MATH 1200 B & C - Problems, Conjectures, Proofs
Professor: Mike Zabrocki
Office: TEL 2028
Office hours: Monday 4-6, Tuesday 4-6pm
Textbook: Mathematical Proofs: A transition to advanced mathematics, by Chartrand, Polimeni, Zhang
As a alternate/optional textbook: Thinking Mathematically, by Mason, Burton, Stacey

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, 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 occasionally 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.


The evaluation will be based on the following criteria
based on attendence and in class assignments

roughly one every 4 weeks
Journal/Investigation Projects
see below
6 total, 3 per term, best 2 from each term
Final Examination
Winter exam period

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 each term.

Assignments: There will be roughly one assignment 3-4 weeks. 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.

Note: Late assignments will be penalized by 20% per day. This will apply to any homework handed in after the class time in which it is due. In addition, assignments which are handed in late are unlikely to be marked in a timely manner.

Journal/Investigation Projects: You are expected to continue working on the problems discussed in class and in the tutorials and to keep a running record of the problems from those exercises (these will be listed on the web page) as well as your progress and the development of a solution for them. The journal will be submitted at the end of each term for grading. Due dates are December 6 or 9 for the Fall and April 4 or 7 for the Winter. Here is a breakdown of some aspects that I plan to evaluate these journals on:
(1) The discussion begins with an explanation of the problem
(2) The explanation should convince the reader that the meaning of the question is understood (e.g. small examples, a clearly labeled table of data, and/or a discussion of the meaning of the question)
(3) diagrams, tables or images that are drawn to aid the reader in understanding the problem are well labeled and explained
(4) Clear statements are made of conjectures that are believed to be true
(5) Explanations of why those conjectures are true are included
(6) An explanation of how the problem solving process proceeded is clear from the explanation
(7) The entries consist of writing which is clear and grammatically correct
(8) A conclusion about the solution to the problem is reached

On both your journals and assignments, I will be looking for evidence of your solutions demonstrating one of the following 4 levels of understanding:
Level 4: Deep understanding of the problem. Complete solution carefully presented. Provides multiple alternative solutions where possible. Considers variations based on the original question (with or without solutions).
Level 3: Good understanding of the problem. Problem solved or a solution provided which can easily be completed, for example, one with a minor error which would be simple to correct. No evidence of engagement beyond finding an answer to the problem as posed.
Level 2: Incomplete understanding of the problem. Limited progress to solution or a solution marred by major errors.
Level 1: Minimal understanding of the problem. Work submitted shows little progress toward solution.

Note that to receive full credit you must go beyond simply solving the problem as posed. Learn to think of your solutions as a starting point.

Quizzes and Final Examination: There will be 3 quizzes per term (dates listed below). A final examination will be scheduled for the April exam period and the date announced in late-February/early March.


Lecture B Tutorials - Mondays 6-7pm
Tutorial 1 - CC 335Tutorial 2- CC 318Tutuorial 3 - CC 109
Varvara NikaVarvara NikaNatasha May
Mark BeiderMark BeiderSeyed Mohammad Tavalla
Mihai Alboiu
Kyle Ali
Jeremiah Bolante
Kevin Chui
Jennifer Daechsel
Brittany Duarte
Kamal Fadlia
Matthew Goodman
Nikolay Karpenko
Doyun Kim
Hoi Lun
Rachael Milwid
Anusiga Nandakumar
Matthew Noon
Chinedum Opara
Alexander Ashbourne
Mayooran Balakrishnan
Karena Cooper
Christopher D'Alonzo
Alexander Keen
Derry Largey
Liting Liang
Kristeen Marshall
Mitesh Mistry
Dhanraj Oomajee
Kratima Shukla
Michelle Swampillai
Diana Talvan
Justin Tong
Qian Wu
Rokhaya Fall
Alfred Ferwerda
Jodie Gonzalez
Yehoshua Komarovsky
Shiyam Pillai
Lavanya Ramanathas
Avrohom Rosenberg
David Shabudin
Pavel Shuldiner
Kent Tam
Shabneez Toorabally
Boyin Wang
Meijin Zhu
Yi Zou
Note: CC = Calumet College

Lecture C Tutorials - Fridays 10:30-11:30am
Tutorial 1 - BC 225Tutorial 2 - CC 335
Elissa RossElissa Ross
Seyed Mohammad TavallaSeyed Mohammad Tavalla
Craig Fernandes
James Fu
Nan Jiang
Jeff Lee
Hongjun Li
Sajeda Mamun
Matthew Mendes
Anna Miadzvedzeva
Cheshta Narula
Hetal Patel
Shashi Ramkeesoon
Abirami Sivalingam
Alyssa Strassler
Sarathambika Sundaralingam
Aditi Tandon
Luca Tarea
Cong Wang
Christina Zakko
Zhexin Zhao
Khrystyna Zhdan
Laura An
Joel Bakole Kalamba
Daniel Booker
Yi Chen
Hinsviraj Gunesh
Sophia Han
Bolong He
Anopan Jeyabalan
Yee Lai
Richard Lax
Yi Liu
Thomas Norman
Narae Park
Adam Podstawka
Ioana Popa
Zekeria Qassem
Wincy Wong
Tony Youbi
Shuaiqi Zhang
BC = Bethune College


(Sept 13, 2010) Homework #1, due in class the week of Oct 4
(Sept 13, 2010) An excerpt from Street Fighting Mathematics by Sanjoy Mahajan.
(Sept 13, 2010) An excerpt from Techniques of problem solving by S. G. Krantz
(Sept 13, 2010) Last year's final exam
(Sept 20, 2010) Tutorial 1 - the first problem for your journals - honey bees
(Oct 4, 2010) Homework #2, due in class the week of Nov 1 (note due date moved from original date)
(Oct 4, 2010) Tutorial 2 - the second problem for your journals - matchsticks
(Oct 30, 2010) Tutorial 3 - the third problem for your journals - shirts
(Nov 8, 2010) Tutorial 4 - the fourth problem for your journals - bit triangles
(Nov 8, 2010) Homework #3 due the week of Nov 29 (Nov 29 for Lect B, Dec 2 for Lect C)
(Dec 14, 2010) Tutorial 5 - the fifth problem for journals - circular sequence
(Dec 14, 2010) Tutorial 6 - the sixth problem for journals - Polya Strikes Out
(Jan 12, 2011) Tutorial 7 - the seventh problem for journals - Postage values
(Jan 19, 2011) Homework #4 due Feb 3 or 7.
(Jan 28, 2011) Tutorial 8 - the eighth problem for journals - tiangular, square, pentagonal, ... numbers
(Feb 17, 2011) Homework #5 due Mar 10 or 14.
(Mar 4, 2011) Tutorial 9 - tilings of a rectangle
(Mar 4, 2011) a binomial identity - my proof of the binomial identity discussed in class.
(Mar 12, 2011) quiz 5 section B
(Mar 12, 2011) quiz 5 section C
(Mar 17, 2011) Tutorial 10 problems - note the special instructions. It will be discussed but not a lot
(Mar 17, 2011) hw#6 : Practice for the final - hand in a copy at the final exam to show me you did it.


(Sept 13, 2010) The textbook for this course "Thinking Mathematically" currently has a new edition. If you have a older edition of this textbook it should suffice.
(Sept 13, 2010) For the first homework assignment you will want to consult the following excerpt from Techniques of problem solving by S. G. Krantz and Street Fighting Mathematics by Sanjoy Mahajan.
(Sept 17, 2010) Check your tutorials. I've listed here the enrollments as of about 10am Sept 17. There has been some shifting to ensure that the number of students in each tutorial is relatively even.
(Sept 24, 2010) Posting from the department: The following information about detailed online resources covering the basics of algebra and trigonometry might prove of some use to those students who are struggling as a result of deficiencies in their high school backgrounds.
(1) There is a quite extensive set of algebra tutorials, covering a wide range of topics, maintained by West Texas A&M University. The URL for the main page of this resource is http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra though a quicker way to get to this page is via the link on the Bethune College Math Help page, http://www.yorku.ca/bethune/math
(2) There is also a detailed online course in Trig basics, which starts right from the beginning, the URL for the main page of the course being http://www.yorku.ca/bethune/math/trig.html (as you can guess from this URL, the Bethune Math Help page has a link which takes you to the Trig course main page). Other potentially useful information is also listed on the Bethune Math Help web page, and further online resources will be added there as they become available.
(Oct 21, 2010) We haven't really discussed the material which is on the homework yet so I will move it back a week and it will be due Nov 1st for lecture B or Nov 4th for lecture C.
(Oct 21, 2010) Those of you who missed question 1 or 2 on the quiz really need to brush up on basic algebra skills. The first two questions were there to indicate if you knew how to expand a polynomial and manipulate exponents. This notation is critically important and is used throughout mathematics. I believe that knowing your algebra will make the difference between you doing well and failing in *all* of your math classes. Find your high school algebra text and/or do the online tutorials that I have posted above. Practice, practice, practice! Go to the math lab on the 5th floor of Ross. Like all of your professors for your math classes I will assume that you have mastered these skills. As you can see we have used them quite a bit in this class (and I imagine in your calculus course too) and we will not go back and review, you will be expected to know that material.
(Nov 15, 2010) You should review what you got wrong on the last quiz and homework assignment. The same sorts of questions that I gave will appear in other contexts throughout this course.
(Nov 30, 2010) I expect lots of questions about what will be on the quiz. I am generally not inclined to give too many clues but I want you to be able understand mathematical statements and tell me if they are true or false (and if so, why). Mostly what I will ask you is what is in chapter 3 and 4 of MP:ATAM but I will be asking questions which are fairly different than what you've seen before, but can be used to convince me that you understand the material.
(Nov 30, 2010) Dont forget that I will be collecting your journals from the fall term next week Dec 6 or 9. You should do ALL the problems from the tutorial.
(Jan 1, 2010) Please check and adjust the calendar if you are in the Thursday class. I had the dates wrong on a previous version (I started the Thursday class after the Monday class and this was not right because classes begin again Jan 4).

(Jan 4, 2010) I realized that my office hours conflicted with my new class schedule. My new Tuesday office hours are 4-6 instead of 3-5pm.

(Jan 15, 2010) There was a mixup with Tutorial 2, Lecture C (see the list above if this includes you) and that group didn't end up meeting. For this group only, tutorial problem 6 is optional. Remember that if you choose to exercise this option, then each of the remaining problems will be worth slightly more.

(Jan 15, 2010) Here are the induction questions that I did on Thursday Jan 13 and (will do) Monday Jan 17. In the following problems, \(r,k,n,a\) are all integers and some of them you will need to determine what values you can prove these values for (some are true only for \(n\) greater than a certain value). FYI, the typeset mathematics below is brought to you by mathjax.
  1. \[ 1+3+5+\cdots+(2k+1) = (k+1)^2 (1) \]
  2. \[ 1^3+2^3+3^3+\cdots+n^3 = \frac{n^2(n+1)^2}{4} \]
  3. \[ 1^4+2^4+3^4+\cdots+k^4 = \frac{k(k+1)(2k+1)(3k^2+3k-1)}{30} \]
  4. \[ 1^2+3^2+5^2+\cdots+(2k-1)^2=\frac{k(4k^2-1)}{3} \]
  5. \[ 1^2+4^2+7^2+\cdots+(3k-2)^2=\frac{k(6k^2-3k-1)}{2} \]
  6. \[ (0a+1)^2+(1a+1)^2+(2a+1)^2+\cdots+(ka+1)^2=\frac{(ak+1)(k+1)+a^2(2k+1)(k+1)k}{6} \]
  7. \[ 1 + 2 + 4 + 5 + 7 + \cdots + (3n-1) + (3n+1) = 3n^2+3n+1 \]
  8. \[ 1 + 3 + 4 + 6 + \cdots + (3n-2) + (3n) = 3n^2+n \]
  9. \[ {{n} \choose {0}} + {{n+1} \choose {1}} + {{n+2}\choose {2}} + \cdots +{{n+r}\choose {r}} = {{n+r+1} \choose {r}} \]
  10. \[ {{r}\choose {r}} + {{r+1} \choose {r}} + {{r+2}\choose {r}} + \cdots + {{n}\choose {r}} = {{n+1}\choose {r+1}} \]
  11. \[ (1-1/\sqrt{2})(1-1/\sqrt{3})\cdots(1-1/\sqrt{n})<2/n^2 \]
  12. \[ \frac{1}{2} \frac{3}{4} \frac{5}{6} \cdots \frac{2n-1}{2n} \leq \frac{1}{\sqrt{3n+1}} \]
  13. If \(x \geq -1\), then \((1 + x)^n \geq 1 + nx\) for all \(n \geq 1\)
  14. $7^{2n} - 48n - 1$ is divisible by $2304$
  15. \[ n^2 < 2^n < n! \]
  16. \[ 2(\sqrt{n+1}-1) < 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots + \frac{1}{\sqrt{n}} \]
  17. \[ 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots + \frac{1}{\sqrt{n}} < 2 \sqrt{n} \]

(Jan 25, 2011) Don't forget the quiz coming up this week. This appeared in a comic called Saturday Morning Breakfast Cereal.
(Feb 11, 2011) For tutorial I've asked the TAs to go back to the last couple of problems that we had and adress some of the questions you haven't finished yet. This means you should bring your writeups and questions that you might have on 'Polya Strikes Out,' 'Stamps' and 'p-agonal numbers' to tutorial. Even if you know how to do these perfectly, I would like you to show off your solutions so that we can see the different ways of approaching the same problem.

(Feb 12, 2011) I feel like I need to make a missing/late test/quiz/assignment policy clearer because recently have had several of these misused. If your assignment is going to be late you will lose 20% per day (I would rather have you hand in it late than not do it at all). If you are UNABLE to get me the assignment on time due to illness (or a large object has fallen on top of you), please contact me by email or phone on the day that the assignment is due telling me when you plan to get it to me. You should deliver the assignment to me at my office as soon as you can along with the appropriate doctors note (or evidence of heavy object preventing you from handing it in on time).

(Feb 14, 2011) The exam for this class is (tentatively) scheduled for Sunday, April 17, 2011 2-5pm.

(Mar 12, 2011) An extra assignment: If you don't do this assignment, you will not get penalized. If you do, I will add points to your quiz grade. I want you to do the quiz for the other class (section B (Monday night) should do section C's quiz and section C should do section B's quiz). I'm not giving partial marks, on the second time around, but I want you to do it all perfectly. Due: Thursday March 17 for section C, and Monday March 21 for section C.

(Mar 30, 2011) So the journals will all be due on Thursday, April 7, 2011. I have arranged for two drop boxes set up on the 5th floor of the Ross building near the elevators. You can depost them there.

(Mar 30, 2011) The last tutorial will be review. It is optional and the TAs have agreed to answer questions on homework 6 and the previous tutorial problems. The tutorials will be held on April 1 in BC 225 and on April 4 in CC 335.

(Apr 5, 2011) Today is the last day to fill out the course evaluation. DO IT NOW!!! It will be on the final.

(Apr 5, 2011) I will have office hours from 2-4pm on Thursday April 14 and (tentatively scheduled for) 3-5pm Friday April 15.

(Apr 6, 2011) Another relevant SMBC:

Lecture meeting schedule

Sept 13 or 16
Intro to course, techniques of problem solving
HW #1 given
Sept 20 or 23
more problem solving, sequences and sums

Sept 27 or 30
sequences and sums, conjecture does not equal theorem

Oct 4 or 7
Quiz 1
HW #1 due, HW #2
Oct 11 or 14
Thanksgiving/reading week

Oct 18 or 21
Go over quiz, logic and English
quiz and HW #1 returned
Oct 25 or 28
Logic, implications, shapes, tautologies

Nov 1 or 4
Quiz 2
HW #2 due
Nov 8 or 11
review quiz, proof by cases
HW #3 given, quiz #2 returned
Nov 15 or 18
properties of real numbers, contrapositive, vacuously true statements
HW #2 returned
Nov 22 or 25
more properties of real numbers

Nov 29 or Dec 2
properties of integers and defintion of \(a \equiv b~(mod~n)\)
HW #3 due
Dec 6 or 9
Quiz 3

Jan 10 or 6
review Quiz #3, review \(a|b\) and \(a \equiv b~(mod~m)\)
return quiz #3, hw #3
Jan 17 or 13

Jan 24 or 20
more induction
HW #4 given
Jan 31, 27

Quiz 4
Feb 7 or 3
review quiz 4, binomial coefficients
return quiz #4, HW #4 due
Feb 14 or 10
binomial coefficients
return HW#4 (Feb 14)
Feb 21 or 24
Reading week

Feb 28 or 17
more binomial coefficients
return HW#4 (Feb 17), HW #5 given
Mar 7 or 3

Quiz 5
Mar 14 or 10
functions, domain, range, 1-1, onto
HW #5 due, return quiz 5
Mar 21 or 17
functions, domain, range, 1-1, onto
HW #6 given
Mar 28 or 24
functions, domain, range, 1-1, onto, relations

Apr 4 or Mar 31
Quiz 6

Tutorial meeting schedule

Sept 13 or 17

Sept 20 or 24
honey bee ancestors
Tut 1
Sept 27 or Oct 1
honey bee ancestors
Tut 2,3
Oct 4 or 8
Tut 1
Oct 11 or 15

Oct 18 or 22
Tut 2,3
Oct 25 or 29
Tut 1
Nov 1 or 5
Tut 2,3
Nov 8 or 12
triangle of bits
Tut 1
Nov 15 or 19
triangle of bits
Tut 2,3
Nov 22 or 26
circular sequence
Tut 1
Nov 29 or Dec 3
circular sequence
Tut 2,3
Dec 6 or 10
Polya Strikes Out
Tut 1
Jan 10 or 7
Polya Strikes Out
Tut 2,3
Jan 17 or 14
Postage denomination
Tut 1
Jan 24 or 21
Postage denomination
Tut 2,3
Jan 31 or 28
-agonal numbers
Tut 1
Feb 7 or 4
-agonal numbers
Tut 2,3
Feb 14 or 11
Tut 1
Feb 21 or 25
Reading week

Feb 28 or 18
Tut 2,3
Mar 7 or 4
tilings of a rectangle
Tut 1
Mar 14 or 11
tilings of a rectangle
Tut 2,3
Mar 21 or 18
practice for final, triangle of numbers
Tut 1
Mar 28 or 25
practice for final, triangle of numbers
Tut 2,3
Apr 4 (CC 335) or 1 (BC 225)
review session
Tut 1,2,3
Apr 7
no class, but journals due 5th floor Ross drop box