MATH PROJECTS

1. MATH PROJECTS Lynda Graham Sheridan College 905 459 7533 (5017) lynda.graham@sheridanc.on.ca

2. 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

3. 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

4. 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.

5. 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

6. 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 .

7. 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@sheridanc.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

8. PEER EVALUATION OF PRESENTATION Evaluation by: ____________________________ Names of presenters:________________________ Date:____________________________________ Rate each below as satisfactory, good, excellentorneeds improvement. ________ correct graph ________ clear, concise explanation and use of mathematical terms ________ correct answer to problem ________ ability to answer questions on the subject Further comments:

9. 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 wl x – 0.5 wx2 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 ?

10. 2nd semester math Your Marks: • correct mathematical calculations 40 • clear, concise writeups which fully explain your group’s thinkings/reasonings: the problem clearly restated and all variables, terminology and notation used defined. 15 • a log of your group’s 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

11. 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.

12. 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?

13. 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)?

14. 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.) • What is the height of the block? Show on a graph.

15. DIFFERENTIAL CALCULUS : a group multi-step project Example: Houdini’s 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).

16. 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)

17. 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?

18. 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.

19. MARKING SCHEME: CALCULUS • correct mathematical calculations 45 • clear, concise writeups which fully explain your group’s thinkings/reasonings: the problem clearly restated and all variables, terminology and any notation used defined 10 • a log of your group’s 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

20. 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.

21. 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)

22. 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 operator’s lack of quality related training was also cited as a possible cause.

23. 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;