Transcript Downloads - Mohawk College

MATH PROJECTS
Lynda Graham
Sheridan College
905 459 7533 (5017)
[email protected]
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WHY PROJECTS?
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noted by National Accreditation Board
shows relevance and the unity of
mathematics
encourages brain-storming and the
creative side of mathematics in openended projects
requires a deeper understanding when
describing the solution precisely in
words
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WHY PROJECTS?
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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:
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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
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about their learning.
HOW?
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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
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Statistics: report on a group quality control project
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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-Calculus
Instructions and Marking Scheme:
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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 [email protected] 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:
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correct mathematical calculations
clear, concise writeups which fully explain your group’s
thinkings/reasonings:
the problem clearly restated and all variables, terminology and
notation used defined.
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a log of your group’s meetings, times and activities
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a knowledgable oral presentation
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correct use of language: spelling, grammar and punctuation
clearly drawn and labelled graphs and diagrams
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10
15
100
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CALCULUS PROJECTS
This 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.
a)
b)
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 bike’s 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: Houdini’s 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.
Houdini’s 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
Houdini’s head? (Neglect Houdini’s volume and the volume of the
block.)
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 What is the height of the block? Show on a graph.
DIFFERENTIAL CALCULUS : a group
multi-step project
Example: Houdini’s Escape (2)
c)
d)
e)
f)
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
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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
group’s 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
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correct mathematical calculations
clear, concise writeups which fully explain your
group’s thinkings/reasonings: the problem
clearly restated and all variables, terminology
and any notation used defined
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a log of your group’s meetings, times and activities
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a written report using technology: a word
processing package/Mathcad/Excel, any references
cited
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correct use of language: spelling,
grammar and punctuation
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clearly drawn and labelled graphs and diagrams
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(presentation )
(10)
100
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STATISTICS: A QUALITY CONTROL
GROUP PROJECT
OBJECTIVE:
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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
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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
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2. Report Writing – grammar, spelling, style, report format 20
3. Mathematics, Statistics, charts
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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 “outof-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
operator’s 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;
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