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MATH PROJECTS

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

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

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

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.

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

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 .

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

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

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 ?

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

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.

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?

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)?

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.

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).

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)

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?

- 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.

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

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.

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)

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.

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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