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-116731-31559500Printable Resources“Excel”ing after the ApocalypseAppendix A:Pre/Post Test Part 1 (Content Knowledge)Appendix B:Pre/Post Test Part 1 (Content Knowledge) - KeyAppendix C:Pre/Post Test Part 2 (Excel Knowledge)Appendix D:Pre/Post Test Part 2 (Excel Knowledge) - KeyAppendix E:Engineering Design ProcessAppendix F:Background StoryAppendix G:Engineering Design Challenge ExplanationAppendix H:“Pandora’s Box” Discussion GuideAppendix I:“Life Under the Bubble” (Discover Magazine, 2010)Appendix J: Desmos Teacher Instructions and Guiding QuestionsAppendix K: Linear vs. Quadratic vs. Exponential worksheetAppendix L: Food Web Simulation HandoutAppendix M: Food Web Simulation Data TableAppendix N: “What is Carrying Capacity” Article –Hard CopyAppendix O: Degree of Impact CardsAppendix P: Lab NotebookAppendix Q: Student CheckpointsAppendix R: Engineering Design Challenge- Part 1: Basic Excel Tutorial Appendix S: Engineering Design Challenge- Part 1: Basic Excel Tutorial KeyAppendix T: Engineering Design Challenge- Part 2: Biosphere Population SimulatorAppendix U: Engineering Design Challenge- Part 2: Biosphere Population Simulator -KeyAppendix V: Engineering Design Challenge- Part 3: Design Optimization Appendix W: Engineering Design Challenge- Part 3: Design Optimization KeyAppendix X: Challenge CardsAppendix Y: Engineering Challenge RubricAppendix Z: Preparing to Present Scientific Conclusions-Student Preparation RubricAppendix AA: Presentation Rubric – Teacher VersionAppendix BB: Presentation Rubric- Student Self Reflection Appendix CC: Peer Evaluation RubricAppendix A: Content Pre/Post Test Indicate whether each graph is showing exponential growth OR exponential decay. Circle one answer. (1 pt each)29413201650050069011162632001. Exponential GrowthExponential Growth Exponential DecayExponential Decay Indicate whether each equation is showing exponential growth OR exponential decay. Circle one answer. (1 pt each)2. Exponential Growth Exponential GrowthExponential Decay Exponential Decay3. Write an exponential function that models the points in each table. (3 pts each)XY1802203541.25XY1-32-93-274-814. State whether each example is an example of linear, quadratic, or exponential. (1 pt each) 3x+2y=104095756223000 b. c. 685800-14224000685800280098500 d. y= x2-6x+13 e. 5. In your own words, how would you describe carrying capacity (you may use pictures, words, or drawings)? (2pts) 6. Which could decrease the human carrying capacity? Circle all that apply. (1 pt)a. governmentsb. energy crisisc. epidemic diseased. unequally distributed resources7. The Earth’s human carrying capacity. Circle all that apply (1 pt)a. Is fixedb. May decrease over time because resources are being depletedc. May increase over time because of technological advancesd. Can be expanded indefinitelye. Cannot be predicted for certain8. Name four processes that determine whether a population will shrink or grow. (4 pts)9. Explain four limiting factors that can contribute to determining a population’s carrying capacity. (4 pts)4789805294640A00A2680335186055B00B-6286518605500Use the graph above to answer questions 10 and 11. (1 pt each)10. Graph A above is showing the… (1 pt)a. Linear growth of a population over 30 yearsb. Logistical growth of a population over 30 yearsc. Exponential growth of a population over 14 years d. Scatter plot of number of individuals over yearly growthe. I don’t know, I would have to guess11. The carrying capacity of a population under growth represented by graph A is ___________ Individuals. (1 pt)952501714500Use the predator-prey graph above to answer questions 12 and 13.12. What would happen to the moose population in year 2015 if the wolf population decreases? Explain. (2 pts)Looking at this graph as a conservationist, why is it important to make sure that the wolf population does not eat all the moose in the population? (2 pts)Appendix B: Content Pre/Post Test-KeyIndicate whether each graph is showing exponential growth OR exponential decay. Circle one answer. (1 pt each)29413201650050069011162632001. Exponential GrowthExponential Growth Exponential DecayExponential Decay Indicate whether each equation is showing exponential growth OR exponential decay. Circle one answer. (1 pt each)2. Exponential Growth Exponential GrowthExponential Decay Exponential Decay3. Write an exponential function that models the points in each table. (3 pts each)4686300100330F (x) = -1 (3)x3 pts 1 pt. = Correct formula1 pt. = Starting value1 pt. = Correct rate 00F (x) = -1 (3)x3 pts 1 pt. = Correct formula1 pt. = Starting value1 pt. = Correct rate 1457325128905F (x) = 320 (.25)x3 pts 1 pt. = Correct formula1 pt. = Starting value1 pt. = Correct rate 00F (x) = 320 (.25)x3 pts 1 pt. = Correct formula1 pt. = Starting value1 pt. = Correct rate XY1802203541.25XY1-32-93-274-814. State whether each example is an example of linear, quadratic, or exponential. (1 pt ea.) 3x+2y=104095756223000 b. 360997546990Answers: LinearExponentialExponentialQuadraticlinear0Answers: LinearExponentialExponentialQuadraticlinear5334004127500 c. d. y= x2-6x+13 5334008699500 e. 5. In your own words, how would you describe carrying capacity (you may use pictures, words, or drawings)? (2pts) PointsStudent Response2The number of individuals of a species that an ecosystem is capable of supporting.1The student partially mentions the population and no mention of how species have to be supported.0The student response demonstrates no understanding of the task. The response may provide incorrect information or be irrelevant to the task. The student may repeat information from the passage or prompt or may have written “I don’t know.”6. Which could decrease the human carrying capacity? Circle all that apply. (1 pt)a. governments (e.g., former one child policy in China)b. energy crisis (e.g., reduces availability of food & resources)epidemic disease (e.g., could wipe out the workforce, reducing food & resources)unequally distributed resources (e.g, large fraction of the population cannot afford to have families)7. The Earth’s human carrying capacity. Circle all that apply (1 pt)a. Is fixedb. May decrease over time because resources are being depletedc. May increase over time because of technological advancesd. Can be expanded indefinitelye. Cannot be predicted for certain8. Name four processes that determine whether a population will shrink or grow. (4 pts)Birth, death, immigration and emigration (1 pt each)9. Explain four limiting factors that can contribute to determining a population’s carrying capacity. (4 pts)4 points: Food availability (predator, prey, diet), water, ecological conditions (pollution, human activity, natural disaster, erosion, desertification, etc.), space (space to hide, competition for food, etc.)3 points: Discussed three of the concepts listed in “4 points.”2 points: Discussed two of the concepts listed in “4 points.” 1 point: Discussed one of the concepts listed in “4 points.”4789805294640A00A2680335186055B00B-6286518605500Use the graph above to answer questions 10 and 11. (1 pt each)10. Graph A above is showing the… (1 pt)a. Linear growth of a population over 30 yearsb. Logistical growth of a population over 30 yearsc. Exponential growth of a population over 14 years d. Scatter plot of number of individuals over yearly growthe. I don’t know, I would have to guess11. The carrying capacity of a population under growth represented by graph A is 1000 individuals. (1 pt)17145012700000Use the predator-prey graph above to answer questions 12 and 13.12. What would happen to the moose population in year 2015 if the wolf population decreases? Explain. (2 pts)2 points: The moose population would increase due to a decrease in predators.1 point: The moose population would increase.0 points: The moose population would decrease.Looking at this graph as a conservationist, why is it important to make sure that the wolf population does not eat all the moose in the population? (2 pts)2 points: A decreased amount of moose in the population would cause the prey of the moose to increase and cascade down through the food web causing an imbalance in the ecosystem.1 point: A decreased amount of moose in the population would cause the prey of the moose to increase. 0 points: There would be no effect on the ecosystemAppendix C: Excel Pre/Post Test Below is an example Excel spreadsheet for predicting the growth of a bacterial population. A linear model, ft=10t, is used to calculate the amount of food drops added to the petri dish. An exponential growth model gt=et, is used to calculate the bacterial population.1. Write the formula that should go into Cell 1 (hint: see the formula bar for the formula used in cell B3). (2pt.)2. Write the formula that should go into Cell 2 (hint: et is written as EXP(t) in Excel’s syntax). (2pt.)3. Which columns were used for the x-axis data and y-axis data to create the plot below, showing the linear increase in the food drops added to the bacteria? (2pt.)4. Which of the plots below gives a better representation of the growth of the bacterial population? Describe at least three features of the plots that support your choice. (3 pt.)-3048007302500Appendix D: Excel Pre/Post Test - KeyBelow is an example Excel spreadsheet for predicting the growth of a bacterial population. A linear model, ft=10t, is used to calculate the amount of food drops added to the petri dish. An exponential growth model gt=et, is used to calculate the bacterial population.1. Write the formula that should go into Cell 1 (hint: see the formula bar for the formula used in cell B3). (2pt.)=10*A91 pt. for the correct use of formula1 pt. for the correct use of data2. Write the formula that should go into Cell 2 (hint: et is written as EXP(t) in Excel’s syntax). (2pt.)=EXP(A5) or =exp(A5)1 pt. for the correct use of formula1 pt. for the correct use of data3. Which columns were used for the x-axis data and y-axis data to create the plot below, showing the linear increase in the food drops added to the bacteria? (2pt.)3190874231140Column A for the x-axis dataColumn B for the y-axis data1 pt. each0Column A for the x-axis dataColumn B for the y-axis data1 pt. each4. Which of the plots below gives a better representation of the growth of the bacterial population? Describe at least three features of the plots that support your choice. (3 pt.)-3619507302500392430070485Plot 2 is clearly superior. It has a title and x- and y-axis labels that give units, which allow us to interpret the plot, and uses a line plot to clearly represent growth over time, compared to deficiencies in the corresponding features of Plot 1.3 pt. The student makes a clearly written response identifying features of the plots that support their choice.2 pt. The student identifies multiple key features of the plots, but their response has misses an important plot feature or is not clearly written.1 pt. The student identifies a feature of the plot in their response; but, misses plot features and does not make a coherent statement.0Plot 2 is clearly superior. It has a title and x- and y-axis labels that give units, which allow us to interpret the plot, and uses a line plot to clearly represent growth over time, compared to deficiencies in the corresponding features of Plot 1.3 pt. The student makes a clearly written response identifying features of the plots that support their choice.2 pt. The student identifies multiple key features of the plots, but their response has misses an important plot feature or is not clearly written.1 pt. The student identifies a feature of the plot in their response; but, misses plot features and does not make a coherent statement.Appendix E: The Engineering Design ProcessThis representation of the engineering process is a first order approximation of what actually happens in an engineering task. The actual process is much less linear, often going from later steps in the cycle back to earlier steps as new information is gathered.34290038735STEP 1 Identify the need or problemSTEP 2 Research the need or problemSTEP 3Develop possible solution(s)STEP 5Construct a prototypeSTEP 4Select the best possible solution(s)STEP 8RedesignSTEP 7Communicate the solution(s)STEP 6Test and evaluate the solution(s)00STEP 1 Identify the need or problemSTEP 2 Research the need or problemSTEP 3Develop possible solution(s)STEP 5Construct a prototypeSTEP 4Select the best possible solution(s)STEP 8RedesignSTEP 7Communicate the solution(s)STEP 6Test and evaluate the solution(s)Identify the need or problemSpecify and prioritize requirements and constraints to better define the need or problemResearch the need or problemExamine current state of the issue and current solutionsExplore other options via the internet, library, interviews, etc.Develop possible solution(s)Brainstorm possible solutionsDraw on mathematics and scienceArticulate the possible solutions in two and three dimensionsRefine the possible solutionsSelect the best possible solution(s)Determine, using simple analysis, which solution(s) best meet(s) the original requirementsConstruct a prototype Model the selected solution(s) in two and three dimensionsTest and evaluate the solution(s)Does it work?Does it meet the original design constraints?Communicate the solution(s)Make an engineering presentation that includes a discussion of how the solution(s) best meet(s) the needs of the initial problem, opportunity, or needDiscuss societal impact and tradeoffs of the solution(s)Redesign Overhaul the solution(s) based on information gathered during the tests and presentationAppendix F: Background Story-8286752603500The Lucky 100You were watching the live feed when the nuclear warheads hit the asteroid. ?You sat with your stomach clenched and your heart pounding in your ears as the ten minutes of signal loss caused by the detonation of ten enormous warheads droned on in silence. ?Even the newscasters had nothing to say during those moments. ?As the grainy picture begins to slowly come back to life, you see the terrifying truth; the asteroid has not been destroyed. ?Instead, several large chunks of rock and debris are still heading toward earth. ?It was as all of the scientists feared, there was not firepower on the planet with enough energy to completely break apart the Texas-sized rock monster. ??You imagine the crews at NASA quickly re-calculating trajectories, making phone calls, wildly entering data into their computers. ?But you know that it is too late. ?The truth of what would happen if plan A failed had been mathematically and irrevocably determined as soon as the asteroid has been spotted. ?Instead, now broken into smaller pieces, it will now hit in multiple locations and the devastation will be the same. ?Earthquakes, tsunamis, resultant flooding, increased volcanic activity, and dramatic atmospheric changes will decimate the planet, wiping out all human life, and other populations as well. ?Climatologists, meteorologists, and geologists have estimated that it will take years for the waters to recede, the atmosphere to re-stabilize, and the earth to become able to sustain human life. ?You reach into your back pocket and take out the plain white key card. ?The card was handed to you by a government official just days ago. ?You were one of the lucky 100 people chosen by lottery to survive the coming global disaster and that credit card shaped piece of plastic is your “golden ticket”. ?You were instructed to be prepared to leave as soon as the official announcement of plan A’s failure was released. ?The transport teams would be coming to collect those lucky, chosen few at midnight. ??From what you were able to discern from the information packet given to you with your keycard, you will be transported to a top secret site out West, where fifty prefabricated biospheres have been placed, one for each state. ?The spheres, once designed for missions to Mars, will be able to protect the inhabitants from the coming devastation. ?The housing inside the spheres is modular (can be moved easily and rearranged when needed); there will be seeds and livestock inside the sphere. ?The spheres have been placed over a large water table, so water is abundantly available. ??Resources from all the residents will be pooled and used as needed within the sphere. -8401055207000Everyone in the sphere has been thoroughly screened and their job assigned. ?You have been assigned to the civil engineering team and must determine how the land will be divided between housing and farming in order to best sustain the population that will have to live and grow inside the biosphere. ?Your job will be to design models in order to analyze population growth, carrying capacity, growth rate, and mortality rate to help determine the best way to divide land and resources and control population growth within the sphere. ?The curt knock at the door startles you back into reality. ?You shoulder your backpack and pull the door open. ?Men with flat expressions and matching black suits usher you to the black sedan, placing the two small suitcases that contain everything you own and that will not be provided for you in the trunk. ?The interior of the car smells new, like the plastic has just been removed from the seats and windows. ?You feel badly placing your old worn backpack on the shiny leather seat. ?Who knows, maybe you will be the first and last person to ever ride in this car? ?As the men climb in the front and the car begins to drive away, you begin to ponder the scope of the task before you. ??The task of determining the best way to divide up land and resources is a necessary and a complicated venture. ?So much of life is unpredictable, and there will definitely be challenges that will change how life in the biosphere will play out. ?As a member of generation one, it will be important that your models are as precise as possible and take into account possible challenges that might arise. With a sigh, you sit back and watch the world pass by out the tinted windows. ?You are ready, you are a leader, and like it or not, you are a survivor; you will do your absolutely finest work in this time of crisis and save humanity.Appendix G: Explanation of Engineering Design ChallengeYour task is to design a biosphere to preserve human life after the apocalypse. You will develop computer simulations in order to predict the performance of different designs. The land inside the biosphere will be partitioned into areas for housing and farming; your goal is to grow and support the largest possible population by the time it will be safe to go outside. You will undergo training in biology, mathematics, and computer science, and combine your skills to complete your mission. You will study biological concepts of population growth and decay. You will study a mathematical model for the population growth inside the biosphere, then, develop Excel spreadsheet simulations. You will consider the logistical growth model, which involves two important parameters: the carrying capacity and the growth rate. These parameters depend on your design choices, which then affect the population, predicted by the model. Your spreadsheets will generate data that you must analyze in order to draw conclusions. You will use plots to visualize and present your data, and explain how they support your conclusions.How many human lives can be taken safely into our future? Perform your task poorly, and humanity is doomed. Do your best to Excel After the Apocalypse! Appendix H: “Pandora’s Box” Discussion GuideIn reference to the selection of the one hundred people for the bio-dome, please understand that this is strictly for starting the experiment with a set population of one hundred individuals. For the purpose of this lesson, we have the one hundred individuals being selected via a lottery. This method of selection is one of many ways the individuals could have been selected and, therefore, presents opportunities for days of discussions. These discussions could be based on education level, specific branches of science or math (psychology, sociology, economics, statistics), scientific and/or engineering experience level, health, diversity, etc. The dynamic nature of the subject matter presents an excellent opportunity for cross-curricular activities and to explore the moral “Pandora’s box” that trying to choose only one hundred individuals, out of a population of millions, would create. We encourage you to work with your Humanities, Language Arts, and Foreign Language departments before you start, to help prepare you , the students, and colleagues for the discussions that will ensue and to develop cross curricular activities that will complement this lesson. Appendix I: “Life Under the Bubble” (Discover Magazine, 2010) Life Under the BubbleBiosphere 2 was one of the most lauded experiments of the 1990s, then one of the most ridiculed. Now it is back, offering a unique way to put theories about climate and environment to the test.By Jordan Fisher Smith|Monday, December 20, 2010Douglas AdeskoBiosphere 2 has stood amid the paloverde, mesquite, and ocotillo southwest of Oracle, Arizona, for less than 20 years, yet it looks decidedly aged. Its skin is mostly glass and lacks window-washing tracks, so the hundreds of panes had to be cleaned by workers hanging on ropes like rock climbers. At one time seven people were employed to do this; today there are none. The desert wind deposits dust on the structure and the rain washes it downward, forming parallel streaks. The rain forest inside pushes against the glass. In 2003 there were about 150 employees on the site. Less than a third remain. Dry leaves collect against the air handlers by the main doorway; whiptail lizards skitter over the concrete paths, and javelinas trot around the grounds at night. A note on a whiteboard in the operating engineer’s office tallies the number of poisonous reptiles encountered on the site, which is greater than the number of maintenance people left to encounter them: “Rattlesnakes: 17.”The café is closed, the mission control building deserted, and inside the row of clear plastic sheds where plants were readied for installation in the main structure, towering exotics—Panama hat palm, angel’s trumpet—stand bleached and lifeless where they perished when the water was turned off. A monochrome monitor displays the last numbers it ever knew, burned into its dead screen. On the shelf below is the 1986 manual for the environmental monitoring system to which it was connected. Nothing ages faster than the future.Constructed between 1987 and 1991, Biosphere 2 was a 3.14-acre sealed greenhouse containing a miniature rain forest, a desert, a little ocean, a mangrove swamp, a savanna, and a small farm. Its name gave homage to “Biosphere 1”—Earth—and signaled the project’s audacious ambition: to copy our planet’s life systems in a prototype for a future colony on Mars. A May 1987 article in DISCOVER called it “the most exciting scientific project to be undertaken in the U.S. since President Kennedy launched us toward the moon.” In 1991 a crew of eight sealed themselves inside. Over the next two years they grew 80 percent of their food, something NASA has never attempted. They recycled their sewage and effluent, drinking the same water countless times, totally purified by their plants, soil, atmosphere, and machines. It wasn’t until 18 years later, in 2009, that NASA announced total water recycling on the International Space Station. At the end of their stay, the Biospherians emerged thinner, but by a number of measures healthier.Despite these successes, the media and the science establishment seized upon the ways in which the project had failed. Chief among these was an inability of Biosphere 2’s atmosphere to sustain human life. As was true outside, the problem was signaled by rising carbon dioxide. By 1996 Biosphere 2 had passed into the hands of Columbia University, and later the University of Arizona took over. Both used it to run scenarios of global climate and atmospheric change. In its later life, “instead of trying to model utopia, Biosphere 2 would actually model dystopia—a future plagued by high carbon dioxide levels,” wrote Rebecca Reider, author of a definitive history of the project. But while most research on impending environmental disaster relied on computer models, Biosphere 2 represented a fascinating alternative mode in which large-scale analog experiments employed real organisms, soil, seawater, and air.The man behind Biosphere 2 was John Allen, a Colorado School of Mines–trained metallurgist and Harvard MBA. In 1963, after two hallucinogenic experiences on peyote, Allen looked out of the Manhattan office building in which he was working and realized he could not open the window. He felt trapped like a bug inside glass—an ironic epiphany for a man who would work so hard to seal up a handful of his followers three decades hence. So he sailed from New York aboard a freighter and traveled the world, seeking wisdom. By 1967 he had become a self-styled esoteric teacher in Haight-Ashbury-era San Francisco, delivering weekly lectures to a group of mostly younger followers and cohabitants. In 1968 he and his students went to New York to set up a theater company, and from there to New Mexico, where they started a commune near Santa Fe. If most such counterculture experiments yielded to entropy and poverty, Allen’s Synergia Ranch is a notable exception. The Synergians were a very hardworking bunch.In 1974 a lanky young Texan and Yale dropout named Ed Bass wandered up the driveway to Synergia Ranch. Like Allen, Bass had a strong interest in the environment. Unlike Allen, he was the billionaire heir to an oil fortune. Later that year Allen and his followers drove an old school bus to Berkeley, California, where they built an 82-foot sailboat. None of them had ever built even a rowboat. In 1975 they began sailing the Heraclitus around the world. They took her up the Amazon River, dove coral reefs in the tropics, and sailed her to Antarctica to do research on whales.With John Allen’s big dreams and Ed Bass’s big money, the Synergians began taking on bigger things. They acquired a huge cattle ranch in Australia, started a sustainable forest in Puerto Rico, built a hotel and cultural center in Kathmandu, and took on other projects in Nepal, the United Kingdom, France, and the United States. Now calling themselves the Institute for Ecotechnics, they began hosting international meetings on ecology, sustainable development, and then space colonization. At a conference in Oracle in 1984, Allen announced his plan to build a prototype Mars colony on Earth before the decade was out. The destiny of human beings was to seed Earth’s life into space, and the first stop would be a working colony on Mars.The principals of the institute broke ground for Biosphere 2 in January 1987. If some of them lacked academic qualifications for the jobs they held, they enlisted real experts to execute the design. Walter Adey, a geologist at the Smithsonian Institution, was in charge of the ocean. The rain forest was the domain of Sir Ghillean Prance, then director of the New York Botanical Garden. These and other experts installed 3,800 species of life inside, even as cranes lifted great sections of white superstructure into place overhead. The majesty and complexity of the project entranced the press, touching on myth and religious narrative, Rebecca Reider wrote. Time called it “Noah’s Ark: The Sequel.” This created expectations that would be hard to meet.Biosphere 2's 91-foot-tall rain forest contains more than 150 plant species and now provides scientists a test ground for ecosystem experiments.Douglas AdeskoIn September 1991, four women and four men in NASA-style jumpsuits entered the air lock of Biosphere 2. Twelve days into the mission, Jane Poynter, a young Englishwoman in charge of the farm, put her hand in a threshing machine while winnowing rice. The group’s doctor sewed the tip of her middle finger back on, but the graft didn’t take and she was evacuated for surgery. She returned in only a few hours to serve out the two-year mission, but when she reentered the air lock, a duffel bag was placed inside with her. It contained nothing of substance, Poynter said—some circuit boards and a planting plan for the rain forest—but the media had a field day with it, along with the fact that someone had left and then reentered, which couldn’t have been done on Mars.More ominous, signs of trouble with the internal atmosphere began within 24 hours. Each morning the crew had a breakfast meeting over bowls of home-grown porridge in Star Trek–style chairs around a polished black granite table. The morning after closure, the crew captain announced that carbon dioxide in Biosphere 2’s atmosphere had risen to 521 parts per million, a 45 percent increase above levels outside at the time. By the following day, the lowest it went was 826. Over the months that followed, the news at the morning meetings got worse. Crew members were feeling tired and began to pant when they climbed stairs.In May 1992 in Palisades, New York, geochemist Wally Broecker got a phone call from someone at Biosphere 2, asking if he would be willing to consult on their atmosphere. Since the late 1970s, when he became the Newberry Professor of Earth and Environmental Sciences at Columbia University’s Lamont-Doherty Earth Observatory, Broecker had been sounding the alarm about a buildup of carbon dioxide in the big atmosphere. An elfish presence with a dried-apple-doll face and wild, tousled hair, he was already one of the great men of atmospheric-change research when he crossed the George Washington Bridge for dinner with John Allen at a Manhattan restaurant. The meeting had a cloak-and-dagger feel. Allen, a handsome, clean-shaven, broad-shouldered man who often wore a fedora, reminded Broecker of Indiana Jones. By Broecker’s account, Allen proffered a graph of the gas composition of Biosphere 2’s atmosphere, then nervously pulled it back, as if someone else might see it. A week later Broecker flew to Arizona and began collecting data.Much attention had been focused on charismatic species when Biosphere 2 was put together. A biologist surveyed the world’s hummingbirds to find one with a bill the right shape to pollinate a variety of plants inside the structure, and without a mating display predisposing it to fatal collisions with the glass. But Broecker and his graduate student Jeffrey Severinghaus discovered that the culprits in the carbon dioxide problem were the tiniest organisms on board: soil bacteria.The process of their subversion was respiration, in which living things release carbon dioxide into the atmosphere. Green plants absorb sunlight and carbon dioxide during photosynthesis, making carbohydrates and releasing oxygen, but they also do the reverse: Plants, too, respire (or breathe), burning carbohydrates to do work like making branches and roots. In the soil around their roots, billions of fungi and soil bacteria respire as well. In fact, the greater part of all “breathing” in terrestrial systems goes on underground.Ever grand in their ambitions, Allen and his people intended Biosphere 2 to be used by rotating crews for 100 years. Feeling they had one shot to invest their world with life-giving nutrients, they had loaded their soils with compost and rich muck from the bottom of a cattle pond. (Agricultural chemicals used inside might end up in their air and water.) When the air locks closed, soil bacteria had a massive party, exhaling carbon dioxide and tipping the balance the wrong way.As oxygen was converted to carbon dioxide, free oxygen in the atmosphere declined. By January 1993, Biosphere 2’s carbon dioxide levels were 12 times that of the outside, and oxygen levels were what mountaineers get at 17,000 feet. The crew’s doctor was having trouble adding up simple figures and disqualified himself from duty. So, a year and four months into the mission, tank trucks containing 31,000 pounds of liquid oxygen started driving up the access road to the site.The story of fresh-faced idealists getting taken down a notch played well in the media. For two years the glass walls of Biosphere 2 were lined with TV cameras and tourists. The crew’s lives turned into reality TV. In fact, the producers of the world’s first reality TV show, Big Brother, which aired in the Netherlands in 1999, acknowledged Biosphere 2 as their inspiration. True to reality TV’s typical plotline, months cooped up together while struggling with their atmosphere and hunger and being filmed by well-fed people led to squabbles among the Biospherians. They emerged from the air lock in September 1993 in two groups of four who weren’t speaking. Organizational cracks opened between them and their advisory scientists and extended into their relationship with Ed Bass. Originally budgeted at $30 million, Biosphere 2 had already cost a reported $200 million. By the time a second crew took its place inside, Bass had had enough. On April 1, 1994, his bankers, accompanied by carloads of armed federal marshals and sheriff’s deputies, swept into the site with a restraining order. The second crew lingered inside Biosphere 2 for another five months and 16 days before terminating its mission.Biosphere 2, it was widely reported, was a catastrophe. In 1999, when Time did its fin de siècle summary of the 20th century, it included Biosphere 2 in its list of the worst 100 ideas.With the biospherians ejected from their eden, Bass’s people began looking for a new entity to operate the facility. Eventually they struck a deal with Columbia University. The new director of research was Wally Broecker, who had coined the term “global warming” two decades earlier. Here was a gigantic laboratory flask with a whole tropical forest and an ocean inside it—models of what many scientists suspected were the two biggest carbon sinks in the world. By 1995, when the deal was closed, Broecker was not alone in his sense of urgency.Appendix J: Teacher instructions for Desmos/Guided QuestionsGo to teacher. If you do not have an account, sign up for free!Search for Polygraph: ExponentialsGet the one made by Desmos itselfClick on new session; this will allow you to set up classes, each with their own class code. Instruct students to student. and have them enter the code. Desmos Description:“This Custom Polygraph is designed to spark vocabulary-rich conversations about exponentials, including how they differ from linear functions. Key vocabulary that may appear in student questions includes: increasing, decreasing, intercept, rate, asymptote, and curve. In the early rounds of the game, students may notice graph features from the list above, even though they may not use those words to describe them. That’s where you can step in. After most students have played 2-3 games, consider taking a short break to discuss strategy, highlight effective questions, and encourage students in their use of increasingly precise academic language. Then ask them to play several more games, putting that precise language to work.”Appendix K: Linear vs. Quadratic vs. Exponential WorksheetFill out the table regarding linear, quadratic, and exponential functions.TypeGeneral ShapeSketch of GraphKey Characteristics of EquationLinearQuadraticExponential-8382032067500Identify each table, equation or graph as linear, exponential, or quadradraticxy-29-140110213449-4826040703500 y = 3x + 1 y = -3x2+ 6x – 1036391854318000xy-170513213-14-35-5Appendix L: Food Web Simulation LabTask A First you'll run a less than "real-life" scenario. Choose only one organism from each trophic level and make sure that the food chain goes in a straight line from one trophic level to the next, i.e., Herbivore A eats Plant A, Omnivore A eats Herbivore A, and the Top Predator eats Omnivore A. Let Plant B survive on its own and see what happens. Predict whether each species will survive, and whether it will increase or decrease in number, as well as whether Plant B will survive to the end. Record your prediction in the Data Table and then run the simulation twice and record your data. Use X for "die out," ↑ for "increase in numbers," and ↓ for "decrease in numbers." Answer the following:Was your prediction correct? How did you arrive at your prediction? What differences were there between your prediction and the simulation?What would happen to this imaginary ecosystem if the producers were to die out?Did any of the species increase in number? What could account for this increase? Which species decreased in number and what might account for this decrease?Which populations would benefit the most from the presence of decomposers?Task BNow try a more "real-life" scenario and experiment with what might happen in an ecosystem that is more like a web. This time click the "all on" button. The model shows who eats whom and the paths by which energy is transferred. Predict which populations will die out, increase in numbers, or decrease in numbers and record your predictions. Run the simulation twice and record the results in your Data Table. Then try to modify who eats whom in order to ensure the survival of all species and record what was changed in your chart. Finally, answer the following:Was your prediction correct? How did you arrive at your prediction? What differences were there between your prediction and the simulation?Were you able to modify the parameters so that each species survived? Explain how you decided what changes to make.Which way does energy flow and how does eating an organism result in energy transfer? Adapted from Annenberg Learner, 2016Appendix M: Food Web Simulation Data TablesData Table Task ADATA TABLE: ECOLOGYLesson 2:Step 1(X, , or )Plant AHerbivore AOmnivore ATop PredatorPredictionSimulation 1Simulation 2Lesson 2:Step 2(X, , or )Plant APlant BPlant CHerbivore AHerbivore BHerbivore COmnivore AOmnivore BTopPredatorPredictionSimulation 1Simulation 2Modifications madeResponses to questions Data Table Task BDATA TABLE: ECOLOGYLesson 2:Step 1 (X, , or )Plant AHerbivore AOmnivore ATop PredatorPredictionSimulation 1Simulation 2Lesson 2:Step 2(X, , or )Plant APlant BPlant CHerbivore AHerbivore BHerbivore COmnivore AOmnivore BTopPredatorPredictionSimulation 1Simulation 2Modifications madeResponses to questionsAdapted from Annenberg Learner, 2016 Appendix N: “What is Carrying Capacity?”By Lindsey Bailey on Thursday, Dec 12th, 2013063500We all know that living things need resources in order to survive. We often, however, don’t make the connection that the amount of available resources dictates the size of a population – that a population will grow when resources are in surplus, decline when resources are scarce, and stabilize when the population is at the maximum level that can be sustained.Making this connection between resource availability and population growth patterns is encompassed in the study of carrying capacity. Carrying capacity is taught in middle and high school biology classrooms around the world and is a crucial concept in the study of population.Because the study of carrying capacity can be complex, it can help to first think of the concept in familiar terms. For instance, you may have seen the phrase “carrying capacity” posted on the side of a school bus. And in fact, “carrying capacity” in reference to a school bus or an elevator is not all that different from carrying capacity in reference to a population. On a school bus, the carrying capacity would be the maximum number of people that could safely fit. In ecological terms, carrying capacity is defined as the maximum number of a species that can sustainably live in a given area. ?In other words, a population’s carrying capacity is the size at which a population can no longer grow due to lack of supporting resources. All populations have a carrying capacity, whether bacteria in a bottle or rabbits in a forest. If we were referring to a rabbit population in a forest, the carrying capacity would refer to the maximum rabbit population that can be supported by that forest’s resources.Biologists often graph populations to show growth trends. A graph that reveals an “s” shape indicates that the population has hit its carrying capacity. For example, in the graph pictured above (taken from the activity,?The Pop Ecology Files), we can see that the population of this particular species was growing until day 23, and then leveled off at a carrying capacity of 2,000.The graph of our human population currently looks like?this, a classic J-curve of exponential growth. The comparison begs the question; does carrying capacity also apply to our human population? And if so, what is it? Appendix O: Degree of Impact CardsFood Availability Food availability in any habitat is paramount to survival of a species. Predators, carnivores, must have prey availability. As long as their prey is available, they usually do not suffer from food stress. Herbivores, plant eaters, have a more complicated diet, and can become stressed from a shortage of food, or a shortage of nutritious food. They will feed first on their preferred foods, and then the staple food that satisfies their nutritional needs. When no other foods are available, herbivores will feed on emergency foods that will fill them up, but not maintain their body weight.WaterAnimals must have water to help with food digestion, to help control and regulate body temperature, and to help eliminate waste products from the body. Usually, the larger the animal, the more water is required to sustain the animal's organ systems. Where water becomes scarce, food may also become scarce as plants die, animals leave or die, and the remaining animals fight each other for whatever water is left. Their bodies become weaker and are less able to fight off disease or predators. Human Pollution Conditions within or adjacent to an environment also affect its carrying capacity. For example, if the environment is located close to a human population, this may affect its carrying capacity. Pollution may also affect an environment's carrying capacity. SpaceAnimals need a place to shelter from poor conditions, and to provide a place for reproduction. Sufficient space within a habitat allows the animals that inhabit it better opportunities to find adequate food and water. Without space, animals cannot ensure a place to hide and raise their young -- or to nest. Animals also need space to rest, even to play. WasteThe more we consume the more waste we produce. By the time a baby born today in the United States reaches the age of 78 years, he or she will have produced nearly 66 tons of garbage. The average American generates 4.4 pounds of solid waste each day, compared to the average person in the U.K. (3 pounds) or Japan (2.3 pounds). The average resident of Nepal produces less than 1 pound per day. Land UseHousing, crops, and room for waste are just some of the land uses for the space on our planet.Appendix P: Lab Notebook“Excel”ing After the Apocalypse Lab Notebook-11164211241000Rules of the Lab Notebook:Include a date for every entryYou may not use personal pronouns. (I think…I believe…)Must be written in blue or black PEN!May not erase any entry in lab notebook. If you make an error, strike a line though the entry. (i.e. error).Personal peer sign-off, upon completion. Day # __________ Title: ____________________ Date: ___/___/___Describe what steps you completed today?What sources did you search today?In what stage in the engineering design process are you?If you could have an expert in any field help you today, what expert would you choose?What question(s) would you ask the expert?Goals for tomorrow:Appendix Q: Student CheckpointsEngineering Design Challenge Part 1 “How long are we going to be trapped?”(Basic Excel Tutorial)StepDateTeacher SignoffStep 1: Protect Your DataStep 2: Data EntryStep 3: Prepare to SimulateStep 4: SimulateStep 5: VerifyStep 6: SimulateEngineering Design Challenge Part 2Creating a Biosphere Population SimulatorStepDateTeacher SignoffStep 1: Organizing Variables and ConstantsStep 2: Prepare to SimulateStep 3: Simulate and VerifyStep 4: Visualize Step 5: InvestigateAppendix R : Engineering Design Challenge: Part 1“How long are we going to be trapped?”(Basic Excel Tutorial)You need to find out how many years the population needs to stay inside of the biosphere, until it’s safe to go outside. The few people outside of the biosphere who have managed to survive are not happy about their situation, and if a biosphere person gets caught outside the dome, they will have their blood drained for transfusion and their flesh eventually eaten. Luckily (for those inside the biosphere), the population outside of the biospheres is declining (due to environmental dangers, tribal conflicts, and automobile-based gladiatorial combat) according to the exponential decay model given by the formula:Pt=P0eatGiven:P0 = 500a = ln?0.5≈ -0.693Step 1: Protect your dataThe very first task is to make sure that your data will be protected. We will save the file, before there is even anything in it! You’ll be sure to keep hitting the “save” button, as you make progress…right? We’ll skip encrypting the file this time.a) Create a new folder in your favorite location. Name it “Title_YourName”.b) Open a new Excel spreadsheet, go to FILE Save As, choose the folder you just created and save the file as “years_trapped.xlsx”.Step 2: Data entryNow we’ll create an area at the top of the spreadsheet to enter the constants. Why? The constants may need to be modified, when you get some new information. We will do the simulation on the left, and make a plot on the right. The organization of the spreadsheet will eventually look like this:170497511620500a) Enter the constants P0 and a using Cells A1, A2, B1 and B2 with labels as follows:b) To change line between “Initial population” and “P0”, hit alt+enter..c) To shrink the zero attached to P, highlight the zero, click on the arrow near “Font” under the “Home” tab. Choose subscript and hit OK. d) Center the labels by highlighting the top two cells by selecting the entire row 1 (click on 1 next to the first row) and pressing the “center” alignment button.e) Adjust the cell widths by going in between column A and B, click when the cursor becomes and drag the mouse left or right.Step 3: Prepare to simulateTo get ready to calculate the population as a function of time, we will create a column of time. a) Make labels for time and population as below. Make them look nice using the techniques you just learned.15754358636000b) Make a column of numbers. Feeling lazy? Excel will work for you! Enter 0 and 1 below the “time” label, in cell A5 and A6. Highlight cell A5 and A6, move the cursor to the lower right corner to make it and drag the mouse down to cell A15. Now we have a column of numbers going from 0 to 10. Step 4: SimulateWe will now enter the formula in the second column to calculate the function values Pt=P0eat, using constants P0 and a we defined in cell A2 and B2. Excel has some mathematical functions built-in, for example, we’ll make use of “=EXP(a*t)” to compute eata) Go to cell B5 and enter “=A2*EXP(B2*A5)”. The relevant cells will be highlighted when you click on the formula bar. Cell B5 now shows the computed value.b) Now we need to copy the formula to the rest of the cells in the “Pt“ column. Feeling lazy again? We can make Excel do the work. We just need to do one thing here before we copy the formula. Why? If you just copy cell B5 (ctrl + c) and paste (ctrl + v) into the cell below, it shows 0, which is wrong. If you pay attention to the formula bar, you can see that it’s trying to use A3 and B3 for constants, but those cells are empty! This is because Excel uses relative cell locations to fetch values, unless you tell it otherwise. To fix this, we need to tell Excel to use absolute cell locations for the constants. To do this, enter $ in front of both the letter and number in the formula, for the corresponding cell. That is, enter “=$A$2*EXP($B$2*A5)” in cell B5. Copy the formula to cell B6 and see what happens.c) Copy the formula to the rest of the column by highlighting cell B5 and B6, get a at the lower right corner and drag the mouse down to row 15.d) Highlight the “Pt“ values and adjust the number of digits you show after the decimal point by clicking the “decrease decimal” button under “Number” in the “Home” tab until the values become integers.Step 5: VerifyThe eighth law of computer programming says “Any non-trivial program contains at least one bug.” <; Let’s run some tests to verify your Excel code. Test 1: Initial value. Calculate the value of Pt at t=0 by hand and compare with the Excel value.Your value: Excel value:Test 2: Cross-check: Your mathematician insists that the population function Pt is equivalent to ft=5000.5tfor the given constants. Calculate the ft value at time t=1 and compare with the Excel value of P1.Your value: Excel value:Optional: Prove that Pt= ft for any arbitrary time t.Step 6: Visualize*Very important* Your statistician reported that the coefficient a in your population decay form was miscalculated, and the correct value is a=-0.06931.a) Change the value of the constant and observe that the function values also change.259080067627500b) Now let’s create a plot to visualize the population decay and find out how long your population needs to stay in the dome. Highlight the region in the spread sheet that contain time and population values. Go to the “INSERT” tab and click on the chart scatter plot icon. It will automatically create a plot using the time column for the x-axis and the population column for the y-axis.c) Make the plot look nice. Change the title to an appropriate one. Insert x- and y- axis labels by clicking the plus button that appears when you click on the plot, selecting “Axis Titles” and changing the axis labels to appropriate ones.d) Now it’s time to figure out how long your population is trapped inside of the dome. You need to find in which year the population becomes zero. From the plot, it looks like the population does not go all the way down to zero by year 10.Create another column to conduct a safety test. In cell C5, type “=IF(B5<=0.5,"Yes","No")” and copy the formula down to the rest of the column. The cell corresponding to the year when the outside population goes below 0.5 will display “Yes” (we’ll consider half a person to be safe). The cell displays “No” otherwise.Copy the time, population and safety test columns further and expand the plot ranges by right-clicking on the plot, selecting “Select Data”, re-selecting the region and pressing “OK”. Repeat until your simulation runs long enough to reach the zero outside population. What year does this happen?We can go outside after Year ____Appendix S: Engineering Design Challenge Part 1 - Key“How long are we going to be trapped?”(Basic Excel Tutorial)You need to find out how many years the population needs to stay inside of the biosphere, until it’s safe to go outside. The few people outside of the biosphere who have managed to survive are not happy about their situation, and if a biosphere person gets caught outside the dome, they will have their blood drained for transfusion and their flesh eventually eaten. Luckily (for those inside the biosphere), the population outside of the biospheres is declining (due to environmental dangers, tribal conflicts, and automobile-based gladiatorial combat) according to the exponential decay model given by the formula:Pt=P0eatGiven:P0 = 500a=ln?0.5≈ -0.693Step 1: Protect your dataThe very first task is to make sure that your data will be protected. We will save the file, before there is even anything in it! You’ll be sure to keep hitting the “save” button, as you make progress…right? We’ll skip encrypting the file this time.a) Create a new folder in your favorite location. Name it “Title_YourName”.b) Open a new Excel spreadsheet, go to FILE Save As, choose the folder you just created and save the file as “years_trapped.xlsx”.Step 2: Data entryNow we’ll create an area at the top of the spreadsheet to enter the constants. Why? The constants may need to be modified, when you get some new information. We will do the simulation on the left, and make a plot on the right. The organization of the spreadsheet will eventually look like this:14382753492500a) Enter the constants P0 and a using Cells A1, A2, B1 and B2 with labels as follows:b) To change line between “Initial population” and “P0”, hit alt+enter.c) To shrink the zero attached to P, highlight the zero, click on the arrow near “Font” under the “Home” tab. Choose subscript and hit OK. d) Center the labels by highlighting the top two cells by selecting the entire row 1 (click on 1 next to the first row) and pressing the “center” alignment button.e) Adjust the cell widths by going in between column A and B, click when the cursor becomes and drag the mouse left or right.Step 3: Prepare to simulateTo get ready to calculate the population as a function of time, we will create a column of time. a) Make labels for time and population as below. Make them look nice using the techniques you just learned.1428750698500b) Make a column of numbers. Feeling lazy? Excel will work for you! Enter 0 and 1 below the “time” label, in cell A5 and A6. Highlight cell A5 and A6, move the cursor to the lower right corner to make it and drag the mouse down to cell A15. Now we have a column of numbers going from 0 to 10. Step 4: SimulateWe will now enter the formula in the second column to calculate the function values Pt=P0eat, using constants P0 and a we defined in cell A2 and B2. Excel has some mathematical functions built-in, for example, we’ll make use of “=EXP(a*t)” to compute eata) Go to cell B5 and enter “=A2*EXP(B2*A5)”. The relevant cells will be highlighted when you click on the formula bar. Cell B5 now shows the computed value.b) Now we need to copy the formula to the rest of the cells in the “Pt“ column. Feeling lazy again? We can make Excel do the work. We just need to do one thing here before we copy the formula. Why? If you just copy cell B5 (ctrl + c) and paste (ctrl + v) into the cell below, it shows 0, which is wrong. If you pay attention to the formula bar, you can see that it’s trying to use A3 and B3 for constants, but those cells are empty! This is because Excel uses relative cell locations to fetch values, unless you tell it otherwise. To fix this, we need to tell Excel to use absolute cell locations for the constants. To do this, enter $ in front of both the letter and number in the formula, for the corresponding cell. That is, enter “=$A$2*EXP($B$2*A5)” in cell B5. Copy the formula to cell B6 and see what happens.c) Copy the formula to the rest of the column by highlighting cell B5 and B6, get a at the lower right corner and drag the mouse down to row 15.d) Highlight the “Pt“ values and adjust the number of digits you show after the decimal point by clicking the “decrease decimal” button under “Number” in the “Home” tab until the values become integers.Step 5: VerifyThe eighth law of computer programming says “Any non-trivial program contains at least one bug.” <; Let’s run some tests to verify your Excel code. Test 1: Initial value. Calculate the value of Pt at t=0 by hand and compare with the Excel value.Solution: P0=P0e0=P0?1=P0=500Your value: 500Excel value: 500Test 2: Cross-check: Your mathematician insists that the population function Pt is equivalent to ft=5000.5tfor the given constants. Calculate the ft value at time t=1 and compare with the Excel value of P1.Solution: f1=5000.51=500?0.5=250Your value: 250Excel value: 250Optional: Prove that Pt= ft for any arbitrary time t.Solution: Take natural log of both functionslnft=ln5000.5t=ln500?ln0.5t=ln500?t?ln0.5=a?t?lnP0lnPt=lnP0eat=lnP0?lneat=lnP0?a?t?lne=lnP0?a?t?1=a?t?lnP0∴ Functions ft=Pt at any arbitrary t.Step 6: Visualize*Very important* Your statistician reported that the coefficient a in your population decay form was miscalculated, and the correct value is a=-0.06931.a) Change the value of the constant and observe that the function values also change.b) Now let’s create a plot to visualize the population decay and find out how long your population needs to stay in the dome. Highlight the region in the spread sheet that contain time and population values. Go to the “INSERT” tab and click on the chart scatter plot icon. It will automatically create a plot using the time column for the x-axis and the population column for the y-axis.c) Make the plot look nice. Change the title to an appropriate one. Insert x- and y- axis labels by clicking the plus button that appears when you click on the plot, selecting “Axis Titles” and changing the axis labels to appropriate ones.d) Now it’s time to figure out how long your population is trapped inside of the dome. You need to find in which year the population becomes zero. From the plot, it looks like the population does not go all the way down to zero by year 10.Create another column to conduct a safety test. In cell C5, type “=IF(B5<=0.5,"Yes","No")” and copy the formula down to the rest of the column. The cell corresponding to the year when the outside population goes below 0.5 will display “Yes” (we’ll consider half a person to be safe). The cell displays “No” otherwise.66992526035100Copy the time, population and safety test columns further and expand the plot ranges by right-clicking on the plot, selecting “Select Data”, re-selecting the region and pressing “OK”. Repeat until your simulation runs long enough to reach the zero outside population. What year does this happen?Solution: year 100We can go outside after Year 100 Appendix T: Engineering Design Challenge Part 2Creating a Biosphere Population SimulatorYour task is to create a population simulator in Excel spreadsheet for the 100 people left in the biosphere for the next ____ years and to decide the optimal way to designate the limited area for necessary resources. There are many factors that influence the population, but let’s start with something simple. We can consider other factors and modify the mathematical model to include more complex scenarios. We will start by making some assumptions and writing them in mathematical forms.Assumption #1: The population follows the logistic model.Based on assumption #1, population as a function of time is written as:Pt=KP0P0+K-P0e-rt(1)K: carrying capacity; P0: initial population; r: growth rateThis is the solution to the differential equation:dPdt=rP1-PK(2)Reference:Department of Mathematics, Duke University #2: The available area will be divided into two sections, one for housing and one for farming.Based on assumption #2, we’ll denote symbols for the design variables asA1: housing area fractionA2: farming area fractionThe constraint on the variables is that housing and farming areas add up to the total area available, i.e., A1+A2=1.This means that we can consider A1 as the independent variable and write A2 as the dependent variable as:A2=_________________Assumption #3: Minimum between housing availability and food resources will be the limiting factor, deciding the carrying capacity.Based on assumption #3, the carrying capacity K is calculated as K=minc1A1,c2A2(3)Carrying capacity coefficients c1 and c2 will be given by your civil and agricultural engineers.Assumption #4: Growth rate becomes half if the food resources support no more than twice the initial population.Based on assumption #4, growth rate r is calculated as r=c3; if c2A2>2P0c3/2; if c2A2≤2P0(4)The reference growth rate c3 will be given by your statistician.Given:Your civil engineer reports that the housing resource coefficient c1 is 400 (#. people per unit area, each year). Your agricultural engineer reports that the food resource carrying capacity coefficient is 600 (#. people per unit area, each year). Your statistician reports that the reference growth rate c3 is 5%. The initial population P0 is 100.Step 1:Before we start generating a simulator, it’s a good idea to make note of what needs to be done. Fill out the table below, organizing constants, independent/dependent variables and functions.SymbolMeaningTypeMathematical Formc1housing carrying capacity coefficientconstant400Step 2:Now let’s start generating a simulator. First, create a new Excel file, name it “population_model_YourTeamName.xlsx” and save the file in the project folder. We will organize the spreadsheet like the one in the figure in the next page. The idea is to have all variables and constants accessible at the top, so that we can run different simulations without messing around too much.Referring to the table created in Step 1, enter the symbol, meaning and type in row 1. Enter the corresponding formula in row 2, referring to the table and equations where necessary.Step 3: Simulate the population growth by creating a column of time, starting at year 0, and typing the population formula in the next column. Be sure to use the absolute locations when using constants and variables from the table. Use A1=0.3 for a test run.Do calculations by hand for the value of Pt=0 and check if Excel gives you the correct value.Pt=0 =Step 4:Make a plot to visualize the population growth. Put a title on the plot and label x-axis and y-axis.Do you see a growth or decay? What is the carrying capacity?Step 5:Using the population simulator you created, investigate the relationship between the land design and the population at year ___ for 10 different combinations of A1 and A2. Note if you observe a decay or growth. A1A2PA1A2PAppendix U: Engineering Design Challenge Part 2- KEYCreating a Biosphere Population SimulatorYour task is to create a population simulator in Excel spreadsheet for the 100 people left in the biosphere for the next 100 years and to decide the optimal way to designate the limited area for necessary resources. There are many factors that influence the population, but let’s start with something simple. We can consider other factors and modify the mathematical model to include more complex scenarios. We will start by making some assumptions and writing them in mathematical forms.Assumption #1: The population follows the logistic model. Based on assumption #1, population as a function of time is written as:Pt=KP0P0+K-P0e-rt(1)K: carrying capacity; P0: initial population; r: growth rateThis is the solution to the differential equation:dPdt=rP1-PK(2)Reference:Department of Mathematics, Duke University< #2: The available area will be divided into two sections, one for housing and one for farming.Based on assumption #2, we’ll denote symbols for the design variables asA1: housing area fractionA2: farming area fractionThe constraint on the variables is that housing and farming areas add up to the total area available, i.e., A1+A2=1.This means that we can consider A1 as the independent variable and write A2 as the dependent variable as:A2= 1-A1Assumption #3: Minimum between housing availability and food resources will be the limiting factor, deciding the carrying capacity.Based on assumption #3, the carrying capacity K is calculated as K=minc1A1,c2A2(3)Carrying capacity coefficients c1 and c2 will be given by your civil and agricultural engineers.Assumption #4: Growth rate becomes half if the food resources support no more than twice the initial population.Based on assumption #4, growth rate r is calculated as r=c3; if c2A2>2P0c3/2; if c2A2≤2P0(4)The reference growth rate c3 will be given by your statistician.Given:Your civil engineer reports that the housing resource coefficient c1 is 400 (#. people per unit area, each year). Your agricultural engineer reports that the food resource carrying capacity coefficient is 600 (#. people per unit area, each year). Your statistician reports that the reference growth rate c3 is 5%. The initial population P0 is 100.Step 1:Before we start generating a simulator, it’s a good idea to make note of what needs to be done. Fill out the table below, organizing constants, independent/dependent variables and functions.SymbolMeaningTypeMathematical FormA1housing area fractionindep. variableA1A2farming area fractiondep. variable1-A1c1housing carrying capacity coefficientconstant400c2food carrying capacity coefficientconstant600c3reference growth rateconstant0.05 (5%)P0initial populationconstant100Kcarrying capacitydep. variable K=minc1A1,c2A2rgrowth ratedep. variabler=c3; if c2A2>2P0c3/2; if c2A2≤2P0Ppopulation growthfunctionPt=KP0P0+K-P0e-rtStep 2:Now let’s start generating a simulator. First, create a new Excel file, name it “population_model_YourTeamName.xlsx” and save the file in the project folder. We will organize the spreadsheet like the one in the figure in the next page. The idea is to have all variables and constants accessible at the top, so that we can run different simulations without messing around too much.Referring to the table created in Step 1, enter the symbol, meaning and type in row 1. Enter the corresponding formula in row 2, referring to the table and equations where necessary.Step 3: Simulate the population growth by creating a column of time, starting at year 0, and typing the population formula in the next column. Be sure to use the absolute locations when using constants and variables from the table. Use A1=0.3 for a test run.Do calculations by hand for the value of Pt=0 and check if Excel gives you the correct value.Pt=0=Pt=KP0P0+K-P0e-r?0=KP0P0+K-P0?1=KP0P0+K-P0=KP0K=P0=100 Step 4:Make a plot to visualize the population growth. Put a title on the plot and label x-axis and y-axis.Do you see a growth or decay? What is the carrying capacity?Solution: Using A1=0.3, the plot should show a logistic growth reaching a carrying capacity of 120, around year 80-100.Step 5:Using the population simulator you created, investigate the relationship between the land design and the population at year 100 (from part 1) for 10 different combinations of A1 and A2. Note if you observe a decay or growth. A1A2PA1A2P-61912519240500Appendix V : Engineering Design Challenge Part 3Design OptimizationYour goal is to allocate area to housing and farming in order to maximize the population of your dome after 100 years using the logistic growth model you used in part 2. However, your demographer gives you three possible scenarios for the population growth, and asks you to examine three different values of constant c3, 1%, 2% and 5%. You could use your simulation from part 2 by guessing and checking a variety of values of the areas, but this is too time consuming! Instead, develop another spreadsheet that sweeps through many possible values of the areas. Like before, make a section at the top of the spreadsheet to enter in the appropriate constants (the time of 100 years is now a constant, too). Make one column in your spreadsheet for the housing area fraction A1, farming area fraction A2, the carrying capacity K, the growth rate r, and the final population after 100 years, P(100). Make a plot to demonstrate your results.This kind of strategy, where we evaluate the performance of many potential values of the design variables, is known as a parameter sweep. It allows us to perform many iterations of the redesign phase of the engineering design process in an automated, labor-saving way.What is the right kind of plot to make?For each population growth scenario, what value of the farming area maximizes the final population?What trends do you notice as c3 changes?sAppendix W : Engineering Design Challenge Part 3-Key-979488188563400Appendix X: Challenge Cards-99822015303500-9588501968500-1060248360800-104711510604500-9572639874300-1049020-1079500 Points 4Accomplished3 Proficient2Developing1 NoviceExplanation of Excel Process(Can you explain what you are doing accurately?)Able to explain performance task to someone elseAble to explain performance task, yet still working on how to articulate a few tasksHas limited ability to explain performance task Unable to explain performance taskData Entry (Connecting variables, reading syntax, etc.)Variables are successfully interrelated, with all correct number values and equations representedVariables are successfully interrelated, with only 1-2 equations or values needing redesignAll equations are in place, but a 3 or more of the equations and values are in need of redesign Equations and number values are not completed or all columns in need of complete redesignData Interpretation (Does your graph accurately interpret the data?)Graphical representation of data includes a scatter plot with all the following:a) x and y axis titleb) correct rangec) correct shape of graphd) descriptive titleGraphical representation of data includes a scatter plot with only 2-3 of the following:a) x and y axis titleb) correct rangec) correct shape of graphd) descriptive titleGraphical representation of data includes a scatter plot with only one of the following:a) x and y axis titleb) correct rangec) correct shape of graphd) descriptive titleGraphical representation of data includes non-scatter plot graphData Analysis(Drawing conclusions on graph and data results)Able to successfully analyze graph by drawing conclusions on population size after 100 years, as well as elaborate on how the graph was used to answer the proposed questionAble to analyze graph by using both variables to draw conclusions, yet is still limited in explanationAble to analyze graph by using only one variable to draw conclusionsUnable to successfully use graph to draw conclusions on future population sizeAppendix Y: Engineering Challenge Grading RubricAppendix Z: Preparing to Present Scientific Conclusions- Student Preparation RubricStudent presentations will include 1 of 5 of the following:Student presentations will include 2 of 5 of the following:Student presentations will include 3 of 5 of the following:Student presentations will include 4 of 5 of the following:Student presentations will include all 5 of the following:Set a themeProvide an outlineMake clear transitionsMake numbers meaningfulUse simple visualsSet a themeProvide an outlineMake clear transitionsMake numbers meaningfulUse simple visualsSet a themeProvide an outlineMake clear transitionsMake numbers meaningfulUse simple visualsSet a themeProvide an outlineMake clear transitionsMake numbers meaningfulUse simple visualsSet a themeProvide an outlineMake clear transitionsMake numbers meaningfulUse simple visuals Appendix AA: Presentation Rubric (Teacher) Points 4Accomplished3Proficient2Developing1NoviceTotalContent KnowledgeStudent thoroughly demonstrates a complete understanding of the concepts covered during the engineering design process.Student demonstrates understanding of the concepts covered during the engineering design process.Student demonstrates some understanding of the concepts covered during the engineering design processStudent demonstrates a lack of understanding of the concepts covered during the engineering design processOrganizationPresentation is well planned and is presented in an interesting and logical order that keeps the audience engaged.Presentation is well planned and presented in a logical order. Audience is able to easily follow the presentation.Presentation demonstrates some planning. Audience has a difficult time following presentation. Presentation does not proceed in a logical order.Presentation demonstrates little to no planning. Audience struggles to follow and/or understand the concepts being presented. .Public Speaking/PresentationPresentation is clear with excellent pacing Student shows tremendous confidence. Presentation is fairly clear and has good pacing. Student shows confidence. Presentation is somewhat clear and has fair pacing. Student shows some confidence.Presentation is not clear with poor pacing.Student lacks confidence.Model/Power PointPresentation is very well designed, flows smoothly, enhances, and supports the concepts being presented.Presentation is fairly well designed, flows well, and enhances and/or supports the concepts being presented.Presentation has some design issues and struggles to enhance and/or support the concepts being presented Presentation is poorly designed, lacks flow, and does not enhance and/or support the concepts being presented.Team WorkAll members of the team are equally responsible for presenting and explaining concepts covered in the engineering design challenge.Multiple members of the group are responsible for presenting and explaining concepts covered in the engineering design challenge. Some members present more (or more often) than other members of the group.One member is responsible for presenting and explaining concepts covered in the engineering design challenge with some support from other members of the groupOnly one member of the team is responsible for presenting and explaining the concepts covered in the engineering design challenge.Appendix BB: Presentation Rubric (Student Self Reflection) Points 4Accomplished3Proficient2Developing1NoviceTotalContent KnowledgeStudent thoroughly demonstrates a complete understanding of the concepts covered during the engineering design process.Student demonstrates understanding of the concepts covered during the engineering design process.Student demonstrates some understanding of the concepts covered during the engineering design processStudent demonstrates a lack of understanding of the concepts covered during the engineering design processOrganizationPresentation is well planned and is presented in an interesting and logical order that keeps the audience engaged.Presentation is well planned and presented in a logical order. Audience is able to easily follow the presentation.Presentation demonstrates some planning. Audience has a difficult time following presentation. Presentation does not proceed in a logical order.Presentation demonstrates little to no planning. Audience struggles to follow and/or understand the concepts being presented. .Public Speaking/PresentationPresentation is clear with excellent pacing Student shows tremendous confidence. Presentation is fairly clear and has good pacing. Student shows confidence. Presentation is somewhat clear and has fair pacing. Student shows some confidence.Presentation is not clear with poor pacing.Student lacks confidence.Model/Power PointPresentation is very well designed, flows smoothly, enhances, and supports the concepts being presented.Presentation is fairly well designed, flows well, and enhances and/or supports the concepts being presented.Presentation has some design issues and struggles to enhance and/or support the concepts being presented Presentation is poorly designed, lacks flow, and does not enhance and/or support the concepts being presented.Team WorkAll members of the team are equally responsible for presenting and explaining concepts covered in the engineering design challenge.Multiple members of the group are responsible for presenting and explaining concepts covered in the engineering design challenge. Some members present more (or more often) than other members of the group.One member is responsible for presenting and explaining concepts covered in the engineering design challenge with some support from other members of the groupOnly one member of the team is responsible for presenting and explaining the concepts covered in the engineering design challenge.Appendix CC: Presentation Peer ReviewPresentation Peer ReviewUse the scale to fill in the chart to evaluate your classmates presentation.1-Strongly Agree 2-Agree 3-Somewhat Agree 4-Disagree 5-Strongly DisagreePresentation CriteriaRatingThe presentation was very well organised and the performance ran smoothly.The information was highly accurate and relevant to the engineering challenge.The presenter(s) face and maintains eye contact with the audience & do not read off the screenStudents use vocabulary and examples relevant to the engineering challenge and concepts covered during the preparation for the challenge.Presenter(s) speak in an understandable voice, using a clear tone, enunciation, and a reasonable pace; message is clearly received.Presentation was colorful, creative, and the information was easy to readGive 1 fact you learned: __________________________________________________The area for improvement is: ______________________________________________Presentation Peer ReviewUse the scale to fill in the chart to evaluate your classmates presentation. 1-Strongly Agree 2-Agree 3-Somewhat Agree 4-Disagree 5-Strongly DisagreePresentation CriteriaRatingThe presentation was very well organised and the performance ran smoothly.The information was highly accurate and relevant to the engineering challenge.The presenter(s) face and maintains eye contact with the audience & do not read off the screenStudents use vocabulary and examples relevant to the engineering challenge and concepts covered during the preparation for the challenge.Presenter(s) speak in an understandable voice, using a clear tone, enunciation, and a reasonable pace; message is clearly received.Presentation was colorful, creative, and the information was easy to readGive 1 fact you learned: _________________________________________________The area for improvement is: ____________________________________________ ................
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