Effects of Rates of Change and Accumulated Change in a Throw-Away Society

byJoseph D. Irizarry


Our society currently encourages production for planned obsolescence and repeated purchase of single-use items, especially paper and plastic packaging, followed by disposal. This pattern has become so normalized in our consumer-product-driven economy that we no longer notice it nor consider its consequences.

I want my students to become aware of how much stuff they routinely throw away. I want them to consider the accumulated effects of a society that throws away so much stuff. How much garbage do we really produce each year? Is that rate constant increasing, or decreasing? Where can we put all our garbage? What happens to those items that we throw away over time? How are we affected by the chemical residues, by-products, and breakdown products of our trash and industrial processes? How do we use calculus to compute cumulative exposure? What are the health effects of cumulative exposure to these substances?


My students are high school seniors taking Advanced Placement Calculus AB in a working class neighborhood in Chicago. Calculus generally deals with two related big ideas: the derivative and the integral. Many of the examples of the derivative and the integral that we encounter in the course come from physics, economics, and population biology. This unit will expand the applications of calculus by making connections to green chemistry. The activities are designed to be especially relevant to residents of a large urban center dealing with municipal solid waste disposal challenges and historically high levels of exposure to persistent bioaccumulative toxins (PBTs).

This unit should take one to two weeks, depending on the amount of contact hours the teacher has with the students and the depth to which the research is taken. I see my twenty-five AP Calculus AB students for ninety minutes each day, five days a week. If this unit is taught after an in-depth study and application of the derivative, and at the point of the introduction of the integral, I predict that it should take five to seven days to teach. For my course outline, this unit falls in the 15 th week of the fall semester, in mid-December.

One element of the College Board Advanced Placement Calculus philosophy, known as the "Rule of Four," states that, "Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations." 1 In this unit, students will be presented with graphical and tabular data representing quantities and rates of change of solid waste production and environmental toxin exposure. Students will use the data to compute or approximate derivatives and integrals. Throughout, they will express the meaning of their answers verbally in the context of the given problems.

The derivative measures the rate of change. It is essentially the slope of a function, the ratio of the change in the dependent quantity to the change in the independent quantity. As such, many activities in this unit are accessible to students in pre-calculus or even algebra. Working from a graph or table of values, students should be able to calculate the rise, or change in dependent variable, divided by the run, or change in independent variable. The independent variable, normally denoted by x, will represent time throughout this unit and may, thus, be denoted by t. The dependent variable, normally denoted by y, may be indicated as a function of x or t in various ways, such as y(x), y(t), f(x), or f(t).

The derivative is actually the instantaneous rate of change of a function, or the slope of a function at a single point. This is defined using the limit of one of the acceptable difference quotients, such as the right-hand, left-hand, or symmetric difference quotient. In essence, the slope of the function is computed using either our target point and a point on the function infinitesimally close behind or beyond our point, or two points straddling our target point. The derivative can be represented symbolically in several ways, such as dy/dx, f'(t), or y'.

It is crucial that calculus students be able to work with any of these representations of the derivative interchangeably. Students should recognize that it is appropriate to use a simple calculation of the slope of the tangent line to a curve or a difference quotient from tabular (numerical) data as an approximation of the instantaneous derivative if no explicit equation is given to define a function or if the function does not have a reasonable analytic derivative. In symbolic terms, students should recognize that


The integral measures the accumulated change given a rate of change. Graphically, the integral is represented by the area under a function curve. Numerically, the integral can be computed using a Riemann sum with rectangles or trapezoids to find the area under the function even if just given a table of values. Analytically, the integral is an antiderivative of the function. Again, it is crucial that students understand and be able to operate interchangeably with the different representations of the integral.

In this unit, students will be provided with numerical data representing the rate at which lead, a significant persistent bioaccumulative toxin (PBT) in Chicago, is taken into the body by various means. Students will use integral calculus techniques to find how much of the substance will have cumulatively been ingested or inhaled.

In this unit, students will also research the persistent organic pollutants (POPs) identified by the United Nations Environment Programme. Students will begin to appreciate the consequences of accumulation of solid waste and persistent toxins in the environment and how mathematical modeling helps to study problems of this scope. They will verbalize steps they can take to reduce their environmental impact and protect themselves from the effects of PBTs.

Background Information

This unit is designed to help students apply the techniques of calculus to the principles of green chemistry. Green chemistry is environmentally benign chemistry, or efficient chemistry when cradle-to-grave costs are considered in both monetary costs for all aspects of production and disposal, as well as the costs to human health and safety. Paul Anastas, the father of green chemistry, and John Warner enumerate twelve principles of green chemistry in their seminal book:

  1. Prevent waste: Design chemical syntheses to prevent waste, leaving no waste to treat or clean up.
  2. Design safer chemicals and products: Design chemical products to be fully effective, yet have little or no toxicity.
  3. Design less hazardous chemical syntheses: Design syntheses to use and generate substances with little or no toxicity to humans and the environment.
  4. Use renewable feedstocks: Use raw materials and feedstocks that are renewable rather than depleting. Renewable feedstocks are often made from agricultural products or are the wastes of other processes; depleting feedstocks are made from fossil fuels (petroleum, natural gas, or coal) or are mined.
  5. Use catalysts, not stoichiometric reagents: Minimize waste by using catalytic reactions. Catalysts are used in small amounts and can carry out a single reaction many times. They are preferable to stoichiometric reagents, which are used in excess and work only once.
  6. Avoid chemical derivatives: Avoid using blocking or protecting groups or any temporary modifications if possible. Derivatives use additional reagents and generate waste.
  7. Maximize atom economy: Design syntheses so that the final product contains the maximum proportion of the starting materials. There should be few, if any, wasted atoms.
  8. Use safer solvents and reaction conditions: Avoid using solvents, separation agents, or other auxiliary chemicals. If these chemicals are necessary, use innocuous chemicals.
  9. Increase energy efficiency: Run chemical reactions at ambient temperature and pressure whenever possible.
  10. Design chemicals and products to degrade after use: Design chemical products to break down to innocuous substances after use so that they do not accumulate in the environment.
  11. Analyze in real time to prevent pollution: Include in-process real-time monitoring and control during syntheses to minimize or eliminate the formation of by-products.
  12. Minimize the potential for accidents: Design chemicals and their forms (solid, liquid, or gas) to minimize the potential for chemical accidents including >explosions, fires, and releases to the environment. 2, 3

The activities of this unit focus centrally on green chemistry principle number ten. However, many of the issues are interrelated. Much of our industrial chemical complex and consumerist society operates directly in opposition to the principles of green chemistry. The question of how to prevent waste is not asked. Instead, it is the question of where to bury or burn our waste that is asked, never mind what environmental effects that disposal has. Many of the substances used in the production of our everyday objects, from plastic bottles to electronic devices, are toxic to human health and the environment. The plastics we use are almost entirely petroleum based and, therefore, derived from a depleting, instead of renewable, feedstock. The plastics we produce will remain in the environment for thousands of years, sometimes leaching compounds that have developmental effects on biological organisms, sometimes breaking down into smaller and smaller pieces that are consumed by smaller and smaller organisms, either killing them or then moving on up through the food chain. Our food and drink is tainted with residues of herbicides and pesticides from our industrialized agriculture and chemical residues from packaging. Some of these residues are persistent, toxic, and bioaccumulative, yet many of their parent chemical compounds continue to be produced and used.

One of the common attacks on green chemistry is the declaration that the cost of doing environmentally friendly chemistry is prohibitively high for business. That statement often does not take into consideration all the real costs of doing business, including energy costs, waste disposal, litigation and liability, and the costs to human health. 4 Additionally, many companies are finding that operating with green chemistry in mind actually saves money up front. Industrial giant 3M realized that it would be easier to prevent pollution than to clean it up, and that everything coming out of a manufacturing plant is a product, a by-product with economic value, or waste, which equals inefficiency. Through their Pollution Prevention Pays (3P) program, which encourages all employees to contribute ideas toward efficiency, 3M calculates nearly 3 billion pounds of cumulative reduced pollution including volatile organic compounds and over one billion dollars in savings calculated from only the first year of implementation of each 3P initiative. 5

Many other companies have programs in place to redesign products to make them biodegradable, non-toxic, or otherwise environmentally sound. Rohner Textil only uses dyes that do not cause cancer or contain PBTs or heavy metals. Herman Miller Furniture replaced chemical plasticizers with a structural honeycomb design to maintain flexibility in a line of chairs without the toxic chemicals. Dow Chemical redesigned its hydrochloric acid scrubbing process to recover an additional 6000 tons of acid waste each year, saving $ 2.4 million dollars annually on reaction inputs and waste disposal. 6

Rachel Carson addressed this issue eloquently in her environmental classic, Silent Spring, in reference to the costs of weed control:

The chemical weed killers are a bright new toy. They work in a spectacular way; they give a giddy sense of power over nature to those who wield them, and as for the long-range and less obvious effects—these are easily brushed aside as the baseless imaginings of pessimists. The 'agricultural engineers' speak blithely of 'chemical plowing' in a world that is urged to beat its plowshares into spray guns. The town fathers of a thousand communities lend willing ears to the chemical salesman and the eager contractors who will rid the roadsides of 'brush'—for a price. It is cheaper than mowing, is the cry. So, perhaps, it appears in the neat rows of figures in the official books; but were the true costs entered, the costs not only in dollars but in the many equally valid debits we shall presently consider, the wholesale broadcasting of chemicals would be seen to be more costly in dollars as well as infinitely damaging to the long-range health of the landscape and to all the varied interests that depend on it. 7

Our problem is short-sightedness. We see one plastic spoon; we throw away one plastic bottle. We do not see all the other waste we produce at one time or think about the difficulties or consequences of its disposal. According to the United States Environmental Protection Agency (EPA), we produced about 30 million tons of plastics in 2007. In total, we produced about 254 million tons of municipal solid waste (MSW) in 2007, or 4.62 pounds per person per day. 8

As an illustration of these enormous numbers, consider building skyscrapers the size of the former World Trade Center towers or Chicago's Sears Tower from garbage. If 7.8 billion tons of waste (municipal and industrial combined) is packed at a density of 1500 pounds per cubic yard, about 2 billion pounds, or 1 million tons, would make a tower of approximately the right size. We could build over 8200 Trash Towers each year, about 23 per day. The reality, for good or bad, is that we don't have to bury that many towers of trash because most of the industrial waste is released into bodies of water, and a third of the MSW is recycled or incinerated. 9 But some of it blows, or is dumped, into the sea.

Planet Earth has seven major tropical oceanic gyres, where high-pressure vortices of hot tropical air swirl above water, endlessly circling a central depression. The largest of these, the North Pacific Subtropical Gyre, is 10 million square miles, about the size of Africa. Oceanographers call it the Great Pacific Garbage Patch. In 1997, the Gyre was calculated to contain 3 million tons of plastic on the surface alone. Although the world's merchant fleet dumps at least 8 million pounds of plastic overboard each year, or 639,000 plastic containers each day, 80 percent of the floating plastic originated on land. The plastic will last practically indefinitely, as there is no current biological process to degrade most plastics. 10

Some of the plastic on the surface and exposed to sunlight does photodegrade, but breaking it into smaller pieces does not necessarily solve the problem. These smaller pieces are mistaken for fish eggs or krill, as are the plastic resin beads known as nurdles and the microscopic plastic exfoliant pellets in hand creams, body scrubs, and other feminine beauty products. Filter feeders and other sea creatures are filling up with plastic pellets and dying, 11 or being eaten and passing the plastic up the food chain.

Eating plastic is not good for you. Besides the physical blockages that can occur in the digestive system, many plastics contain compounds that are endocrine disruptors. These compounds mimic the hormone estrogen and, thus, destroy the balance that controls much of the development and functions of the organs of the reproductive system. Pthalates, bisphenol A (BPA), and dioxins are three of the compounds in this class that are used in, and leach out of, plastics. Additionally, the plastic particles bind with other toxins such as dichloro-diphenyl-trichoroethane (DDT). Plastic particles such as nurdles can attract and magnify the concentration of toxins from the surrounding water. 12 The plastic pellets attract the toxins, which accumulate in the tissues of the plankton, and then the concentrations of toxins are magnified again as consumers eat these simpler organisms at the base of the food chain.

These processes are known as bioaccumulation and biomagnification. Bioaccumulation is the process in which relatively long-lived compounds, from exposure to contaminated air, food, and water, build up in the body because they are absorbed but not rapidly metabolized or excreted. 13 Similarly, biomagnification occurs when these long-lived, generally fat-soluble toxins accumulate in higher concentrations as lower organisms are eaten by those higher on the food chain. These two terms are often used somewhat interchangeably. 1 4 Probably the most well known example of this effect is the near-extinction of the American bald eagle because of infertility and fragile-egg syndrome caused by the accumulation of DDT. Other examples are similarly catastrophic. In 1967, one study found DDT in sea water at a concentration of 50 parts per trillion, plankton at 40 parts per billion (ppb), 1000 ppb or 1 part per million (ppm) in fish, and 26 ppm in cormorants, a predatory bird at the top of the food chain—a magnification by a factor of 520,000. In a similar case in 1957, western grebes (an aquatic bird) were killed by the application of DDD to kill gnats on Clear Lake north of San Francisco. DDD was applied twice to the water in concentrations of 1/70 th ppm and 1/50 th ppm, but plankton contained a DDD level measured at 5 ppm, the fish between 40 and 300 ppm, and the fatty tissue of the dead grebes 1,600 ppm. 15 Biomagnification of toxins can be disastrous for predators at the top of a food chain.

People are top predators for many food chains. A 1969 sampling of water intake pipes from Lake Michigan into the Chicago supply found lindane, heptachlor epoxide, aldrin, and DDT. 16 FDA food samplings between 1982 and 1984 commonly found the pesticides DDE, malathion, dieldrin, pentachlorphenol, BHC-alpha, diazinon, HCB, and heptaclor epoxide. A later FDA sample from 1984-1991 found the fungicide captan, the insecticide chlorpyrifos, and the pesticide daminozide (Alar) in foods commonly eaten by children. 17 Although some of those compounds have been banned in the U.S., the compounds have already accumulated in the fatty tissues of those who were exposed and may not be removed for a very long time. John Wargo writes, "If exposure were stopped, it would take the average person twenty years to eliminate all DDT residues, and it appears that the DDE residues can never be fully eliminated." 18

Even for students born after the banning of these substances in the United States, bioaccumulation is still occurring. Much of our food is grown abroad from countries that still use some of these compounds. Some of these banned compounds are produced in the United States and then exported to these same countries that grow the food we import. The United Nations Environment Programme identified twelve persistent organic pollutants (POPs) for continued monitoring in 1995: aldrin, chlordane, DDT, dieldrin, endrin, heptachlor, mirex, and toxaphene (pesticides), PCBs and hexachlorobenzene (industrial chemicals), and dioxins and furans (industrial by-products). 19 Even if students never encountered any of these POPs in their food, air, or water, they would still carry a load of these toxins from the most intimate source of all—their mother's milk.

The human breast is largely fatty tissue with a high blood supply and, thus, tends to accumulate fat-soluble toxins. These toxins are released during significant weight loss and milk production in nursing mothers. The organochlorine toxins are even able to pass from mother to child across the placenta before birth. 2 0 Paradoxically, an extreme example of mothers passing their bioaccumulated toxins to their infants has occurred in a population seemingly far removed from industrial pollution, the indigenous arctic peoples. Volatile POPs evaporate in warmer climates and are transported by wind and water to the arctic. Arctic peoples have a traditional diet rich in fat and blubber from animals that may already have biomagnified toxins. Research in 1987 and more recently revealed that women in the arctic have higher levels of PCBs in their breast milk than those living farther south, as well as higher levels of other POPs and heavy metals like mercury and lead. 2 1

Although lead is not a synthetic organic compound or fat soluble, it does accumulate in body tissues. The portion of ingested or inhaled lead that is absorbed by the body depends upon the source and mechanism, as well as the age of the individual and many other factors. Some of the absorbed lead is stored in blood and soft tissues with a relatively short half-life of 1 to 2 months and 70% of the excretion occurring via urine. What lead is not quickly excreted, generally 50% of the absorbed lead, is then incorporated into bone with a half-life of years to decades. Via these processes, about 90% of the stored lead in adults is located in the skeleton. 22

Chicago is one of the cities with historically high levels of lead poisoning in children. According to a 2002 report from the Loyola University Chicago School of Law, 23 "Illinois has the largest number of identified cases of childhood lead poisoning in the nation. In some communities in Chicago, one-third of neighborhood children test positive for elevated blood levels…. Almost 7% (340,000) of all housing units in Illinois are considered at high risk for lead-based paint hazards. Of these units, over 120,000 are in Chicago." These at-risk units are in older buildings, many built before 1950. More than three quarters of the homes built before 1980 contain some lead-based paint. Additionally, Chicago has high lead soil levels from automobile exhaust during the period of leaded gasoline use from 1928 to the late 1970's. 24 According to the Chicago Department of Public Health, "While lead poisoning continues to decline, over 11% of children tested in Chicago were identified with lead poisoning in 2001." 25 However, by 2007, the overall rate for children ages 0-6 years in Chicago testing positive for elevated blood lead levels was 2.5%. 2 6

The major source of lead exposure for children is lead dust from lead paint. Paint on weathered and deteriorated surfaces, especially around windows and doors, flakes and crumbles. Children ages 0-3 years may directly eat the paint, but more commonly inhale or ingest lead dust picked up throughout the residence through normal hand-to-mouth activity. This same activity will cause ingestion of lead from contaminated soil around the home and throughout the neighborhood. 27 Minor sources of lead also come from fossil fuel air pollution, water in lead or lead-soldered pipes, bullets, mining operations, and the work clothing of adults in lead-related industries. 28

In 1991, the Centers for Disease Control and Prevention (CDC) established 10 Μg/dL as the lead blood level that would define lead poisoning in children. However, there is no known threshold concentration at which lead's deleterious effects begin. The dominant effect of lead poisoning is upon the brain and nervous system. Children with lead poisoning exhibit decreased cognitive functioning and developmental and behavioral difficulties and delays. Lead has detrimental effects on multiple systems and processes in the body—interfering with the function of other metal ions in the body such as calcium, zinc, and iron, altering gene expression, causing kidney damage, inhibiting certain enzymes, increasing blood pressure, and possibly causing reproductive damage and promoting cancer. 29

In this unit, it would be desirable to be able to accurately measure and model student exposure to environmental lead and compute accumulated lead levels. In practice, this is extremely difficult. Statistical models such as the U.S. EPA's integrated exposure uptake biokinetic (IEUBK) model for childhood lead exposure make many assumptions regarding exposure and absorption rates based on environmental and biologic measurement sampling. However, the interactions between all the components of the model are quite complex, as illustrated in Figure 1 below. 30,31 Because of the complexities of the interactions, the models may not accurately predict actual blood lead levels (BLLs). 32 Biomonitoring is an alternate, complementary approach that avoids the inaccuracies of interpretation in environmental modeling by taking actual blood or urine level measurements of environmental chemicals to assess exposure. 33


EPA's IEUBK computer model can be downloaded 34 and used to model student lead exposure and BLLs if one has the resources to accurately measure the necessary inputs. In this unit, we will use the default settings 35 from the IEUBK as our exposure and absorption data rates for students to use calculus integration techniques to compute cumulative amounts of lead in an individual child at various ages.


Stage one is for reading, research, reflection, and reporting. Students will be given a few articles and/or short chapters dealing with the problems of municipal solid waste (MSW) disposal to read. The reading material can be introduced in class and then assigned as homework. Students will write in their calculus journals their personal reactions to the articles, as well as predictions or insight regarding how the readings connect to calculus. At this stage, the objectives are to build student awareness and begin to change attitudes toward the current state of MSW disposal. Additionally, students should recognize that issues dealing with rates of production or disposal are real world applications of the derivative.

Students will then be assigned to one of seven cooperative groups. Each group will be given a subtopic to research within the broader theme of municipal solid waste disposal. Teams will then use the internet to research their topic, answer a set of focus questions, and prepare for a presentation to the class. The following day each team will present the results of its research in a creative, original way. Students will be encouraged to use technology such as PowerPoint and the SmartBoard (a touch-sensitive whiteboard and projection screen linked to a computer and digital projector) appropriately.

In stage two, we will begin data mining the figures supplied by the 2007 EPA MSW Report. This report presents results graphically, numerically in table form, and verbally in summary comments. Students will compute derivatives from each of these representations. Students will calculate the slope of the line tangent to the graph of the total MSW generation from 1960 to 2007. Students will use a difference quotient (slope) calculation to find average rates of change and approximate instantaneous rates of change of MSW generation, recovery, and disposal. Students will express their answers in appropriate units and explain what the numerical values mean in the context of MSW processing. Students will express verbal statements from the report in symbolic form as well.

As time permits, one extension of this process is to calculate the derivative of MSW production, recovery, or disposal analytically (symbolically) from the numerical data provided. This will require first calculating a regression curve from the raw data. Various statistical analysis tools could be used for the task. In my class, students would use Texas Instruments graphing calculators from the TI 83/84 family to compute the function that best models the data. Analytic calculations of the derivative can be compared with the results obtained through numerical approximations earlier in this stage.

A second extension of this process could be used to set the stage for the introduction of the integral as a Riemann sum if it has not been introduced earlier. Students could plot the data with years as the independent variable and quantity as the dependent variable, and then calculate the rate of change of MSW production for each and every reported year of one specific category of waste, as begun earlier in this stage. Students would record the rate of change data in tabular form, in the format previously used for the total change, and plot the points with years as the independent variable and the rate as the dependent variable. Then, using rectangular and/or trapezoidal regions, students would back-calculate the accumulated change each year. An interesting teaching opportunity arises in noting that a starting value is required to have the quantity values each year match the original data set.

Stage three will return to research and reporting. Students will be assigned to pairs or trios to research each of the persistent organic pollutants (POPs) of concern identified by the UN Environmental Programme. Each team will complete a worksheet of focused questions on the nature and effects of its POP. Each team will then prepare for a brief report to the class. As a tie-in to the next stage, one team could be assigned lead as a toxin of concern, although it is not an organic pollutant. Alternatively, the teacher could present a report on lead at the end of the student presentations.

Stage four will apply integral calculus to the accumulation of lead as a persistent, bioaccumulative toxin (PBT). The teacher will present information about lead poisoning in Chicago, including the breakdown of exposure by community. 36 The teacher will also show the CDC's Flash multimedia presentation 3 7 about biomonitoring and the impact it has had on lead poisoning rates in the United States. Students will be given the default exposure rates used by the EPA's IEUBK computer model for ages 0 to 7. Using rectangular Riemann sums, students will calculate the cumulative exposure, which emphasizes the integral as a measure of accumulated change. Significant attention must be paid to units at this stage. The exposure levels are in micrograms per day, while the time intervals are in years. This will afford many chances to emphasize the importance of units in context when dealing with rates of change.

Some of the exposure rates in the model are broken down into interrelated units that would require dimensional analysis to generate a rate in micrograms per day. This would provide for a profitable extension. Additionally, the IEUBK adult data could be combined with O'Flaherty's data on bone calcium and lead uptake and turnover 3 8 in adult years to perform an integration for a simulated adult. A further extension would be to apply the activity of mercury accumulation designed by Frederick Atkins. 39

The final phase of this unit is to allow students the opportunity to reflect and respond. A culminating lesson will introduce the principles of green chemistry, especially principle number ten—"Design chemicals and products to degrade after use: Design chemical products to break down to innocuous substances after use so that they do not accumulate in the environment." 40 Students will again write in their calculus journals their feelings and reactions to what they have learned about the need for green chemistry. They will describe how differential and integral calculus apply to green chemistry. Finally, each student will write down two or three changes that he or she has determined to make in his or her own life in response to the things learned about the state of the environment during this unit. Students will be invited to share their decisions with the class.

Classroom Activities

Activity #1: Reading and Research

Part A

Instructions to students:

Read the following three items:

•"High Tech Trash," National Geographic, January 2008, by Chris Carroll, (available online at http://ngm.nationalgeographic.com/2008/01/high-tech-trash/carroll-text)

•"Chapter 9: Polymers Are Forever" in The World Without Us, by Alan Weisman

•"Plastics" in What We Leave Behind, by Derric Jensen and Aric McBay

In your journal, respond to the following two questions:

  1. As you reflect on your readings, how do you feel, and/or what impressed you most about what you learned?
  2. How do the readings relate to calculus? In other words, can you think of any real-world contexts in what you read that are connected to the themes we've been talking about in class?

Part B

Instructions to students:

With your research team, you are going to find basic information about one of the seven groups of plastics identified by recycling code.

•Start your research by visiting the two web sites listed below. As you have time, you may extend your search to other sites, but maintain your focus.



•Complete each of the sections of the worksheet below.

•After you have completed the worksheet, plan and begin working on a brief presentation for the whole class about your plastic type. If you choose to do a multimedia presentation, you may use the projector and SmartBoard in class. Please also send a copy to your teacher via email so that your presentation can be posted on the class site.

Recycling Code (Number and abbreviations) Common name(s)
Is used to produce… Can be recycled into…
Basic chemical unit(s) Other interesting information

Activity #2: Rates of change in the 2007 EPA Municipal Solid Waste Report

Part A

Use the graph below (Figure 2) to answer the questions about the rate of change.


  1. Use a difference quotient to find the average rate of change in plastics generation from 1986 to 2003. Make sure to include appropriate units.
  2. Find the average rate of change in plastics recovery from 1960 to 1985. What does that answer mean?
  3. Find the average rate of change in plastics recovery from 1987 to 2007.
  4. If G(x) represents the number of millions of tons of plastic generated in year x, and G(1999) = 24,
    1. find G'(1999).
    2. Explain, with correct units, what G'(1999) means.
    3. Describe how you found your answer.
  5. If R(x) is the quantity of plastic recovered (by recycling or incineration) in year x, and W(x) = G(x) - R(x) is the quantity of plastic discarded as waste, find W'(1999).
  6. Is W'(x) positive, negative, or zero on the interval 1996 x 2004? What does that mean in the context of the situation? What implications does that trend have for society?


Part B

  1970 1980 1990 2000 2004 2005 2006 2007
t 0 10 20 30 34 35 36 37
G(t) 10 490 960 1520 1650 1650 1650 1830
R(t) 0 10 160 500 520 520 510 600
W(t) 10 480 800 1020 1130 1130 1140 1230

This table provides information about plastic beverage bottles, in thousands of tons. Let t represent the number of years since 1970. G(t) represents the weight of plastic generated for drink bottles in year t, while R(t) is the weight of bottles recovered and W(t) is the weight of bottles discarded as waste.

  1. Find the average rate of change of production of plastic drink bottles from 1980 to 2000. Express your answer in correct units.
  2. Approximate the instantaneous rate of change of production of plastic drink bottles in the year 2002. Explain why you used the method you chose.
  3. Approximate W'(35), the derivative of W when t = 35. What does this value mean?
  4. Approximate W'(37). Based on other values in the table, how does this rate compare to others? What are the implications in the real world?


Part C (Extension using graphing calculators to calculate a regression curve.)

Instructions to students:

  1. Using your graphing calculator in STAT mode, enter the values from the data table from Part B for years and plastic beverage bottles discarded as waste. Use the values for t as the independent variable in L1 and W(t) values for L2.
  2. Plot the data as a scatter plot using the STATPLOT menu and ZOOM STAT.
  3. After looking at the shape of the scatter plot, choose a regression model from the STAT CALC menu. Identify your independent and dependent variables as L1, L2 and graph the result of your regression calculation. If it is not a good fit, try a different model.
  4. When you think you have a good match, record the equation of the regression model: y = __________________________
  5. Use the rules of derivatives to calculate y'.
  6. Find y'(25) and y'(37). How do these results compare to W'(25) and W'(37)?

Worked Examples:

The following series of screens captured from a TI-84 + graphing calculator illustrate the steps the students should take to plot the data and calculate a regression model.


Activity #3: Research

Break the students into twelve teams. By drawing strips of paper, randomly assign each team one of the twelve persistent organic pollutants (POPs) identified for continued monitoring in 1995 by the UN Environmental Programme: aldrin, chlordane, DDT, dieldrin, endrin, heptachlor, mirex, toxaphene, PCBs, hexachlorobenzene, dioxins, and furans. Students may find most of the needed information directly from the UN report 41, but they may be encouraged to search for additional information. Each group should answer the following questions and include this information in a brief report to the class, as in Activity #1.

  1. What is the name of the POP?
  2. When was it first produced?
  3. What is it used for?
  4. How are people exposed to the POP?
  5. What are the hazards of exposure?
  6. Is it still produced in the US? If not, when was production stopped?

Activity #4: Accumulated change from the EPA's Integrated Exposure Uptake Biokinetic (IEUBK) Model for Lead in Children

t 0 1 2 3 4 5 6
E(t) 2.26 1.96 2.13 2.04 1.95 2.05 2.22

This table gives the default values for rates of childhood exposure to lead in food. The child's age is given in years and indicated by t, while E(t) is the exposure rate in micrograms (Μg) per day.

Part A

  1. Graph the data in the first quadrant of a coordinate grid using correct labels and an appropriate scale.
  2. Draw a rectangle for each interval. Let t mark the right-hand edge of each rectangle, and let E(t) be the height of each rectangle. In other words, the point (t, E(t)) is the upper right corner of each rectangle.
  3. Find the area of each rectangle, using appropriate units. (Hint: multiply the width of your rectangles by 365 to convert years to days.)
  4. Add up all the areas. This is called a Riemann sum. Record your result.
  5. What does this result represent in the context of the problem?

Part B

  1. Repeat the process from Part A using trapezoids instead of rectangles. You may recall that the formula for the area of a trapezoid is A t r a p = (h/2)*(b 1 + b 2). In this case, h is the width of the interval and b 1 and b 2 are the values of E(t) at the upper left and right corners of the trapezoids.
  2. Compare the sum of the areas from Part A with the sum of the areas from Part B. What do you notice? Try to explain the reason for this result.

Part C

We want to approximate the total amount of lead a child would ingest in food up to the day he or she turns 7 years old. Devise a procedure for doing this that could be somewhat better than what you did in Parts A and B. Explain your procedure in words, draw a picture to support it, and show your calculations.

Worked Examples: The graphs for Parts A and B are shown below.


Note that, using right-hand endpoints, there is no rectangle represented by the value E(0)=2.26, and no area represents the lead accumulation from age 6 to 7 years.

For Part B, the width of the intervals is the same, and, using trapezoids, the area is


We notice that this area is very similar to, but slightly larger than, the previous result of 4.508 mg. Students might mention that using trapezoids averages out highs and lows, or that the upper boundary appears to more smoothly match a function curve if we assume the exposure rate is really continuous instead of discrete.

Two possible graphs for part C are shown below. The crucial idea is that our previous methods neglected the interval from 6 to 7 years of age. We can let t mark the left-hand edge of each rectangle, and let the point (t, E(t)) be the upper left corner of each rectangle. With trapezoids, we can duplicate the last value and add a rectangle. Either of these methods more closely matches the intent of the IEUBK than our methods in Part A and B. Students might devise other similar, reasonable methods.



Adkins, F.A. and Gyasi, W.K. "A Calculus Module for Modeling Bioaccumulation, Biomagnification, and Elimination of Mercury", http://hawk.ma.iup.edu/mercury/ (accessed May 16, 2009). An explanation of bioaccumulation and classroom activity to compute individual student mercury levels.

Air Quality Criteria for Lead. Research Triangle Park, NC: U. S. Environmental Protection Agency, June 1986. http://cfpub2.epa.gov/ncea/cfm/recordisplay.cfm?deid=32647 (accessed August 13, 2009). Scientific review of the literature through 1985 of the effects of lead as an environmental pollutant.

Anastas, Paul T., and John C. Warner. Green Chemistry: Theory and Practice. New York: Oxford University Press, USA, 2000. The work that codified the principles of green chemistry.

Binns, H.J. "Lead Poisoning: Still a Common Problem in Chicago-Child's Doctor Spring 2001." Child's Doctor, Winter 2003, Children's Memorial Hospital, Chicago. http://www.childsdoc.org/spring2001/leadpoisoning.asp (accessed July 12, 2009). A discussion of the most significant causes of childhood lead poisoning in Chicago.

"Biomonitoring: Making a Difference." Department of Health and Human Services, Centers for Disease Control and Prevention, National Biomonitoring Program. http://www.cdc.gov/biomonitoring (accessed July 14, 2009). A multimedia presentation and transcript about biomonitoring and blood lead levels.

Boardman, Michael. et al. The College Board AP Calculus AB Calculus BC Course Description May 2010-2011. New York: The College Board, 2009. The definitive curriculum guide for the AP Calculus program.

Carson, Rachel. Silent Spring. New York: Houghton, 1962. This eloquent yet thoroughly researched book is credited with starting the modern environmental movement.

"Chicago Department of Public Health, Childhood Lead Poisoning Prevention Program." City of Chicago. http://egov.cityofchicago.org (accessed July 13, 2009). Information about lead poisoning in Chicago neighborhoods and lead abatement programs.

Esty, Daniel, and Andrew Winston. Green to Gold: How Smart Companies Use Environmental Strategy to Innovate, Create Value, and Build Competitive Advantage. New York, NY: Wiley, 2009. A guide to making green chemistry principles profitable in business.

Jensen, Derrick, and Aric McBay. What We Leave Behind. Seven Stories Press, 2009. An environmental screed against an unsustainable culture of waste.

Municipal Solid Waste in the United States: 2007 Facts and Figures. U. S. Environmental Protection Agency, Office of Solid Waste, November 2008. http://www.epa.gov/osw/nonhaz/municipal/msw.99.htm (accessed July 6, 2009). Complete statistics on MSW from 1960-2007.

Mushak, P. "Uses and Limits of Empirical Data in Measuring and Modeling Human Lead Exposure," in Environmental Health Perspectives, Vol 106, Supplement 6, Dec. 1998, 1467-1484. This paper examines the challenges in modeling lead exposure.

O'Flaherty, Ellen J. "Physiologically Based Models for Bone-Seeking Elements. V. Lead Absorption and Disposition in Childhood," in Toxicology and Applied Pharmacology, Vol 131, 1995, 297-308. A detailed discussion of the differences in lead absorption over time.

Ritter, L., Solomon, K.R. and Forget, J. Persistent Organic Pollutants, An Assessment Report on: DDT-Aldrin-Dieldrin-Endrin-Chlordane-Heptachlor-Hexachlorobenzene-Mirex-Toxaphene-Polychlorinated Biphenyls-Dioxins and Furans. International Programme on Chemical Safety, 1995. A report to the United Nations Environment Programme on the chemistry, toxicology, relevant transport pathways and the origin, transport and disposition of the common persistent organic pollutants.

Third National Report on Exposure to Environmental Chemicals. Atlanta: Department of Health and Human Services, Centers for Disease Control and Prevention. National Center for Environmental Health, July 2005. Detailed information about the health effects of dozens of chemicals monitored by the CDC.

"Twelve Principles of Green Chemistry | Green Chemistry | US EPA." U.S. Environmental Protection Agency. http://www.epa.gov/greenchemistry/pubs/principles.html (accessed July 24, 2009). A reworded online version of Paul Anastas' green chemistry principles.

User's Guide for the Integrated Exposure Uptake Biokinetic Model for Lead in Children (IEUBK) Windows?, U. S. Environmental Protection Agency, Office of Superfund Remediation and Technology Innovation, May 2007. www.epa.gov/superfund/lead/products.htm#ieubk (accessed July 13, 2009). Software model to calculate blood lead levels for individuals or communities.

Wargo, John. Our Children's Toxic Legacy: How Science and Law Fail to Protect Us from Pesticides. New Haven: Yale University Press, 1998. A primer on the history of pesticides, their effects, and what we can do about it.

Weinberg, Anita. "Childhood Lead Poisoning: Issues, Policies, Initiatives," in Public Interest Law Reporter, Spring 2002. www.luc.edu/law/academics/special/pdfs/leadpoison.pdf (accessed July 12, 2009). A summary of the causes and current state of lead poisoning in Chicago with litigation and other efforts to remediate conditions in disadvantaged neighborhoods.

Weisman, Alan. The World Without Us. New York: St. Martin's Press, 2007. An accessible book about what would, and would not, remain if humanity suddenly disappeared.

Student Reading List

"High Tech Trash," National Geographic, January 2008, by Chris Carroll, (available online at http://ngm.nationalgeographic.com/2008/01/high-tech-trash/carroll-text).

"Chapter 9: Polymers Are Forever" in The World Without Us, by Alan Weisman.

"Plastics" in What We Leave Behind, by Derric Jensen and Aric McBay.

"Is Recycling Worth It?" Popular Mechanics, December 2008, by Alex Hutchinson

Classroom Resources (web resources accurate as of August 13, 2009)

CDC, "Biomonitoring: Making a Difference," http://www.cdc.gov/biomonitoring/presentation.htm. A multimedia presentation about biomonitoring and blood lead levels.

Chicago Department of Public Health, http://egov.cityofchicago.org. "Childhood Lead Poisoning Prevention Program," "Data on Lead Poisoning in Chicago," "Elevated_BLL_by_CommArea_2007.pdf." A neighborhood-by-neighborhood map of the percentages of children with unsafe blood lead levels.

EPA. User's Guide for the Integrated Exposure Uptake Biokinetic Model for Lead in Children (IEUBK) Windows?, www.epa.gov/superfund/lead/products.htm#ieubk Software model to calculate blood lead levels for individuals or communities and user's guide with printed exposure rates.

EPA, Municipal Solid Waste in the United States: 2007 Facts and Figures. Complete statistics on MSW from 1960-2007.

Websites with information about the different plastic polymer types, organized by recycling code number.



Appendix: Implementing District Standards

This unit is intended to be used in an 11 th or 12 th grade mathematics course with integrated green chemistry content. Below are the Illinois State Goals for Mathematics that are addressed significantly in this unit.

STATE GOAL 6: Demonstrate and apply a knowledge and sense of numbers….

A. Demonstrate knowledge and use of numbers and their representations in a broad range of theoretical and practical settings.

C. Compute and estimate using mental mathematics, paper-and-pencil methods, calculators and computers.

STATE GOAL 7: Estimate, make and use measurements of objects….

C. Select and use appropriate technology, instruments and formulas to solve problems, interpret results and communicate findings.

STATE GOAL 8: Use algebraic and analytical methods to identify and describe patterns and relationships in data, solve problems and predict results.

This unit addresses goal 6 in that one of the fundamental purposes is to extend the range of theoretical and practical settings to which calculus is applied to include more chemistry. Data sets regarding municipal solid waste disposal and human exposure to persistent environmental toxins are used as a real world context for students to estimate derivatives and integrals in a variety of ways. This unit directly addresses goals 7.C.5a (use dimensional analysis to determine units and check answers in applied measurement) and 7.C.5b (determine how changes in one measure may affect other measures) for the late high school grades. This unit strongly implements goal 8 in providing "verbal, symbolic, and graphical formats for representing [diverse] settings," and challenging students to "examine tables of values; to interpret the relationships expressed by patterns in these tables; to relate change and variation in graphs and formulas; to reason about changes in quantities and the relationships involved in changes; and to find solutions to everyday problems using algebra's symbolic manipulation and formulas."

Note that this unit also addresses the ACT College Readiness Standards for Mathematics at the 33-36 level for Graphical Representations in that it requires students to "solve problems integrating multiple algebraic and/or geometric concepts," and to "analyze and draw conclusions based on information from graphs in the coordinate plane."

Finally, it implements most of the major goals of the AP Calculus AB curriculum. The unit requires students to apply both the derivative and the integral to a model of a physical situation. Students must work with the model and communicate results in a variety of ways. The unit connects the derivative to the integral, encourages the use of technology to solve problems and interpret results, and requires students to determine the reasonableness of results expressed with proper units.


1 2008-09 Development Committee and Chief Reader, The College Board AP Calculus AB Calculus BC Course Description, 5.

2 P.T. Anastas and J.C. Warner, Green Chemistry: Theory and Practice, 30.

3 EPA, http://www.epa.gov/greenchemistry/pubs/principles.html.

4 Anastas, 10.

5 D.C. Esty and A.S. Winston, Green to Gold: How Smart Companies Use Environmental Strategy to Innovate, Create Value, and Build Competitive Advantage, 106-107.

6 ibid, 197-199.

7 R. Carson, Silent Spring, 68-69.

8 EPA, Municipal Solid Waste in the United States: 2007 Facts and Figures, 1.

9 D. Jensen and A. McBay, What We Leave Behind, 292-293.

10 A. Weisman, The World Without Us, 115-126.

11 Ibid.

12 Jensen and McBay, 108-115.

13 F.A. Adkins and W.K. Gyasi, "A Calculus Module for Modeling Bioaccumulation, Biomagnification, and Elimination of Mercury", http://hawk.ma.iup.edu/mercury/ (accessed May 16, 2009).

14 J. Wargo, Our Children's Toxic Legacy, 139.

15 Ibid., 140-141.

16 Ibid., 147.

17 Ibid., 158-159.

18 Ibid., 168.

19 L. Ritter, K.R. Solomon, and J. Forget, "Persistent Organic Pollutants, An Assessment Report on: DDT-Aldrin-Dieldrin-Endrin-Chlordane-Heptachlor-Hexachlorobenzene-Mirex-Toxaphene-Polychlorinated Biphenyls-Dioxins and Furans".

20 Wargo, 168-169.

21 Jensen and McBay, 144.

22 CDC, Third National Report on Exposure to Environmental Chemicals, 40.

23 A. Weinberg, "Childhood Lead Poisoning: Issues, Policies, Initiatives," www.luc.edu/law/academics/special/pdfs/leadpoison.pdf.

24 H. J. Binns, "Lead Poisoning: Still a Common Problem in Chicago," www.childsdoc.org/spring2001/leadpoisoning.asp.

25 Chicago Department of Public Health, Childhood Lead Poisoning Prevention Program, http://egov.cityofchicago.org.

26 Chicago Department of Public Health, Childhood Lead Poisoning Prevention, http://egov.cityofchicago.org, "Map of Childhood Lead Poisoning in Chicago," Elevated_BLL_by_CommArea_2007.pdf.

27 Binns.

28 CDC, Third National Report on Exposure to Environmental Chemicals, 38-39.

29 Ibid., 39-40.

30 P. Mushak, "Uses and Limits of Empirical Data in Measuring and Modeling Human Lead Exposure," 1469.

31 EPA, Air Quality Criteria for Lead, Vol. 1, 12

32 E.J. O'Flaherty, "Physiologically Based Models for Bone-Seeking Elements. V. Lead Absorption and Disposition in Childhood," 297-308.

33 CDC, "Biomonitoring: Making a Difference," transcript, www.cdc.gov/biomonitoring/pdf/flash_transcript.pdf.

34 EPA, www.epa.gov/superfund/lead/products.htm#ieubk.

35 EPA, User's Guide for the Integrated Exposure Uptake Biokinetic Model for Lead in Children (IEUBK) Windows?, 11-13.

36 Chicago Department of Public Health, Childhood Lead Poisoning Prevention, http://egov.cityofchicago.org, table of "Data on Lead Poisoning in Chicago," Table_BLL_by_CommArea_2007.pdf, "Map of Childhood Lead Poisoning in Chicago," Elevated_BLL_by_CommArea_2007.pdf.

37 CDC, "Biomonitoring: Making a Difference," Flash multimedia presentation, http://www.cdc.gov/biomonitoring/presentation.htm.

38 O'Flaherty.

39 Adkins and Gyasi.

40 EPA, http://www.epa.gov/greenchemistry/pubs/principles.html.

41 Ritter et al, www.pops.int/documents/background/assessreport/en/ritteren.pdf.

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