Math 1200: Problems, conjectures and proof

FW 2009-10 Math 1200: Problems, Conjectures and Proof

Instructor: Mike Zabrocki (email: mylastname atsign mathstat period yorku point ca)

MZ's Office hours: Tuesdays 1-2:30pm in TEL 2028 (phone x 33980)

TA: Dorota Mazur

DM's Office hours: by appointment in Ross N611 (email: dorota@yorku.ca)

Lecture: Tuesdays 10-11:30pm in room TEL 007

Tutorials: Wednesdays 10:30-11:30am in room Ross S125

* Tutorials will meet in alternate weeks, both will meet in Ross S125.

Course Description: Students entering a university level mathematics program often lack the experience to deal with questions and problems when there is no obvious method to apply. One purpose of this course is to enable students to develop the confidence and ability to attack richer and more demanding problems. The attempt to check work and to explain one's discoveries to others leads naturally to the need for explanation and proof. Learning how to present convincing reasoning - or proof - is another course outcome. This course is about thinking and about communicating.

To do well in upper division courses at York University, students will need to be proficient in these types of skills and Math 1200 is a required first year course to help students succeed in their later courses. Class and tutorial attendance is mandatory and active participation is expected of all students.

The course textbook will be Mathematical Proofs: A Transition to Advanced Mathematics. The text is useful because it has lots of examples and problems. We will be covering Chapters 2-7 and occassionally digress in to subjects that appear in the other chapters. We will also be working with the most recent edition of J. Mason, L. Burton, and K. Stacey, Thinking Mathematically (Prentice Hall). The problems in this book are easily accessible while at the same time allowing for rich and varied investigations.

With an emphasis on communication/convincing argument, there is a critical contribution to be made by: group work, reading a proposed 'proof' including other student's work, presenting and discussing as a whole class. There is also value in working through several different approaches to solve a problem, and taking the time to understand an alternative approach offered by a peer in the class. Doing mathematics well includes talking and listening to mathematics and there will be assignments that require collaborative work with another student in the class, as well as support for forming study groups.

Prerequisite: 12U Advanced Functions and Introductory Calculus or equivalent.

Announcements:

(May 2, 2010) Here are the raw scores that I have for this class. The first column of this tables is the last four digits of the student id numbers. If you notice any discrepancies, please let me know.

(March 31, 2010) To study for the final you should practice problems in the book Mathematical Proofs from Chartrand, Polimeni and Zheng from chapters 0 through 9. My suggestion for practice is to do problem after problem after problem and that you should read the book only in response to working on the problems. Remember, this is a math text and not a history or medical text and you are not asked to memorize almost anything in this course. Many of the problems that I did for review over the last two days. Last years exam is available on the web page for the other course run by Eli Brettler. While some of the tutorial questions on that web page are similar, most of the homework problems are not and are another good source to go for practice.

(March 31, 2010) Look familiar?

If you look in the rollover in the comic on the xkcd page it says "I have never been totally satisfied with the explanation why e to the ix give a sinusoidal wave." I hope you are now...

(March 17, 2010) Course evaluations are online. I would really like your feedback for this class and I am going to make it worth 2% on the final. Is it beneficial to have a course like this starting as a math major? Are there subjects that we should have spent more time on? What could have been better about the class? Be specific with your comments. I am going to make 2% on the final exam if you have filled out the online course evaluations. You must do the evaluation by April 5, 2010! Please visit the site at: http://courseevaluations.yorku.ca/

(February 26, 2010) I posted the instructions for the second investigation project. This is due March 3 for Tutorial 1 and March 10 for Tutorial 2.

(February 23, 2010) The two comics that I showed in class about tautology and the principle of explosion are from xkcd.com. They are timely because they appeared during reading week.

(February 23, 2010) I am not printing out a copy of the homework #8, but I have posted it online and I showed it to you in class on the overhead projector. I will post in the next few days some more details about the investigation project and the due dates. They should be correct on the calendar below.

(February 8, 2010) Plan on getting your first draft of your second investigation project done by March 2. You should write up clearly and completely the solution to one of the tutorial problems. Then you should extend one of those problems in the same way that you did for the first investigation project. Please pick one of the tutorial problems 4 through 9 (I don't want to revisit 1 through 3) for this second investigation project.

(February 3, 2010) I posted a summary of the discussion that we had in class (with a few more details) about how the differences of a sequence can be used to find a polynomial formula for the sequence. There are two exercises at the end of the discussion and you should be able to solve both of them.

(November 20, 2009) The mathematics department organizes a team to participate in the mathematics modeling contest that will be held February 18-22, 2010. I will try to remember to mention this in class, but in case I forget, if you are interested in participating contact Jane Heffernan. This is the message that we were asked to circulate from her:

The MCM is a contest where teams of undergraduates use mathematical modeling to present their solutions to real world problems. Each team can have a maximum of three members who work together to find a solution to one of three posed problems. The solution may include mathematics as well as computer simulation. The team must also write a report on their solution. Problems are designed to be open-ended and are unlikely to have a unique solution. Attention must be focused on clarity, analysis, and design of the solution.

The MCM will take place on February 18-22, 2010. If students have any questions regarding the MCM they may email me (jmheffer@mathstat.yorku.ca) or look at the MCM website http://www.comap.com/undergraduate/contests/. If they are interested in participating they should email me by January 10. There will be an information/training session in late January.

(November 19, 2009) Tutorial 2, your investigation projects have been graded and are now sitting outside my office. I will not be here next week (Dorota will be administering the quiz). Please come by and pick them up when you can.

(October 27, 2009) Make sure you read through Chapter 2 of the book and do the problems at the end of the section. Look through the problems that we worked on in the tutorial and the lecture and try to explain the solutions to all the the problems that we discussed on those days all clearly. At the end of class today I asked you about two truth valued statements (we rushed out the the class a bit quickly) forall x in R, there exists y in R, x+y+3=8 vs. there exists a y in R, forall y in R, x+y+3=8.

(October 27, 2009) The second HW assignments are expected to be ready to be returned tomorrow (that is, ready for tutorial). I still have some of the first homework assignments. If you would like to pick up your homework at any time I am leaving them in a return box outside of my office door (TEL 2028).

(October 27, 2009) I am going to announce in class today about the component of the grade 'Investigation Projects.' I have told you all along that you will be responsible for following up with one of your tutorial problems. I want you to write up one of your tutorial assignments. There are three to choose from that have been posted on the web page. You should take one of these three and write a clear and complete solution to the problem. Then you should extend the problem itself in a non-trivial manner and then try to solve that problem.

(October 6, 2009) The assignments are graded and were ready to be handed back in class today but I wanted to take a look at them first. If you want your assignment you may come by my office (TEL 2028) to pick it up. There is a pickup box mounted to the wall next to my office and I will leave them in there. I will also bring them to tutorial tomorrow and class on the 20th.

(Sept 29, 2009) One day I will learn to use a calendar correctly. The homework assignment is due October 20.

(Sept 22, 2009) Here is a list of good study habits. It is a good idea to follow these for all your courses but a number of these suggestions are techniques to do well in math courses in particular.

(Sept 22, 2009) We have started learning the material in Chapter 2 of Chartrand, et. al. 'Mathematical Proofs'. If you want to read ahead a bit we will be looking more at more of the chapter in coming weeks. Try the problems at the end of the chapter for practice. I will give you specific problems starting next week.

(Sept 22, 2009) Here is the shorter syllabus that I will hand out next week (when I get the paper unjammed from the copier).

(Sept 22, 2009) The logic puzzle that we started in class is called "First Past the Post."

(Sept 16, 2009) I originally said that the homework given on the first day would be due on Sept 22. After discussing it a little in class I will make it due in 2 weeks rather than 1. The new due date is Sept 29.

(Sept 16, 2009) Those from tutorial 2 that showed up to tutorial 1, we will be doing roughly the same exercise next week. I marked you down so you don't need to come both weeks but if you want the extra practice and would like the time to discuss how to finish the problems then please come again.

(Sept 15, 2009) Math Lab will be opening this Wednesday September 16 from 10:30 am - 3:30 pm and will continue to be open Monday to Friday 10:30 am - 3:30 pm until the end of the term, Tuesday December 8. In December an alternate schedule will be posted for the Math Lab during exams.

(Sept 7, 2009) Note that the first class meeting will be September 15, 2009. All tutorials will meet in Ross S125 (note in the university schedule there is a different room for Tutorial 1 and Tutorial 2 but this is only for scheduling purposes).

Handouts:

(Sept 15, 2009) Homework 1- due September 29
(Sept 22, 2009) short syllabus that I will hand out on the 29th.
(Sept 23, 2009) Tutorial 1- rectangles in a square grid
(Sept 29, 2009) Homework #2- logic and the definitions of shapes
(Sept 29, 2009) An island with knights and knaves (and gold, if I'm not lying)
(Sept 29, 2009) Tutorial #2- CSI logic puzzle
(Oct 20, 2009) Homework #3- drill and skill because these looked a little weak on the quiz
(Oct 21, 2009) Discussion for the 3rd tutorial- open and closing locker doors
(Nov 3, 2009) Homework #4- justifying math statements
(Nov 15, 2009) Discussion for the 4th tutorial- Square take away
(Nov 16, 2009) Homework #5- sequences and patterns
(Dec 1, 2009) Discussion for the 5th tutorial- Hot dogs and buns
(Dec 1, 2009) Homework #6- sequences and patterns
(Dec 1, 2009) Discussion for the 6th tutorial- pick up sticks
(Dec 1, 2009) Discussion for the 7th tutorial- information around a table
(Feb 2, 2010) Homework #7- sets and relations, due Feb 9
(Feb 2, 2010) Discussion for the 8th tutorial- string around the Earth
(Feb 3, 2010) kth differences of sequences and polynomials
(Feb 23, 2010) Homework #8- statements about mod and divides
(Feb 24, 2010) Discussion for the 9th tutorial- dividing up a triangle
(Feb 26, 2010) instructions for the second investigation project
(Mar 17, 2010) Discussion for the 11th tutorial- exercises on relations, induction and summation notation

Evaluation:

The evaluation will be based on the following criteria

Participation	based on attendence and in class assignments	10%
Assignments	roughly one every 2 weeks	25%
Investigation projects	see below	20%
Quizzes	6 total, 3 per term, best 2 from each term	15%
Final Examination	Winter exam period	30%

Do your own work. Don't look for a solution on the web or take one from another student's work unless you already have found your own solution and intend to review another to make a comparison. Work that is not original will be graded accordingly. Presenting someone else's work as your own without proper citation is academic dishonesty. You must cite any internet sources which you have consulted. You will be required to take the York University Academic Integrity Tutorial.

Participation: You are expected to show your commitment to this course and your fellow students by sharing your mathematical knowledge and your feelings about the material. Attendence at the weekly classes and the tutorials is obligatory and you will lose 2 points from your course grade for each class or tutorial that you miss in excess of two each term.

Assignments: There will be roughly one assignment every other week. Most assignments will require explanation beyond a simple one or two word/numerical answer. It is good practice to RECOPY THE QUESTION EVERY SINGLE TIME when you do the assignment. This makes it possible to understand what the assignment when it is handed back to you and it attempts to reduce the error of answering a different question than is on the assignment. Full credit is given to papers which demonstrate deep understanding of the problem by providing multiple solutions and considers variations based on the original question when this is appropriate. Your assignment should include complete sentences and explanations and not just a few equations or numbers. A solution will not receive full credit unless you explain what your answer represents and where it came from. You may discuss the homework with other students in the class, but please write your own solutions.

Investigation Projects: After each tutorial, you are expected to continue working on the problems discussed. Each project will consist of the results of deep and sustained investigations of your choice of three (of the six or more) tutorial problems considered each term. As with the homework, you should consider multiple solution methods, extensions of the problems, relationships with other related problems. You should include a report on your experience (how you felt) during the process of investigation. Each term you will be required to hand in early (as an indication of your progress) your report for one (of the three) problems.

Quizzes and Final Examination: There will be 3 quizzes per term (dates listed below: Oct 6, Nov 3, Nov 24, Jan 13, Feb 10, Mar 10). A final examination will be scheduled for the April exam period and the date announced in late-February/early March. Quizzes

Meeting dates:

Details of the class meetings below will be filled in as the term progresses.

Lecture Schedule	Topics	Remarks
Sept 9/09 - Tutorial		does not meet (wait until after first lecture)
Sept 15/09 Lecture #1	discuss class, lines breaks the board into regions	homework #1 distributed
Sept 16/09 Tutorial 01	rectangles in a square grid
Sept 22/09 Lecture #2	problem solving strategies, logic, first past the post
Sept 23/09 Tutorial 02	rectangles in a square grid
Sept 29/09 Lecture #3	Chapter 2.1-2.9 in Chartrand, logic, tutologies, knights and knaves	hw #1 due, homework #2 distributed
Sept 30/09 Tutorial 01	Solve the crime or do the time
Oct 6/09 Lecture #4		Quiz #1
Oct 7/09 Tutorial 02	Solve the crime or do the time	hw #1 to return
Oct 13/09	No class	Reading week
Oct 14/09	No class	Reading week
Oct 20/09 Lecture #5	quiz #1 review, pentagonal numbers and telescoping sums	quiz #1 returned, hw #2 due, homework #3 distributed
Oct 21/09 Tutorial 01	opening and closing lockers
Oct 27/09 Lecture #6	'Investigation projects', exists and forall
Oct 28/09 Tutorial 02	opening and closing lockers	hw #2 to return
Nov 3/09 Lecture #7		Quiz #2, hw #3 due, homework #4 distributed
Nov 4/09 Tutorial 01	Square take-away	draft 1 of invest proj due
Nov 10/09 Lecture #8	justifying exists and forall statements	return quiz #2, return tutorial 1 invest proj
Nov 11/09 Tutorial 02	Square take-away	draft 1 of invest proj due
Nov 17/09 Lecture #9	sequences and patterns	hw #4 due, homework #5 distributed
Nov 18/09 Tutorial 01	Hot dogs and buns
Nov 24/09 Lecture #10		Quiz #3
Nov 25/09 Tutorial 02	Hot dogs and buns
Dec 1/09 Lecture #11	polynomials and sequences, tower of Hanoi	final invest proj due for tutorial 1, hw #5 due, hw #6 distributed
Dec 2/09 Tutorial 01	Pick up sticks
Dec 8/09 Lecture #12	Induction	return quiz #3, final invest proj due for tutorial 2
Jan 5/09 Lecture #13	Induction and tower of Hanoi revisited	hw #6 due
Jan 6/09 Tutorial 02	Pick up sticks
Jan 12/09 Lecture #14	Induction and differences of sequences	return invest proj
Jan 13/09 Tutorial 01	Information around a table	return hw#6
Jan 19/09 Lecture #15	Differences of sequences	Quiz #4
Jan 20/09 Tutorial 02	Information around a table	return hw#6
Jan 26/09 Lecture #16	sets, subsets and relations	hw #7 given
Jan 27/09 Tutorial 01	string around the earth
Feb 2/09 Lecture #17	relations	return quiz #4
Feb 3/09 Tutorial 02	string around the earth
Feb 6/09		Last day to drop the course without receiving a grade
Feb 9/09 Lecture #18	Forall/exists statements about relations	Quiz #5, hw #7 due
Feb 10/09 Tutorial 01	Dividing up triangle
Feb 16/09	No class	Reading week
Feb 17/09	No class	Reading week
Feb 23/09 Lecture #19	modulo equivalence (sec 8.5), go over quiz #7, swan logic	hw #8 given
Feb 24/09 Tutorial 02	Dividing up triangle
Mar 2/09 Lecture #20	more mod equivalence
Mar 3/09 Tutorial 01	factorial representation	first draft of 2nd investigation project due
Mar 9/09 Lecture #21		Quiz #6, hw #8 due
Mar 10/09 Tutorial 02	factorial representation	first draft of 2nd investigation project due
Mar 16/09 Lecture #22	one-to-one and onto functions	hw #9 given
Mar 17/09 Tutorial 01	Exercises on relations, induction and summation notation
Mar 23/09 Lecture #23
Mar 24/09 Tutorial 02	Exercises on relations, induction and summation notation
Mar 30/09 Lecture #24	review	hw #9 due, final draft of 2nd investigation project due
Mar 31/09 Tutorial 01

The following is a list of students enrolled in the two tutorials (as of Oct 27, 2009 3:00pm):

Tutorial 1	Tutorial 2
Stephanie Athayde Roman Ayala Deepali Bhikajee Sukhjit Brar Philip Christian Natalie Drumonde Meilin Duong Ming-Chieh Fan Suzette Fernandes Bao Huynh Dmitry Kryukovskiy Yee Lai Haein Lee Sanghyun Lee Laavanya Maheswaran Chia-Wei Mo Abdoul Niang Reubinder Sidhu Rajdeep Virk Elijah Wong Siu Wong	Lynn Cao Cheng Chen Andrew Choly Diane Da Costa John Galinaitis Fei Guo Robert Jordan Ilana Khmurov Colman Ladouceur Robert Levy Kristeen Marshall Bebe Mustafa Trung Ngo Saayma Rangrez Ibrahim Shakul Patrick Sin Mitchell Williams Daniel Zinn

There are three sections of this course. You may want to know what is going on those classes as reference and because of the resources provided by those instructors are relevant to our course as well.
Section A - Ami Mamolo
Section C - Eli Brettler