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Presented by Holli Adams hadams@pcc.edu 503 978-5677 Jerry Kissick jkissick@pcc.edu http://spot.pcc.edu/~jkissick 503 614-7606 Portland Community College AMATYC Math Conference 2009 Technical Writing as a Vehicle to Learn Math

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Part 1: Using writing assignments to learn mathematical concepts Holli Adams Part 2: Using projects to enhance student understanding of mathematical concepts Jerry Kissick AMATYC Math Conference 2009 Technical Writing as a Vehicle to Learn Math

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Using Projects To Enhance The Learning Of Mathematics Use projects in all classes Projects require use of material covered in the class Require the use of technology to create reports Word processing Equation editor (MathType) Graphical software Calculator Maple Winplot Excel Students choice Give students problem statement, and in some cases, a project of their own choosing, and writing guidelines. Tell students their report is to be like a paper for a writing class only it contains math.

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Writing Guidelines These vary by class, but all contain essentially the same material. This project will be a group project. Groups will consist of at least 3, and no more than 4 members. Each group will collectively submit a report. (Papers must be word- processed. The mathematics in the paper must be created with some sort of math equation editor. Graphs must be created with a graphing program or be a downloaded calculator screen. These graphs must be incorporated into the body of the paper and not enclosed at the conclusion. The names of all group members shall be clearly marked on the cover page of the report with their individual assignments.

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Writing Guidelines The goal of the project is to document analyses and resultant solutions in a written report that is logically coherent, technically correct, and aesthetically pleasing. The report shall be written as a “ stand alone ” document. Therefore, reference to a copy of the problem statement shall be unnecessary. The reader of your report should be able to infer the problem statement simply by reading the solution. The project shall be graded on two criteria: Content and Presentation

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Content for the project shall include the following attributes: Mathematical calculations that clearly enumerate how a solution was derived. All relevant variable names and associated units shall be declared. Calculator key strokes need not be listed unless they add to the clarity of the analysis presentation. Graphs and figures shall show relevant and detailed information. Graphs axes shall be clearly labeled with variable names and, where appropriate, proper units. Where more than one graph is presented on shared axes, they shall be labeled such that each curve is clearly distinguishable. Graph scales should be clearly indicated on appropriate axes.

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Presentation for the project shall include the following attributes: Problem Statement/Introduction Elements of the problem being addressed should be clearly stated. An overview of the solution method may be appropriate. All relevant assumptions should be clearly stated. Analysis Presentation An overview solution method may be appropriate. Derivations of calculations shall be explained. All figures, graphs, and diagrams must be assigned an identifier. This may be a figure number or letter. Where appropriate, a descriptive caption should be added (e.g., “Figure 1: Distance as a function of Time:). All references to these figures in the text should include these identifiers. Conclusions and Results Final conclusions shall be stated clearly, together with supporting remarks.

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The following is a sample schedule given out in math 112 (Trigonometry) during the winter quarter. This project is to be performed in groups of 3 or 4 students. One report (two copies, one paper and one electronic) will be submitted for each group. Details of what is expected in the report are included in the following pages. Time period: ItemDate In class time Project description handed out.2/6/065 min Ask questions about what is expected.2/8/0610 min Each member bring their rough plan and discuss2/20/0610 min Group discussion on progress. 2/27/0610 min Draft report ready for review by group members.3/6/0610 min Group members agree on final version of report.3/8/0610 min Report submitted--2 copies. 3/15/06 If you cannot meet with your group members outside of class, it would still be a good idea to exchange phone numbers and/or e-mail addresses. Intragroup communication is probably the most important key to success.

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What Do You Do If A Student Does Not Contribute To The Group? The Following Page Is Now Included In My Projects. It is optional for you to turn in this page. If you feel the some members of your group did not contribute their share of the work, fill out the form indicating how much you feel each member of your group contributed to the completion of the project. 1. Participation. The purpose of the project is to integrate solving mathematical problems with writing and team working. Allocation of credit will be based on a group score and an individual score. Make sure each team member has an opportunity to make a significant contribution to the project. Each team member will submit a completed rough report and contribute to the final report. 2. Evaluation. A top notch report is clear, concise, complete, convincing and correct (logically, grammatically, mathematically, etc.). Your team will receive a grade on the basis of these qualities. Your individual grade will depend on the quantity and quality of your contribution. The project is due Wednesday March 15, 2006. 3. Written Report One complete report (each team member must contribute) and each team member’s supporting work (rough work for each part of the project). Submit 2 copies of the complete report. The second copy must be a computer file. You must include for each question all supporting mathematical calculations that clearly enumerate how the solution was derived. (This includes definitions of variables, intermediate calculations, graphs, scattergrams, and units of measurement.) 4. Individual Participation. Everyone may submit this sheet confidentially with an estimate of individual participation for each team member. (Scale ranges from 0 =”No show” to 3 = “Actively participated”). The individual grade will be determined by the amount of participation in the group project.

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Sample Project Assignments Beginning Algebra Grade Point Average (GPA) Students learn how to calculate GPAs and to estimate what grades they need to get to achieve a desired GPA. Uses basic equation solving and the concept of finding a weighted average. Source: Holli Adams Vertical Distance Viewed Through Tubes Students choose a tube and make measurements of the vertical distance viewed through the tube at different distances from a wall. Involves creating the data, making a graph of the data points, finding a best fit line and its equation. Students then predict the vertical distance that will be viewed at a given distance and then go measure the actual distance observed. Source: Joan Waldvogel PCC

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Sample Project Assignments College Algebra Fish Hatchery Students investigate average and instantaneous growth rates in a fish hatchery where there is a mandate to harvest a certain number of fish each week. This project involves a logistic function and is given early in a college algebra course where the students have studied exponential functions and been introduced to the graph of a logistic function. Requires table construction of average rate of change and instantaneous rate of change from given functions (the functions actually require calculus to create, so they are given in this project), function composition, creating graphs and interpreting the data created. Source: Holli Adams

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Sample Project Assignments College Algebra Yellow Light Students determine the length of time a traffic light should remain yellow under certain conditions involving vehicle speed, length of intersection, reaction time and braking time. Project involves the determination of a cross/don’t cross decision point when a signal turns yellow. Involves creating a function to model the stopping distance and the time to cross an intersection. Source: various, including modeling workshop in Tennessee in 1995. Projects for Precalculus, Saunders (Developed under NSF Grant)

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Sample Project Assignments Trigonometry Detecting Speeders Using Radar Using the concept of Radar and how it measures distance, calculate the speed of a vehicle. Requires the use of the law of cosines to determine distance measured by a radar gun. Project is usually given out immediately after covering the Laws of Sines and Cosines. Source: Projects for Precalculus, Saunders (Developed under NSF Grant)

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Sample Project Assignments Trigonometry Daylight Model and Seasonal Affective Disorder (SAD) Model the hours of daylight over a year and determine when and how much artificial daylight is needed to counteract SAD. Requires creation of a model of the data for daylight hours over a years time. Model is used to answer SAD questions. This project is given out right after covering the material on the meaning of all the constants in the equation. Source: various, Chemeketa CC, modified several times by PCC faculty

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Sample Project Assignments Differential Calculus Building a Smooth Bridge Design connections from a bridge to existing roadways to make smooth connections between the two. Calculate region where a barge with a crane can pass below the bridge. Use the derivative of a function to match a bridge to land so that their tangents are the same. Source: Washington Center Source Book for Revitalized Calculus (1995) Bike Tracks Given that the rear wheel of a bike traces out a sine curve, determine the path of the front wheel. Solution involves determining where the tangent line from the sine curve is located 1 meter in front of the rear wheel. Source: Forest Simmons PCC

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Sample Project Assignments Differential Calculus - cont Building a Better Roller Coaster Design the connections between pieces of a roller coaster so they are smooth. Involves solving a system of equations with 11 unknown variables. Source: Calculus, Concepts & Contexts by James Stewart. Most cost effective pipeline Given cost figures, determine the most cost effective route for a pipeline around and/or through a wetlands region. This is an optimization problem involving a cost function and a look at various pipe runs. Source: Washington Center Source Book for Revitalized Calculus (1995)

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Sample Project Assignments Integral Calculus Trout Reintroduction to Clover Creek Given data on trout requirements in terms of water volume and current flow rate, determine whether or not to re-introduce trout to 2 different parts of a creek. This problem involves integration and approximate integration and the creation of volume and flow rate functions. Source: Washington Center Source Book for Revitalized Calculus (1995)

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Sample Project Assignments College Algebra (Math 111C) Project Description A fish hatchery begins operations with an initial population of fish and a mandate to harvest 15 fish per week. The biologist who set up the hatchery estimated that the fish population can be modeled by the equation: where is the number of weeks the hatchery has been in operation. Unfortunately, the biologist took another job out of the country and you are now in charge. Your immediate task is to familiarize yourself with what is in place and write a report to the Fish Hatcheries Bureau in Washington D. C. explaining how the hatchery was set up and what is expected to happen. Your report should address the following: a. What is the initial population? b. Make a table which gives the fish population every 10 weeks for 4 years. Based on the table, describe what happens to the fish population over the 4 year period. c. When the value of is small, which parts of the equation are providing the strongest influence on the population? Explain. d. When the value of is large, which parts of the equation are providing the strongest influence on the population? Explain. e. What is the maximum population and when does it occur? Explain. f. Expand your table to include the average rate of change in the fish population for every 10 week period for 4 years. Based on the table, describe what happens to the average rate of change in the population over the 4 year period. g. What is the maximum average rate of change and when does this occur? h. Using the population equation and calculus, the biologist generated the following function which gives the instantaneous rate of change in the fish population: Expand your table to include the instantaneous rate of change in the population every 10 weeks for 4 years. i. Compare the values for average rate of change and instantaneous rate of change over the 4 year period, and comment on what you observe. j. Use function composition to write an equation which gives the instantaneous rate of change in the fish population as a function of time.

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Samples of how students begin this project Given the Equation: P (t) = 2250 + 750e^ (t/25) 9 + e^ (t/25) This represents the amount of fish the fish hatchery began when it started up and harvest 15 more fish per week. Representing the number of weeks the hatchery has been running is (t). This fish hatchery started with the initial population of 300 fish. Using Table #1 you can see the total fish population for every 10 weeks at a total range of 4 years.

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Samples of how students begin this project # of weeks# of fish 10321.095 20349.128 30384.744 40427.489 50475.427 60525.26 70573.146 80615.802 90651.312 100679.243 110700.249 120715.52 130726.32 140733.896 150739.089 160742.633 170745.038 180746.663 190747.758 200748.495 Table #1

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Samples of how students begin this project. Over this four year period the population increased rapidly and then began to increase at a much slower rate. In this given formula, 750e^ (t/25) in the numerator influences the results more when (t) is smaller and e^ (t/25) influences the population

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Samples of how students begin this project. Our fish hatchery was asked to harvest 15 fish per week. The fish population can be modeled by the following equation: Where P is the population of fish and t is time in weeks. The initial population of fish was 300. Table 1 shows how the fish population grew every 10 weeks for 4 years.

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Samples of how students begin this project. WeekFISH population Average rate of change Instantaneous rate of change 0300--1.8 103212.12.4 203492.83.2 303853.63.9 404274.24.6 504754.85.0 605255.0 705734.84.6 806164.33.9 906513.53.2 1006792.82.4 Table #1

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Samples of how students begin this project. 1107002.11.8 1207161.61.3 1307261.0.91 140734.80.62 150739.50.62 160743.40.23 170745.20 180747.20.12 190748.10.08 200748.10.08 2107490.04 The instantaneous rate of change is calculated using the following model:

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Examples Introduction This fish hatchery was established by a biologist who we will call Mr. Fischer, who took another job in another country and I was called to take charge of the hatchery. In a nutshell, this project is set up to explain to the Fish Hatcheries Bureau in Washington D. C., how this hatchery was set up and what is expected to happen. The biologist, Mr. Fischer estimated that the fish population could be modeled by the equation: … where t is time in weeks that the hatchery has been in operation. There is also a mandate to harvest 15 fish per week.

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Examples Purpose This report is basically to predict what will happen to the fish population during the span of four years. At what rate is the fish population growing, if there is any correlation between the average rate of change and the instantaneous rate of change. The fish hatchery was established with an estimated initial population of 300 which was arrived at using the equation:

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Examples I am writing to you to inform you of my findings, having recently replaced the old biologist who was handling the coordination of the fish harvesting program at this fish hatchery and what is my perception of the program. This report contains the original biologist ’ s set up, its operational goals and my analysis of this program. The initial population of fish was 300. The goal of the hatchery was to harvest 15 fish per week. My predecessor created the following model for determining the fish population: P(t)= 2250+ 750e t/25 9+e t/25 where t is the number of weeks the hatchery is in operation.

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Examples (Please note: when the values of t are small, the constants 2250, 750, and 9 provide a stronger influence on the model. However, as t becomes larger, thus making the exponential larger, multiplying it by the value of e, increases the product 750*e t/25 dramatically. This is also true for e t/25 in the denominator.) Using this population model we are able to create the following table which gives the fish population over every 10 weeks for 4 years:

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Examples Figure 1: Fish Population in 10 week intervals for 4 years Weeks in Number of FishWeeks in Number of FishOperation 0300 10321.0946256 20349.1284493 30384.7437262 40427.4894616 50475.4265302 60525.2604325 70573.1455111 80615.8019549 90651.3119684 100679.2432249 110700.249234 120715.5199938 130726.3496769 140733.8955808 150739.0890256 160742.6331562 170745.0377533 180746.6627875 190747.7580664 200748.4949621 208748.9062166

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Example of Project of Student ’ s Choosing How long does it take to fill and heat a hot tub? Introduction: Combining methods from both Physics and Calculus, this project sought to determine the time it takes to fill and heat a hot tub. In order to model the time it takes, not just one, but several differential equations are required. Some Physics: According to thermodynamics, in an enclosed system, the heat by an object is equal to the heat gained by another, thereby conserving energy within the system. We use the equation:

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Example of Project of Student ’ s Choosing Where Q is the heat absorbed or lost, c is the specific heat, and T is the temperature. In our case, c, the specific heat, is that of water, which is 1.00 cal/g · K. Step One: The primary source of water from the hot tub is from the cold, unheated water from the hose, and the hot water provided by the water heater. In this problem, the water heater has a 189.25 Liter capacity, and an output of 11.355 Liters/min. Also, the water has the ability to heat the water inside the tank 0.56° C per minute. Beginning with a the tank full of hot water, and with the temperature of the cold water refilling the tank as it outputs hot water as 10° C, we can set up a few differential equations to model the output temperature of the water heater.

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Example of Project of Student ’ s Choosing Our first differential equitation is to find the amount of cold water in the water heater at any given time. Assuming that the cold water coming into the hot water heater was always evenly mixed with the hot water already existing in the tank, we took an approach to first find the concentration of cold water in the tank.

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CALCULUS I LAB ASSIGNMENTS Calculus I has a lab which meets for three hours per week. Lab Assignments have been developed over the years by PCC faculty. The problems range from practice with computations to problems requiring a deep understanding of the of the meaning of limits and derivatives and the interpretation of what a derivative actually means.

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