Holli Adams

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AMATYC Math Conference 2009Technical Writing as
a Vehicle to Learn Math

• Presented by
• Holli Adams
• hadams_at_pcc.edu
• 503 978-5677
• Jerry Kissick
• jkissick_at_pcc.edu
• http//spot.pcc.edu/jkissick
• 503 614-7606
• Portland Community College

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AMATYC Math Conference 2009Technical Writing as
a Vehicle to Learn Math

• Part 1 Using writing assignments to learn
  mathematical concepts
•  
• Holli Adams
•  
• Part 2 Using projects to enhance student
  understanding of mathematical concepts
•  
• Jerry Kissick

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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
• Item Date In class time
• Project description handed out. 2/6/06 5 min
• Ask questions about what is expected. 2/8/06 10
  min
• Each member bring their rough plan and
  discuss 2/20/06 10 min
• Group discussion on progress. 2/27/06 10 min
• Draft report ready for review by group
  members. 3/6/06 10 min
• Group members agree on final version of
  report. 3/8/06 10 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 members 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/dont 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
Table 1
100 679.243
110 700.249
120 715.52
130 726.32
140 733.896
150 739.089
160 742.633
170 745.038
180 746.663
190 747.758
200 748.495
of weeks of fish
10 321.095
20 349.128
30 384.744
40 427.489
50 475.427
60 525.26
70 573.146
80 615.802
90 651.312
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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.
Table 1
Week FISH population Average rate of change Instantaneous rate of change
0 300 -- 1.8
10 321 2.1 2.4
20 349 2.8 3.2
30 385 3.6 3.9
40 427 4.2 4.6
50 475 4.8 5.0
60 525 5.0 5.0
70 573 4.8 4.6
80 616 4.3 3.9
90 651 3.5 3.2
100 679 2.8 2.4
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Samples of how students begin this project.
110 700 2.1 1.8
120 716 1.6 1.3
130 726 1.0 .91
140 734 .80 .62
150 739 .50 .62
160 743 .40 .23
170 745 .20 .20
180 747 .20 .12
190 748 .10 .08
200 748 .10 .08
210 749 0 .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 biologists 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 750et/25
9et/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 750et/25 dramatically. This is also
true for et/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
Fish Weeks in Number of Fish Operation Opera
tion
0 300
10 321.0946256
20 349.1284493
30 384.7437262
40 427.4894616
50 475.4265302
60 525.2604325
70 573.1455111
80 615.8019549
90 651.3119684
100 679.2432249
110 700.249234
120 715.5199938
130 726.3496769
140 733.8955808
150 739.0890256
160 742.6331562
170 745.0377533
180 746.6627875
190 747.7580664
200 748.4949621
208 748.9062166
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Example of Project of Students 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 Students 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/gK.
• 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 Students 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.