MATH PROJECTS

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MATH PROJECTS

• Lynda Graham
• Sheridan College
• 905 459 7533 (5017)
• lynda.graham_at_sheridanc.on.ca

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WHY PROJECTS?

• noted by National Accreditation Board
• shows relevance and the unity of mathematics
• encourages brain-storming and the creative side
  of mathematics in open-ended projects
• requires a deeper understanding when describing
  the solution precisely in words

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WHY PROJECTS?

• ability to recognize mathematics as a way of
  thinking and speaking about quantities,
  qualities, measures, and qualitative and
  quantitative relationships and to extend beyond
  to a level where you model your applications
• "preparation for" and "ability to" work with
  others in group activities and problem solving
  situations with an understanding of group
  dynamics for innovative decision making as well
  as conditions of "groupthink" that lead group
  problem solving astray
• ability to use a general problem solving
  technique and incorporate computer and graphing
  calculator technology to facilitate problem
  solving

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When selecting an existing project, or creating
one of your own, consider the following

• Does the project come with classroom
  instructional materials (e.g., teacher resources,
  student activities, rubrics and assessment
  tools)?
• What is the total time for project completion?
• Is the project collaborative in nature?  A
  collaborative project, particularly involving
  students outside your own school setting, will
  take more time and monitoring to help students
  learn how to be a part of a team and communicate
  appropriately with others.
• How will students benefit both academically and
  personally from their involvement in the
  project?  Their participation in an actual real
  world activity might encourage them to do their
  best work, and see the relevance of mathematics
  in their daily lives.  If students have input
  into project selection, and like the topic, they
  will tend to become more involved and excited
  about their learning.

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HOW?

• First semester pre-calculus a simple group (2 or
  3) word problem presentation
• Second semester pre-calculus a group one-step
  project report
• Differential calculus report on a multi-step
  group project
• Integral calculus report on a multi-step group
  project
• Statistics report on a group quality control
  project
• Reference Technical Mathematics Calter Calter,
  Calculus An Active Approach with Projects The
  Ithaca College Calculus Group

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1ST SEMESTER PRE-CALCULUS a simple group (2 or
3) word problem presentation Example

• The formula for the pressure loss h in a pipe is
  where f is the friction factor, L is the length
  of the pipe in feet, Q the flow rate in cubic
  feet per second and D the pipe diameter in
  inches. Calculate the pipe pressure drop in a
  pipe with a diameter of 2.84 in. and a length of
  124 feet. In this pipe, f 0.022 and the
  flow rate is 184 gal/min .

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1st Semester Pre-CalculusInstructions and
Marking Scheme

• In groups of 2 or 3, you will solve the problem
  assigned to you.
• Then on the specified day, you will give a brief
  presentation to the class the solution on your
  laptop and the whiteboard, if needed. Use the new
  graphing calculator Graphmatica in Downloads for
  a computer graph. Be prepared to field any
  questions from other students.
• A brief, computer-written report, showing your
  solution, is emailed to me at lynda.graham_at_sherid
  anc.on.ca on the due date or put in the
  assignment drop-box in Vista.
• Marks are for
• correct written answer to problem
• ability to explain the process of how the answer
  was obtained
• ability to answer questions from other students
• participation by every member of the group during
  the presentation
• any additional questions to pursue that you might
  have about the original problem

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PEER EVALUATION OF PRESENTATION

• Evaluation by ____________________________
• Names of presenters________________________
• Date____________________________________
• Rate each below as satisfactory, good, excellent
  or needs improvement.
• ________ correct graph
• ________ clear, concise explanation and use
  of mathematical terms
• ________ correct answer to problem
• ________ ability to answer questions on the
  subject
• Further comments

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2nd SEMESTSER PRE-CALCULUS a group (2 or 3)
simple project

• Example BENDING MOMENT
• The bending moment M at any distance x for a
  simply supported beam carrying a distributed load
  w N/m and length l is M 0.5 w l x 0.5 w x2
• a) What conic shape is the bending moment when
  w 1360 N/m and l 3.00 m ?
• b) Graph the conic on Graphmatica and estimate
  the zero bending moment and the maximum bending
  moment.
• c) Show on the graph the points of zero bending
  moment.
• d) Show on the graph the point of maximum bending
  moment.
• e) At what distance from one end of the beam
  will the bending moment be 1000 N/m ?

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2nd semester math Your Marks

• correct mathematical calculations 40
• clear, concise writeups which fully explain your
  groups
• thinkings/reasonings
• the problem clearly restated and all
  variables, terminology and
• notation used defined. 15
• a log of your groups meetings, times and
  activities 5
• a knowledgable oral presentation 15
• correct use of language spelling, grammar and
  punctuation 10
• clearly drawn and labelled graphs and
  diagrams 15
  100

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CALCULUS PROJECTSThis will be a culminating
application of derivatives in a multi-step
project.

• Objective
• You are to write a clear, concise solution to the
  problem.
• In the introductory paragraph(s), outline the
  problem and the major steps in your solution.
• Pictures and diagrams are essential and should
  be integrated into the solution.

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DIFFERENTIAL CALCULUS a group multi-step
project Example Bicycle Race (1)

• Jessica is a local bicycle racing star and today
  she is in the race of her life. Moving at a
  constant velocity k metres per second, she
  passes a refreshment station. At that instant (
  t 0 seconds) her support car starts from the
  refreshment station to accelerate after her,
  beginning from a dead stop. Suppose the distance
  travelled by Jessica in t seconds is given by the
  expression kt and distance travelled by the
  support car is given by the function
• (10t2-t3) where distance is measured in metres.
  
• This latter function is carefully calculated by
  her crew so that at the instant the car catches
  up to the racer, they will match speeds. A crew
  member will hand Jessica a cold drink and the car
  will immediately fall behind.
• How fast is Jessica travelling?
• How long does it take the support car to catch
  her?

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DIFFERENTIAL CALCULUS a group multi-step
project Example Bicycle Race (2)

• c) Suppose that Jessica is riding at a constant
  velocity k , which may be different than the
  value found in part (a). Find an expression for
  the times when the car and the bike meet which
  gives these times as a function of her velocity
  k . How many times would the car and the bike
  meet if Jessica were going faster than the
  velocity found in part (a)? or slower than the
  velocity found in part (a)?
• d) Consider a pair of axes with time measured
  horizontally and distance vertically. Draw
  graphs that depict the distance travelled by
  Jessica and by the car plotted on the same axes
  for the original problem (parts (a) and (b)) and
  for the questions of part (c). You should have
  three graphs one for the bikes velocity found
  in part (a), one for a faster bike and one for a
  slower bike. If Jessica had been going any
  faster or slower than the velocity you found in
  part (a) passing the drink would not have been so
  easy. Why? Justify your answer.
• e) A cubic polynomial P(x) has a double root at
  x a, then PN(a) 0. How does this relate to
  your answer for part (a) and to your graphs in
  part (d)?

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INTEGRAL CALCULUS a group multi-step project
Example Houdinis Escape (1)

• Harry Houdini was a famous escape artist.
  Houdini had his feet shackled to the top of a
  concrete block which was placed on the bottom of
  a giant laboratory flask. The cross-sectional
  radius of the flask, measured in metres was given
  as a function of height, y, from the ground by
  the formula with the bottom of the flask at y
  0.3 m .
• The flask was then filled with water at a steady
  rate of 2 m3/min. Houdinis job was to escape
  the shackles before he was drowned by the rising
  water in the flask.
• Now Houdini knew it would take him exactly 10
  minutes to escape the shackles. For dramatic
  impact, he wanted time to escape so it was
  completely precisely at the moment the water
  level reached the top of his head. Houdini was
  1.8 metres tall. In the design of the apparatus
  he was allowed to specify only one thing the
  height of the concrete block he stood on.
• Your first task is to find out how high this
  block should be. Express the volume of water in
  the flask as a function of the height of the
  liquid above ground level.
• What is the volume when the water level reaches
  the top of Houdinis head? (Neglect Houdinis
  volume and the volume of the block.)
• What is the height of the block? Show on a graph.

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DIFFERENTIAL CALCULUS a group multi-step
project Example Houdinis Escape (2)

• Let H(t) be the height of the water above ground
  level at time t. In order to check the progress
  of his escape moment by moment, Houdini needs to
  derive the equation for the rate of change as a
  function of h(t) itself.
• Derive this equation.
• How fast is it changing when the water just
  reaches the top of his head?
• Express h(t) as a function of time t.
• Houdini would like to be able to perform this
  trick with any flask. Help him plan his next
  trick by generalizing the derivation of part b) .
  Consider a flask with cross-sectional radius
  r(y) and a constant inflow rate . Find as a
  function of h(t).

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GUIDELINES FOR CALCULUS GROUP PROJECTS

• This project is an important part of this course.
  
• You will work in groups of two or three (no
  more) students.
• All members will receive the same mark for the
  group portions of the project.
• It should take at least two weeks to complete.
• (You will give a brief presentation to your
  fellow class members and they in turn will give
  you feedback)

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Checklist

• Does this paper
• Clearly (re)state the problem to be solved?
• State the answer in a few complete sentences
  which stand on their own?
• Give a precise and well-organized explanation of
  how the answer was found?
• Clearly label diagrams, tables, graphs or other
  visual representations of the math?
• Define all variables, terminology and notation
  used?
• Give acknowledgement where it is due?
• Use correct spelling, grammar and punctuation?
• Contain correct mathematics?
• Solve the questions that were originally asked?

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• 1. Group Work. Start early, since projects
  require development of ideas and clear, concise
  writeups. It is important that everyone in the
  group understands how the problem is being solved
  and any group member may be asked to report on
  the groups progress. There should be a group
  leader/secretary and as a group you may want to
  rotate this position.
• 2. Consultations. Feel free to consult me about
  your project. I will try to help with
  difficulties without giving away the solutions.
  If you submit your report a few days before it is
  due, I will read it to detect any major problems
  and return it for revisions before the due date.
• 3. Formal Writeup. A word processing package
  could be used for the writeup. Equations and
  graphs may be neatly hand written or produced on
  a computer. Be sure that names of all group
  members appear on the cover page.
• 4. Meetings. Meetings should have a structure
  and a time limit. Think about the project before
  the meeting. Before the end of any meeting
  decide on what is to be done and who is going to
  do it.
• 5. Log. Your group should keep a log. It
  should include (at least) times you met, members
  who attended, summary of decisions reached, etc.
  
• 6. Oral Presentation. Everyone in your group
  should demonstrate a thorough knowledge of the
  problem and solution. Your peers will fill in a
  sheet marking you on what they liked and what
  they had learned.

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MARKING SCHEME CALCULUS

• correct mathematical calculations
  45
• clear, concise writeups which fully explain your
• groups thinkings/reasonings the problem
• clearly restated and all variables, terminology
• and any notation used defined 10
• a log of your groups meetings, times and
  activities 5
• a written report using technology a word
• processing package/Mathcad/Excel, any
  references
• cited 10
• correct use of language spelling,
• grammar and punctuation 10
• clearly drawn and labelled graphs and
  diagrams 10
• (presentation )
  (10)
• 100

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STATISTICS A QUALITY CONTROL GROUP PROJECT

• OBJECTIVE
• TO TELL A STORY THAT IS CLEARLY UNDERSTOOD, ABOUT
  HOW THE PROBLEM WAS IDENTIFIED AND ABOUT HOW
  YOU ARRIVED AT YOUR RECOMMENDATION OF A
  SOLUTION, WHICH HAS BEEN VERIFIED THROUGH THE
  USE OF STATISTICAL TOOLS
• The report must describe all phases of the
  project and provide the reader with a clear
  picture of your process, as well as of the model
  results.

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MARKING SCHEME STATISTICS

• 1. Analysis explanations, conclusions 25
• 2. Report Writing grammar, spelling, style,
  report format 20
• 3. Mathematics, Statistics, charts
  55
• 100
• Marks in more detail
• 1. A thorough description/story of a quality
  improvement process from start to finish (10)
• Summary/Objectives/Analysis/Conclusion
  Recommendations (15)
• 2. Title Page/ Table of Contents/ Appendix, as
  needed/ Bibliography (9)
• Page numbers and titles on graphs (6), spelling,
  grammar (5)
• 3. Charts Cause Effect Chart, Pareto Chart,
  Control Charts (25)
• Frequency Distribution, Histogram, Measures of
  Central Tendency and Spread (15)
• Identification of patterns and problems in your
  analysis i.e., Control Charts (5)
• Statistics supporting decisions in control
  and capable (10)

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STUDENTS STATISTICAL SUMMARY (1)

• The following is a technical report of a quality
  control sampling research conducted on April 10,
  2040 at machining center 1 at MelFaJo
  Technologies Incorporated, located at 1202
  Sheridan Way, Jamaica, Mars.
• The ISO department commissioned the research
  after a number of complaints by the operator at
  machining center 2 concerning out-of-spec
  parts received from machining center 1. A total
  of 100 samples were taken. Statistical methods
  such as Frequency Distribution, Histogram Graphs,
  Control Charts, and Central Tendency Measurements
  were used in the analysis of the sample data.
• The report showed that there were dimensional
  inconsistencies in the range of samples taken. A
  loose, defective bolt on the clamping device was
  found to be one of the contributing factors,
  therefore it was replaced. Another reason was
  found to be that an aging, out of line machine
  was being made to do a high precision job. The
  operators lack of quality related training was
  also cited as a possible cause.

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STUDENTS STATISTICAL SUMMARY (2)

• Recommendations were made to
• Assign the process to a newer, more precise
  machine located elsewhere in the plant
• Mandate more frequent measurement checks by the
  operator
• Mandate more frequent measurement checks by the
  supervisor
• Mandate more quality control training for both
  the operator and the supervisor