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Q&A #3790 |
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Hi Renae, A project that has worked well for me has students working across the topics of parametric equations, composition of functions, data collection and analysis, and technology to foreshadow the study of related rates in calculus. I begin the unit by helping students tie together parametric equations and composition of functions. Throughout the unit students will use graphing calculators. A typical problem I might provide: Each of Frosty's parts is a spherical snowball. When the sun comes out, his head (volume = 6 cubic feet) begins to melt at the rate of .25 cubic feet per minute. Frosty's corncob pipe falls from his mouth when the radius of his head is 9 inches. How soon after the sun pops out does Frosty lose his pipe? Solution: The volume is a function of time. V = f(t) = 6 - t/4 The radius is a function of the volume. R = g(V) = ((3V)/(4pi))^(1/3) Now compose the functions the functions. R = g(V) = g(f(t)) = (g o f)(t) = g(6 - t/4) =((3(6 - t/4))/(4pi))^(1/3) We now have 2 parametric equations in terms of t. Graph these equations on the graphing calculator and trace to radius = .75 feet. Frosty's pipe will fall when time equals 16.9 minutes. The next dimension I add in the unit is the use of real data and the regression capabilities of the graphing calculator. A typical problem I might provide: Using data from the United States National Center for Health Statistics showing birth years and the life expectancy for a selected population such as white males, black females or others, students are asked to enter the data in their calculator. They are to discover the graph and regression formula and graphically estimate the life expectancy for a selected person born in the year 2020. Then estimate in what year the life expectancy would be 100 years. Most of my students are in physics classes where they have used paper and pencil to find linear regressions. They were surprised during this phase of our work to find that with their graphing calculator a natural log, power, or exponential regression may be more appropriate. Next I provide students with data and ask them to analyze it and then students themselves propose several useful problem situations. The first time I asked students to design problems I was surprised to find how difficult it was for students to write clear solvable problems. However, as students develop their problem writing skills I am always amazed at the variety and creativity they display in their problems. A typical situation I might provide: From the American Journal of Public Health, a list of several countries, the cigarette consumption per adult per year in each country, and the coronary heart disease mortality per 100,000 people in that country is given. Students are to propose and solve several practical problems involving the information. One cooperative group proposed that the United States embark on a campaign to decrease the average cigarette consumption by 50 cigarettes per year with the goal being to establish when the mortality from coronary heart disease would be cut in half. Another group was quick to point out that even though the data for cigarette consumption and coronary heart disease showed a high positive correlation there could possibly have been other factors involved. The heated discussion that followed helped students see the importance of questioning hastily arrived at conclusions. The foundation has now been laid. Students can work with parametrics and composition of functions. They can use regression capabilities of graphing calculators to analyze data and look at sets of data and suggest meaningful ways for using the data. It is now time for students to step outside the classroom and use their skills. I challenge my students to use these skills to solve real life problems. As a unit project, working in teams, students will scatter throughout the school and community to gather data of related events and design appropriate problems. Some cooperative groups select a department in our high school to work with and some work with businesses in our community. Students will prepare ideas for projects before meeting with their resource persons. Contacts will be made and meetings held. The resource person will discuss the team's ideas and perhaps describe needs they have for data that could be gathered. Students design a method for gathering the data and the data is gathered. Students write a useful problem from the data that can be solved employing parametrics, composition of functions and technology. In past years, students have written problems investigating related rates such as: -school attendance vs. outside temperatures -number of plate lunches sold in the cafeteria vs. weather conditions -number of books checked out of the school library vs. number of students entering library -school baseball team wins vs. ERA over several seasons -pop rivet (Maurer Industrial Supplies) sales and the Dow Jones -furnaces (church plumbing, heating, and air conditioning, etc) sold and serviced vs. temperatures Through class discussions students design effective modes for communicating results. During this discussion many students come to realize that their future careers will very likely entail this same need for communication. Most students choose to include portions of their project in their portfolio. -Jim and Carolyn, for the Teacher2Teacher service |
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