Instructor: Mike Zabrocki
Office: TEL 2028
Telephone: 416-736-2100 x33980
E-mail: mylastname at mathstat pt yorku pt ca
web page:

Math 1200 Section B
Problems, Conjectures and Proofs

Below is a list of remaining dates in the year. Hopefully I have this one right.

Date Section Notes
Feb 2/09 Cat Ladies REMEMBER: class is in Ross 525!
Feb 9/09 Apple Pi HW 7 due
Feb 18/09 Cat Ladies (Note: Wed) make up date for Fall, Quiz 3, HW 8 due
March 9/09 Apple Pi First day of 'winter' term for us
March 16/09 Cat Ladies ninth homework assignment due
March 23/09 Apple Pi tenth homework assignment due, talk about projects
March 30/09 Cat Ladies Quiz 4
April 6/09 Apple Pi project descriptions are due in person/by email BEFORE this date
April 13/09 Cat Ladies twelfth homework assignment due (and by 12th, I mean 11th)
April 20/09 Apple Pi Quiz 5
April 27/09 Cat Ladies thirteenth hw assignment due (12th)
May 4/09 Apple Pi
fourteenth assignment due (13th)
May 11/09 Cat Ladies Project Presentations
May 20/09 Apple Pi (Note: Wed) Make up day for Winter, Quiz 6

Holidays and Important dates:
Sept 29, 2008 Rosh Hashanah begins
Oct 13, 2008 Thanksgiving
March 10, 2009 Last day 2 drop w/out penalty
Feb 16, 2009
Family Day
May 18, 2009
Victoria Day
May 20, 2009
Makeup day
Oct 6, 2008
Quiz 1
Nov 3, 2008
Quiz 2
Feb 18, 2009
Quiz 3
March 30, 2009
Quiz 4
April 20, 2009
Quiz 5
May 20, 2009
Quiz 6

May 11, 2009 project presentations

Tutor: Dorota Mazur :
Class: Ross 525 from 7:30-9pm

Tutorials: The class has been divided into two groups and will meet in alternate weeks. Those that are in Tutorial 1 are now Cat Ladies and Tutorial 2 is Apple Pi. The list of who should attend which tutorial is below (I will update it after a week or so). If you are confused about how you became a Cat Lady or an Apple Pi please ask your classmates (they are to blame).

Cat Ladies and Apple Pi will meet in South Ross 525.
Note that I switched the class meetings after the first day because September 29 is when Rosh Hashanah begins and it is listed on the Important dates only as a footnote. Since we won't meet the lecture at 7:30 that evening it is better if we don't meet the tutorials either.

Cat Ladies
Apple Pi
Patryk Ambrozy
Nadine Mohammed
Ata Munim
Jonathan Weltman
Mithika Jegasothy
Robert Jordan
Phu Liem
Reuben Weltman
Shaofang Xu
Katryna Sa
Weiyang Shi
Claudia Diaconescu
Irena Dikushin
Reynaldo Isip
Lee-Ann Attong
Brett Bridges
Alvin Chou
Kajethra Umathevan
Iru Shah
Zakir Khoja
Marlon Walker
Huyanh Tran
Sanjam Suri
Ji hyun Kim
Bianca Zeppa
Lei Zhao
Naamah Jacobs
Sunhye Lee
Diana Malandrino
Adil Durrani
Jimmy Lim
Natasha Lunardo
Darek Dufaj
Victoria Panthaky
Jason Larabee
Divya Na
Brant Nanton
Ranindupal Singh
Adil Durrani

If you are not a first year student or transfer student who is starting this year AND a math major then it is unlikely that you need to enroll in this course. If you do not need to be in this course then I suggest that you go to the undergraduate office on the 5th floor (speak to Janice).

Text: John Mason with Leone Burton and Kaye Stacey, Thinking Mathematically.
Gary Chartrand, Albert D. Polimeni, Ping Zhang, Mathematical proofs : a transition to advanced mathematics

Statement of Purpose: This is a critical skills course. Here are some questions to consider.

  • Just what are the objects which you consider when you do mathematics?
    What is meant by the fraction one half? How does it represent a ratio? How does it represent a quantity? Are these conceptions different? Can you reconcile them?
    How would you describe a triangle to someone (for example a blind person) who has never seen one.
    How would you describe a circle?
    What conventional conceptions do you have which inform your own thinking about these and other mathematical objects?
  • What is meant by a proof?
    How you convince yourself, and how do you convince others that an answer is correct? What are the conventions for presenting concise mathematical proofs? How well does the presentation reflect the means by which a particular mathematical discovery was made? What does it mean for an ordinary language argument (mathematical or otherwise) to be valid? What is a counterexample? How does one make conjectures and how does one go about trying to assess whether they are correct?
    It is pretty easy to convince oneself or others of the correctness of answers which seem intuitively correct. What is much harder is to convince when answers while correct are counterintuitive. An example some of you may have seen is the "Monty Hall Problem".
  • Can you learn problem solving?
    Most of the problems you solved in High School were done mechanically or by mimicking solutions to similar problems in the textbook? What means are available to deal with problems which are genuinely novel?
    The text, "Thinking Mathematically" by John Mason has a rich selection of problems for consideration. Most require minimal technical background but almost all require hard thinking. Mason suggests a way of working strongly grounded in self awareness both in terms of what you are doing, and how you feel while doing it.
  • Are there techniques which extend your problem solving and proving capabilities?
    You will learn about combinatorial proofs which are arguments based on the analysis of situations rather the manipulation of formulas.
    You will learn about recursive methods and mathematical induction as a tool in calculations and in proofs.
    You will learn to use representations from other branches of mathematics (for example, geometric models to solve probability problems) to help obtain answers.
    You will learn to present proofs and explanations which are concise and logically correct.

  • Evaluation:

    Participation Based on attendance and convener presentation 10%
    Individual Investigation and Writing Assignments Normally one each week 30 %
    Group Investigation Project Winter term 15%
    Quizzes 3 Fall, 3 Winter 15%
    Final Examination Winter examination period 30%

  • Participation: Participation is how you show your commitment to the course and to the other students taking the course with you. You are expected to share both of your mathematical knowledge and the feelings you have as you engage in doing mathematics.
    Attendance at the weekly classes and at the tutorials is obligatory. You will lose 2 points from your course grade for each class or tutorial in excess of two which you miss each term.
    The participation grade normally is divided 5 points for participation in class and tutorial and 5 points for work as convener.
    The convener receives all student solutions for one assigned problem and is expected to prepare a report on those solutions (different approaches and solution methods, what worked, what didn't, what difficulties arose both in finding a solution and in finding a way to justify and explain it) for presentation to the tutorial. The presentation should include the convener's (original) solution as well as an indication of how his thinking about the problem may have changed after reading the other solutions. Each student will have one opportunity during the course to be convener.

  • Individual Investigation and Proof Assignments: Problems for investigation and solution will be posed weekly. Solutions are to be handed in. You may be asked to include a journal style discussion of how your solution or solutions were discovered.
    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.
    Some assignments will be designed so that they can be handed in a second time with corrections. These will be submitted electronically using the online course forum. Assignments may be accepted with penalty and will be given a weight of 10% less for every day that it is late. Late assignments should be handed in to the receptionist at TEL 2005 to be placed in my mailbox.

  • Group Investigation: Groups normally will normally consist of three students, with groups of size two or one permitted. This is a project assignment and must include a creative component. There will be a written report and a poster presentation. Researching a known subject is okay only if the presentation has original elements. More detail will be available at the beginning of the winter term.

  • Quizzes: There will be 6 in class 20 to 30 minute quizzes, 3 per term.

    Here are some sample quiz question types:

    1. Given a problem and a sketch of a solution, formulate a more complete solution and present it with justification.
    2. Given a proof of some result, find any errors and correct them.
    3. Given various conjectures, find counterexamples if false, proofs if true.
    4. Provide simple proofs including direct proofs, indirect proofs, proofs by mathematical induction.
    The grade will be obtained by taking the average of the best 2 quiz grades from each of the terms. There will be no makeups for missed quizzes.

  • Final Examination: This will be a conventional timed, closed book exam, scheduled during the University Final Examination period. Question types would be similar to those examples given for the quizzes.

  • Check this web page frequently for announcements, postings and updates.


    (Sept 7/08) The tutorials will not meet before the first class. Note that Cat Ladies will meet Sept 15.

    (Sept 8/08) For homework do the first 3 problems on the handout that was given in class and is posted below. Also read the first chapter of the textbook Thinking Mathematically. I am hoping it will give you ideas on what to do if you are stuck on these problems.

    (Sept 9/08) Someone pointed out to me that we do not have lecture the night of September 29 because of Rosh Hashanah and I had scheduled the first quiz that evening and so it has been rescheduled and almost all dates for the Fall term have changed.

    (Sept 16/08) On wikipedia the multiplication principle is called 'rule of product' instead of the multiplication principle and the addition principle is called 'rule of sum.' More links: link1, link2, link3 and link4. The reason why we had to spend so much time last night discussing it is because many of your explanations will appeal to it.

    (Sept 16/08) Usually I will be in my office Mondays 3-5 but email me if you are planning on coming during this time. I am usually available, but I may leave if no one is there. You may email me or the TA (Dorota Mazur : if you would like some extra help or to meet outside of class.

    (Sept 16/08) Several people asked how to hand in late homework last night after class. Late homework should go to the receptionist at TEL 2005 and they will place it in my mailbox. Please note my policy on late assignments that is listed above (namely that there is a penalty of 10% per day).

    (Sept 16/08) I will be giving homework assignments that include some problems that everyone should be able to do and some problems that should challenge everyone. Your goal should be to write and explain yourself as clearly as possible. Make sure that you understand the problem completely and try examples and then try to explain the examples. The textbook has some good techniques for how to identify the problem when you are stuck and how to get around the block. If you cannot give me a solution to the homework problems you should at least be able to explain what it is you are stuck on.

    (Sept 19/08) As you write your homework keep in mind that it should be clear enough that you will be able to reread it long after you have finished the assignment. This means that you should include the question (either rewrite it or make sure that the question is with the assignment). You should use complete sentences and not just some equations, and state clearly what it is you have shown at the end. You may think this is useless and tedious to add this detail but imagine if you are the person who is reading these papers part of your assignment for the class will be to do exactly that for the homeworks of your classmates in your tutorial.

    (Sept 23/08) I've posted the homework that will be due on Oct 6. In general, the homework will be due two weeks after I assign it. I've also posted the convener instructions. I will post my convener solutions for the first homework assignment shortly.

    (Sept 23/08) Someone asked me immediately after class about the first question on the homework assignment because the "Answer" and perhaps even the question that I give there is unclear. This is the type of thing that I would like you to critique about the solution. What is vague? What about the solution needs to be made clearer? Note that a "word" in that question is any sequence of letters and not necessarily a word that you will find in a dictionary.

    (Oct 20/08) I forgot to put on the 4th homework assignment on the web page (yes, there was one). It is now posted below. Remember that when you do the convener part, you should be writing solutions that everyone in the class can read and understand.

    (Oct 20/08) Dorota corrected the quizzes this past week and the average was 2.91/6 for Apple Pi and 3.66/6 for Cat Ladies.

    (Oct 20/08) I posted the homework assignment that is due on October 27 on the web this evening for those that will not be making it to class tonight.

    (Oct 27/08) Please remember that when you are writing the convener summaries you are essentailly writing the solutions for that homework. It needs to be neat and legible and clear enough to tell what the answer is.

    (Oct 27/08) Remeber that there is a quiz on November 3. We have a larger room booked for that evening in North Ross 203 for the lecture only. 7:30-9:30pm

    (Oct 27/08) We will be holding all tutorials and lectures (except when we have quizzes) in South Ross 525. This room is bigger and more comfortable.

    (Oct 28/08) There is a mistake on the 3rd homework problem. It needs to read "where 0<=c_i<=i." I will correct the page if you want to print out another copy.

    (Oct 28/08) It was pointed out that the quiz link was broken last week. This has been corrected. I didn't write up solutions for the quiz because Dorota went over in class on Oct 20.

    (Oct 29/08) Someone asked me about problem 2 on the 6th homework assignment. When n=0, the left hand side is an empty sum (the number of terms in that sum is n).

    (Oct 29/08) Check out Club Infinity on the 5th floor of Ross building across from the undergraduate office (Ross N537). This is a good place to go and hang out to work or meet other York students and math majors (hey, all first years need to take this course, why not commiserate together?). I think that tomorrow is the annual halloween party.

    (Nov 3/08) Don't forget that we have the quiz this evening and are in North Ross 203 tonight starting at 7:30pm. Apple Pi will be meeting in South Ross 525 again at 6pm. I have seminar at 3pm today so I won't be available between 3-4 this afternoon.

    (Nov 3/08) I said that I think that the homework assignment is a little long (although still only 4 questions). I said you can do problem 1 and 2 and then one of 3 or 4. For pratice you should do both 3 and 4 but you don't need to write up both when you hand it in.

    (Nov 4/08) I did (what I thought) was one of the harder problems from homework 6 and typed it up and posted it below. Read it carefully because I had to modify the problem slightly in order to justify the statement and you should convince yourself that it does answer the question from the homeowork after the modification.

    (Nov 6/08) Someone pointed out to me that the first question on the homework should be clarified because it does not state that the a_i need to be integers. This is a good point but it is implicit because of the definition of polynomial requires that the exponenets be integers.

    (Nov 6/08) Bad news. CUPE 3903 voted to strike and it didn't look like the vote was even close. Until the strike is over the University has cancelled classes. I am not on strike but I won't be going to my office much. Let me know if you have any questions on the homework. Watch the York University web site for news on updates on the strike. If the strike goes on for a while I will expect you to have done all four of the homework problems and not just three. :)

    (Nov 10/08) It doesn't look like this strike is going to be resolved any time soon. It seems like a good idea to implement the bulletin board system. The math department has a math forum board where you can post message about this class, vent about the strike, discuss your favorite proof of question #5 of homework 7, swap chocolate chip cookie recipes, etc. Let me know if you have any problems logging in (this is an obvious issue the first time that we use new technology). You will need to activate your AML account through passport York - select Accounts-Manage My Services. I have contacted computing services about activating the AML account so if this is not an option within passport York for you yet, then it should be soon.

    (Nov 12/08) I've posted a set of solutions to the second quiz below.

    (Jan 28/09) Cross your fingers that we will be going back to class soon. We will have to restructure this class a bit when we go back. Where we had the luxury of time before because we were covering the course over an extended period, we will now be learning the same amount of material with much less class time than I had hoped ( if we didn't already have challenges with a course like this and no strike). We will have a new textbook when we go back. Also I expect to change the convener process. It wasn't working the way I expected, partly because there was too much lag time in returning homeworks.

    (Jan 30/09) We are starting back on Monday! The 7th homework assignment will be due on FEB 9 (not the first day back). Cat Ladies meets on Feb 2. If you want to recall which section you are in look at the list above. If you have any questions feel free to email me and also post messages on the forum. I am looking forward to seeing you all.

    (Jan 30/09) I will post a revised calendar here when I get the chance to work through the dates. For now we will be following the revised calendar
    This means that we have 3 class meetings to finish the Fall term on February 2, 9, 18 (Feb 16 is Family Day, Feb 18 is a makeup day for Monday classes). Cat Ladies meets Feb 2. Apple Pi meets Feb 9. Cat Ladies meets Feb 18.

    (Feb 2/09) I've placed a list of revised dates above. I hope I have this right now but last term I found out only later that a few dates I had did not match the school's calendar.

    (Feb 10/09) Don't forget that there is no class on Monday (Feb 16) and we have a quiz on Wednesday (Feb 18). We will have the quiz in the math lab. Some people have asked me what it will cover. It should look similar to the last quizzes and homeworks. You can expect topics like: logic, coin weighings, paths in an nxk rectangle, induction, 1-1/onto, binomial coefficients. Please don't leave your questions to the last minute, make sure that you understand the most recent homework assignments.

    (Feb 13/09) I have been searching online for the textbook, Mathematical proofs : a transition to advanced mathematics by Gary Chartrand, Albert D. Polimeni, Ping Zhang, and I find it costs about $125. We will be using the 2nd edition, but I imagine that the first edition is close enough for our purposes. A copy has been placed on reserve at the library (first edition). I am working on getting another copy placed on reserve (second edition) soon.

    (Feb 13/09) I have written up an example of using paths in a rectangle to arrive at binomial identities. Try out the example problems if you want to practice more for the quiz on Wednesday because there is likely to be a question of this type there.

    (Feb 19/09) I posted the quiz and I will post the solutions when I get the chance. There is no homework due when we get back but what I am going to do is start assigning homework problems that will be due for the 'convener' part of the grade soon (because the past 'convener' assignments were not perfect homework solutions that I expected).

    (March 10/09) The quiz average was roughly 10 out of 18. There were three questions on the quiz and you should have been able to answer 1 out of the 3 correctly to pass this course (6 out of 18). On this one quiz you can count 13-18 is B+ to A+ range, 7-12 is C to B range, 0-6 is C- or below. A similar distribution seems to be holding for the other two quizzes. If you are consistently scoring in the bottom 33% on these quizzes please come in and see me. I can help you do better.

    (March 10/09) The proof that I gave that the square root of 2 is irrational is roughly the same as the one posted here. After you have seen it 20 times or so you start to know how it is done as soon as you start writing it. When Ravi asked "why do you need to assume that a and b have no common factors?" the answer I wanted to say was "I'll remember why before I get to the end of the explanation. I just remember that I need to do that." My memory does not work very well and I don't remember how I did answer his question.

    (March 16/09) The book I refer to on the 10th homework assignment is Mathematical Proofs by Chartrand/Polimeni/Zhang. I left a second copy at the library to go on reserve last night. You don't need the book to do the assignment. The questions are the (a) through (f) part of what is on the page.

    (March 22/09) There are a few typos on the homework assignment. I corrected the pdf posted here after someone pointed out that part (d) especially is unclear. If should read "if 1-n^2>0."

    (March 23/09) I am posting the project proposal instructions. I want a paragraph or two that explains what you are planning to do for your project BEFORE April 6 (don't leave it until the last minute because I am going to ask you to do it again if it doesn't meet the criteria that I outline in the project description). For March 30th I expect that you will have found a partner and given some thought to the piece of mathematics that you are going to adopt. Please notice that the class projects and quiz 6 have switched dates in the calendar above.

    (March 23/09) Please feel free to check out the other Math 1200 course page to see what they are doing. It is very close to ours and sometimes they have interesting material and problems that we don't do.

    (April 13/09) We don't have a homework assignment this week but there is lots to do. Work on your project, and when you do write a journal entry about what you learned and what you expect to learn. The quiz next week will cover some of Chapter 3 and 4...especially chapter 4.3 and 4.4 were parts that we talked about in class (did you notice that one of your homework problems was on page 92 of the book?), chapter 8 and chapter 11.1 through 11.4.

    (April 23/09) I recommend that you write a journal entry every week and every time you meet with your partner to keep track of your progress. It is not necessary that this be long, but it is necessary that your journal be a personal record of your progress made on the project. It should not be like your partners except that you happen to be writing about the same thing).

    (April 28/09) The exam schedule has been announced and the exam for math 1200 will be Sunday May 24, 2009 from 12-3pm in CLH A. More details will be announced as we have them available.

    (May 12/09) Course evaluations are online. We really need feedback on this course in particular because it is the first year we are running it. 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 evalations. You must do the evalation by MAY 21!
    Please visit the site at:

    (May 18, 2009) Dorota took photos last week and I have put them together on a web page.

    (May 18,2009) Note that the quiz for May 20th is cancelled. Make sure you are there because we will be reviewing for the final exam and we have two presenations.

    Class handouts

    (Sept 8/08) First homework assignment
    (Sept 15/08) Second homework assignment
    (Sept 22/08) Third homework assignment
    (Sept 22/08) This is the current version of the convener instructions for Cat Ladies
    (Sept 22/08) This is the current version of the convener instructions for Apple Pi
    (Sept 22/08) First homework convener (this summary was done by me) solutions
    (Oct 20/08) Fourth homework assignment - this was given out in class Oct 6.
    (Oct 20/08) First in class quiz
    (Oct 20/08) Fifth homework assignment
    (Oct 27/08) HW 2 solutions Cat Ladies
    (Oct 27/08) HW 2 solutions Apple Pi
    (Oct 27/08) Sixth homework assignment
    (Nov 3/08) HW 3 solutionst Apple Pi
    (Nov 3/08) HW 3 solutionst Cat Ladies
    (Nov 3/08) Quiz 2
    (Nov 3/08) Seventh homework assignment
    (Nov 4/08) Problem number 6 from homework 6.
    (Nov 12/08) Quiz 2 solutions
    (Feb 10/09) Eighth homework assignment
    (Feb 11/09) The logic questions that I gave you in class on Monday
    (Feb 13/09) An example of counting paths in an n x k rectangle
    (Feb 19/09) Quiz 3
    (Feb 22/09) Quiz 3 solutions
    (March 9/09) Ninth homework assignment
    (March 10/09) A solution to the 12 coin problem
    (March 16/09) Tenth homework assignment
    (March 23/09) Instructions for project
    (April 7/09) Quiz 4
    (April 7/09) Twelfth homework assignment (ummm, eleventh?)
    (April 10/09) A table of criteria that I will use to evaulate your final projects
    (April 12/09) A solved version of quiz 4
    (April 23/09) Thirteenth homework assignment (12th)
    (May 4/09) Fourteenth homework assignment (13th)
    (May 4/09) Discussion from the other 1200 class about sqrt(2) is irrational and square take away problem
    (May 6/09) Quiz 5
    (May 6/09) Quiz 5 solutions