with the list of options there are to Decrease the Human Population, then it appears that we, as well as current and future generations, must make a choice from that list, or nature will choose for us.

Choices to Decrease the Human Population on Earth Abstention Abstention Contraceptives Contraceptives Abortion Abortion Small Families Small Families Stop Immigration Stop Immigration Disease Disease War War Murder and Violence Murder and Violence Famine Famine Accidents Accidents Pollution Pollution Smoking Smoking ??- which would you choose?

Excerpts from Arithmetic, Population and Energy a presentation by Dr. Albert Bartlett, Prof. Emeritus Dept. of Physics University of Colorado, Boulder Note: You need QuickTime 7.1 to view movie clip. It can be downloaded for free at http://quicktime-downloads.com/ Note: You need QuickTime 7.1 to view movie clip. It can be downloaded for free at http://quicktime-downloads.com/http://quicktime-downloads.com/ Reproduced with permission of the University of Colorado Reproduced with permission of the University of Colorado

Other Possible Investigations of Exponential Growth Coal Production and Consumption Coal Production and Consumption Natural Gas Production and Consumption Natural Gas Production and Consumption Electrical Energy Production and Consumption Electrical Energy Production and Consumption Sustainability of Natural Resources Sustainability of Natural Resources Aluminum, Copper, Silver, etc. Aluminum, Copper, Silver, etc. World Food Production and Consumption World Food Production and Consumption Health Care Costs Health Care Costs Insurance Rates Insurance Rates Inflation Inflation Increase in Internet Hosts Increase in Internet Hosts

Curriculum Collaboration Many “real world” applications of exponential growth could also be topics of investigation in science, social studies, technology and communication arts. Many “real world” applications of exponential growth could also be topics of investigation in science, social studies, technology and communication arts. The integration of topics increases a student’s interest, understanding and retention of knowledge. The integration of topics increases a student’s interest, understanding and retention of knowledge. The retention of knowledge will help students make more educated decisions in their adult lives. The retention of knowledge will help students make more educated decisions in their adult lives.

In Conclusion “ We must educate all of our people to an understanding of the arithmetic and consequences of growth, and especially in terms of population and the earth’s finite resources.” “ We must educate all of our people to an understanding of the arithmetic and consequences of growth, and especially in terms of population and the earth’s finite resources.” Dr. Albert Bartlett “Arithmetic, Population and Energy”

Disclaimer It is important to note that the calculations in this presentation are not predictions of the future. They simply give us first-order estimates of life expectancies should conditions of steady growth go unchecked, and provide us with potential consequences should we choose to ignore their implications.

Thank You! Your DVD copy of Arithmetic, Population and Energy a presentation by a presentation by Dr. Albert Bartlett, Prof. Emeritus, Dept. of Physics University of Colorado, Boulder As well as the Implications of Exponential Growth workbook Have been generously provided by US, Inc. Publishers of The Social Contract 445 E. Michigan Avenue Petoskey, MI 49770-2623

Classroom Educational Materials The Implications of Exponential Growth workbook was created and assembled by Bill Aten Boyne Falls, MI baten@centurytel.net Reference citations are included on each page of the workbook as well as web site information. ALL materials are the property of US, Inc., Petoskey, MI and William Aten, Boyne Falls, MI. Permission for reproduction provided by request.

Implications of Exponential Growth Bill Aten Mathematics Consultant Charlevoix-Emmet ISD Charlevoix, MI With collaborative support provided by US, Inc.

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Implications of Exponential Growth Bill Aten Mathematics Consultant Charlevoix-Emmet ISD Charlevoix, MI With collaborative support provided by US, Inc.

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