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“Go with the Flow: Using Derivatives to Describe Storm Water Runoff Rates” Grade Level & Duration: Grade 12 Two 50 minutes lessons+ Field Trip Subject: AB Calculus Prepared By: Regina Lamendella Analyze Learners Overview & Purpose (STEMcinnati theme) Education Standards Addressed (Ohio). The purpose of this unit is for students to discover how engineers use derivatives to solve realworld engineering problems (Careers). Students will utilize urban, sub-urban, and rural storm water runoff data, to generate three different mathematical functions. Using the derivative students will discover differences in instantaneous rates of change of water runoff within these different settings (Application to the real-world). Students should be able to correlate water runoff rate data to watershed characteristics, identify challenges associated with increased runoff rates in an urban setting such as Cincinnati (Societal Impacts), and suggest appropriate best management practices for its control. Finally, this unit will conclude with a trip to the national, award-wining Sanitation District 1, for a tour of storm water management technologies utilized for reduction of urban runoff. At the end of this unit, students should be able to feel comfortable using derivatives to model change within environmental systems. (Understanding Technology) Predict how decisions regarding the implementation of technologies involve the weighing of trade-offs between predicted positive and negative effects on the environment and/or humans. Science: Students will be exposed to hypothesis driven research, data collection, and data analysis. Understand the meaning of the derivative in terms of a rate of change and local linear approximation and they should be able to use derivatives to solve a variety of problems. Technology: Students will be using Microsoft Excel to generate graphical representations of their collected storm water data and to generated the slope of the tangent line. Engineering: Students will learn how environmental and civil engineers design best management practices to reduce stormwater runoff. Students will also learn how engineers use derivatives to answer questions about environmental phenomena. Mathematics: Students will apply the derivative to describe differences in storm water runoff rates in urban, sub-urban, and rural watersheds. (Scientifics Inquiry) Derive simple mathematical relationships that have predictive power from experimental data (e.g., derive an equation from a graph and vice versa, determine whether a linear or exponential relationship exists among the data in a table) College Board AP Calculus Select Goals and Objectives Goals and Objectives (Specify skills/information that will be learned.) Teacher Guide Goals: Students should understand the meaning of the derivative in terms of a rate of change and they should be able to use derivatives to solve a variety of real-world environmental engineering problems. Objectives: Students will be able to: Explain how engineers use derivatives to solve engineering problems. Generate a graph representing storm water runoff data to create a graph representing the derivative of that function. Compare instantaneous rates of change in runoff across three different graphs, representing different watershed scenarios and relate these different values to watershed characteristics. Recommend appropriate storm water management technologies for reduction of urban runoff in a city such as Cincinnati, and how these management practices might impact the hydrographs they have created. Select Instructional Strategies – Information (Give and/or demonstrate necessary information) Direct Instruction: Powerpoint to show engineering disciplines that use derivatives to solve everyday problems. Presentation of storm water runoff and combined sewer overflows in the greater Cincinnati area. Indirect Instruction: Inquiry learning using three different watershed models for discovery of how derivatives are useful in understanding environmental phenomena, such as storm water runoff rates. Student Guide Materials Needed Microsoft Excel Microsoft Powerpoint Sharpie markers (3) Three Aluminum Baking Pans (11’’x16’’) Topsoil, Gravel, Sand (<1 cubic foot) Roofing Shingles (2) Graduated cylinders (3, 500 mL) Trash bags (3) Glue Acrylic paint Balsa Wood Drill of sharp puncturing tool Water (4 Liters) Watering cans (3) Packaging boxes (3) Utilize Technology Microsoft Excel- for plotting storm water runoff data and derivative of this graph Other Resources (e.g. Web, books, etc.) Microsoft Powerpoint- Direct instruction on how combined sewer overflows are a problem in the greater Cincinnati area and what type of engineers and scientists use derivatives to solve environmental problems Calculus: Graphical Numerical, and Algebraic. 3rd edition. AP edition. Finney, Demana, Waits, and Kennedy. 2006 Prentice Hall. Boston, MA. Storm water management technology-Students will be observing best management technologies (i.e. green roofs, porous pavement, riparian zones etc.) that Sanitation District 1 uses to reduce storm water runoff http://apcentral.collegeboard.com/ apc/public/repository/ap08_calculus _coursedesc.pdf http://www.sd1.org/education/ fieldTrips.asp#f4 For detailed “Catch” see attached “Catch Activity for AP Calculus” For Powerpoint presentation please see “Storm Water Presentation.ppt” For detailed description of constructing watershed models, please see attached “Constructing Watershed Models” For Student worksheet see attached “Student Storm Water Handout” For methods of storm water activity see attached “Methods for Storm Water Activity” Require Learner Participation Activity (Describe the independent activity to reinforce this lesson) Lesson 1 CATCH: Begin class with a “rainfall observation”. Have two prepared models. One model is blacktop and one model is turf. (5 minutes) Discuss observations. (2-3 minutes) PRE-ASSESSMENT: See below (5 minutes) Powerpoint presentation 1) How engineers need to use derivatives. 2) Why storm water runoff is a problem, especially in the Cincinnati area. Presentation of the class problem/activity (10 minutes). Three model Activity: Separate class into three groups. Each group works on a different watershed (urban, sub-urban, and rural) Students simulated rainstorm event and measure cumulative volume of runoff over time. See stormwater procedure for details (20 minutes). REVIEW: Make sure each group has water runoff data for their model. Refer to essential questions below. (i.e How will they graph the data they have collected? (dependent/independent variable) How will they determine which watershed scenario had highest rate of runoff? (5 minutes) Two students will be invited to come up and pour 500 mL of water out of a watering can onto two different models. See attached document for detailed description of CATCH. Lamendella_R_08_Major 1_Catch Students should be writing down observations. Students should observe that rates of runoff are different between the two models. Students have chance to ask questions about what engineers do and help formulate the methods for experimental design of class activity. Need a timer, a marker, a recorder, an overseer, and a rain-pourer. Collect water runoff data. Students record runoff water volume for each time interval in student worksheet. See attached document to find pre-assessment for this lesson. Lamendella_R_08_Major1_PrePostassessment See attached for the Powerpoint presentation. Lamendella_R_08_Major1_Powerpoint See attached for description of how to make models. Lamendella_R_08_Major1_WatershedModelDesign See attached for a description of the procedure for the activity. Lamendella_R_08_Major1_ActivityProcedure See attached for runoff volume data sheet Lamendella_R_08_Major1_DataSheet Lesson 2 CATCH: Short Video “Troubled waters”. This video will help students understand how combined sewer overflows impact human health and the environment (5 mins). Review of problem and data collected during Lesson 1. Discuss how to approach our objective (i.e graphing the runoff, curve fitting, graphing derivative, and identifying differences in instantaneous rates of change of runoff among three different watershed models (5 mins). Plot cumulative volume (y-axis) vs. time (x axis), plot values of slopes of tangent lines to create graph of the derivative in Excel Spreadsheet (15 mins) Combine storm water graphs and derivative graphs from each of the watershed scenarios (10 mins). Review (5-10 minutes): Discuss conclusions and environmental implications of water runoff in these Each group will plot their respective water runoff graph and derivative graph. Then one representative will bring their data to me and we will plot all three storm water graphs and derivative of those graphs together. Students will answer questions on their student handouts, comparing these graphs from the different watershed scenarios. http://www.youtube.com/watch?v=MJMtUa8i4Jc See attached for storm water runoff student worksheet. Lamendella_R_08_Major1_StudentHandout See attached for Excel spreadsheet of storm water cumulative volume graph and derivative graph. Lamendella_R_08_Major1_RunoffGraphs Require Learner Participation Lesson 2 (Cntinued) Activity (Describe the independent activity to reinforce this lesson) POST-ASSESSMENT (See below) (5 mins) REVIEW: Present the trip to Sanitation District 1 and some of the stormwater management practices they use. How will these practices impact stormwater runoff rates? (5 mins) Have students come to chalk board and draw the graphs. Essential Questions: 1. Name at least two differences in the derivative of the storm water runoff function for each of the three watershed scenarios. 2. What watershed characteristics result in the differences rate of change of storm water runoff within each of these different watershed scenarios? 3. Explain how you might use derivatives to solve other watershed related problems? Lesson 3 Field Trip to Sanitation District 1. Leave Norwood HS at 8:30am. Arrive to SD#1 at 9:30am. 9:30 am: Begin Guided tour of SD #1 to see storm water management practices (2 hours). Students will observe impacts of storm water management practices (i.e green roofs, porous pavement, detention ponds, wetlans, riparian zones) Lunch 11:30-12:15 (Donated by Kroger) 12:15 pm: Meeting with scientists and engineers that work at SD 1 to learn their educational goals, training, experience, and professional development. Students will also gain a better understanding of what environmental scientists and engineers do for the county. Students will gain and understanding of everyday problems that scientist and engineers solve within the environmental field. Evaluate (Assessment) See pre/post assessment sheet. (Steps to check for student understanding) See Student Handout sheet. Please see attached sheet for rubric for grading. Additional Notes Misconceptions: Discuss some assumptions/limitations of the models Measuring runoff at one point (real-world multiple points) No pipelines in models. In the realworld what would we measure? Student have trouble understanding the difference between discrete data (data point they collected) and a continuous function. Stress that we can only take derivatives of continuous functions. This is why we generate a best-fit curve, that generate a continuous function. Name:_____________________ Bell: ______ Date:_______ Pre/Post assessment on Applications of Derivatives 1. Define a derivative in your own words. 2. Each of the following professionals uses derivatives to solve real-world problems in their respective discipline. Match the professional (on left) with how they might use derivatives to solve a real-world problem (on right). Professional A. Chemical Engineer B. Environmental Engineer C. Civil Engineer Can use a Derivative to: i. Describe how fast a reactor can Make a new petroleum product ii. Describe how efficiently their Global Position System works iii. Describe how a fluid moves inside a reactor iv. Describe concentration of Chlorine in water distribution Pipes. v. Describe strength of material D. Electrical Engineer 3. You are a hydrologist who has been calculating storm water runoff data within the Mill Creek Watershed before, during, and after a significant rainstorm. Using storm water runoff data how you could calculate when the maximum water runoff rate occurred during this storm event? 4. List three different storm water management practices that could be used to reduce storm water runoff in the greater Cincinnati area. a. b. c. Reflection for Lesson #1. The three watershed models held up fine for two class periods and can be reused in subsequent classes. However, I would suggest using plastic tuberware stands in place of the cardboard stands, which became wet and subsequently weakened. Students were very excited to perform the group work, in which they measured runoff from the models. If I were to do this lesson again, I would use smaller groups, (i.e. 3-4 people instead of 5-6 people) as I noticed a few people standing around without a task. The additional people could measure storm water runoff from a fourth model, which could represent a modified Cincinnati with green roofs, porous pavement, etc. I would shorten the lecture on calculus and engineering to no more than 10 minutes, as I saw I was losing some of my audience. It might be more effective to ask the students their current understanding of what engineers do and how they use calculus, in order to break them of any misconceptions. Make each student record volume, so as to have multiple copies of data and to keep the students engaged in the twenty minute activity. Reflection for Lesson #2. Instead of working in teams for the Microsoft excel, make each student plot the data. I noticed that students who were more proficient at using excel took over the exercise and the less experienced simply watched. Instead of plotting the discrete data and taking the slope between two successive points on the graph, and using that to represent the derivative, I would fit the runoff data to a continuous function, generate an equation for the function, and take the derivative of the equation to produce a graph of the derivative. Reflection for Lesson #3. While the field trip was useful in giving the students a clearer understanding of what environmental engineers and scientist do, the trip could have been improved in the following ways. Since SD#1 is actually collecting data on storm water runoff from different scenarios, this lesson could be repeated with the students actually collecting data from their test sites, plotting it, and using the derivative to describe differential rates of runoff in a real-world scenario. Additionally, it would be beneficial to describe some of the limitations of the watershed design, and how scientist and engineers use computer models to estimate runoff in various watersheds (i.e SWIM 5.0)