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G ENERATING K NOWLEDGE AND P ASSION FOR S CIENCE AND M ATHEMATICS THROUGH M ODELING

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A recently released report 1 addressesd the challenges America faces in STEM fields. The report suggested that specialty science and math schools have demonstrated an ability to educate future scientists and engineers. However, even these schools face challenges, which include public support, funding, and quality curriculum. Atkinson et al. (2007). Addressing the STEM Challenge by Expanding SM&SHS.

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With respect to quality curriculum, how have we at the North Carolina School of Science and Mathematics (NCSSM) approached this particular issue? Our approach has been through modeling because it provides opportunities for students to engage real-world applications that are both challenging and motivating.

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C ALCULUS Standard set of optimization problems Finding the maximum volume of an open box constructed from a rectangular sheet of cardboard. Rectangular plot of land bounded by a river/cliff and 3 pieces of fence. Minimum distance between a stationary object and some other object that travels along a given path.

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W HAT DO STUDENTS / TEACHERS GAIN ? Students Practice with derivatives and solving equations. Superficial understanding. Passion? Teachers Minimal information about students’ ability to think.

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C OMMON S TUDENT QUESTIONS Who makes up these problems? Why do I want to know that? When will I use this?

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W HAT ’ S GAINED FROM A M ODELING APPROACH ? A deeper appreciation for mathematics. Exercises technical writing and communication skills. Generates enthusiasm and interest in the applications of mathematics.

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H OW DOES ONE BEGIN ? Do small labs, which provide opportunities to work with software (Mathcad) and basic ideas. Identify independent and dependent variables. If a max/min is known, what should graph look like? Then what information can we obtain about the characteristics of the function? Move on to a bigger project in groups of 3 to 4.

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E XAMPLE Researchers study the way salmon swim upstream to spawn. In theory, the energy the salmon expend in swimming against the current depends on the speed at which they swim as well as the amount of time they spend swimming. We have a function of two variables, but we can rewrite this as a function of one variable since time is equal to distance divided by rate.

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The graph begins with a point that we know to be the minimum. Based on that, what should the graph look like?

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Now, what function could possibly model this behavior? What do we need to consider? Distance the fish has to swim. Fish will expend a lot of energy if it swims slightly faster than the water. Fish will also expend a lot of energy if it swims really fast. Some constant of proportionality.

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c is a positive constant of proportionality D is the distance the fish has to swim r is the rate of the water v is the velocity of the fish p>0

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S UBWAY D ESIGN Suppose that you were building a new subway system. How far apart would you place the stations in order to minimize the time that a typical subway user takes to get from his point of origin to his final destination? Creating a model that will allow you to answer that question is the purpose of this investigation. Consider these two questions before you begin creating your model.

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QUESTIONS TO PONDER? What is/are the advantage(s) (in terms of a commute of shorter duration) of having subway stations close together? What is/are the advantage(s) (in terms of a commute of shorter duration) of having subway stations far apart?

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Note: The previous slides are only a portion of the problem we give to students. They are given guidelines (See Subway Design Student Version), which may be tailored to give more or less guidance. After reading the details of the problem, students then are asked to identify the independent and dependent variables as well as make some assumptions. Word of caution: Some students can go overboard with the assumptions and create a situation they have difficulty modeling. KEEP IT SIMPLE!!!

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T RAFFIC F LOW Everyone has at some time been on a multi-lane highway and encountered road construction that required the traffic to occupy only one lane each way. Naturally, the Department of Transportation would like to maximize the flow of traffic through this stretch of highway. What speed limit would be set for such a stretch of road to ensure the greatest traffic flow while also maintaining safety? In this investigation, you will explore how you might model traffic flow and answer a number of questions regarding this using calculus.

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I NFECTIOUS D ISEASE How does a disease like SARS spread? Groups to be considered are “people currently infected”, “people not infected now but who are susceptible”, “people who had the disease but recovered” and “people who died from SARS”. Once you have this modeled, how can you use the model to think about how to control the spread of the disease? A nice feature of this investigation is real data on spread of various recent diseases.

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P OLLUTION IN THE G REAT L AKES Water flows from Lake Huron to Lake Erie to Lake Ontario to the Atlantic Ocean. If a factory on any lake dumps pollution into the water, the “down stream” lakes are also polluted. What’s the long term behavior of this system? A nice enhancement to this investigation is to learn a new technique of integration so you can solve the system analytically.

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C HALLENGES FOR EDUCATORS ? Problems suitable to level of students. Finding new problems. Match to student interests.

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Richard Noble III (919) 416-2741 noble@ncssm.edu

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted in Theorem 1, the sign of the second derivative on an interval indicates the concavity.

Rogawski Calculus Copyright © 2008 W. H. Freeman and Company As noted in Theorem 1, the sign of the second derivative on an interval indicates the concavity.

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