# WARNING: This is an out of date web page from 2011-12 and is for reference only

 MATH 1200 section B - Problems, Conjectures, Proofs Professor: Mike Zabrocki email: Office: TEL 2028 Office hours: April 2, 5-6pm; April 5, 2-3pm; April 9 at 3-4:30pm Textbook: Mathematical Proofs: A transition to advanced mathematics, by Chartrand, Polimeni, Zhang As a alternate/optional textbook: Thinking Mathematically, by Mason, Burton, Stacey Course meets CLH 110 7:30pm-9pm Monday's (dates below) and tutorials meet IN ALTERNATE WEEKS in Ross S156 and S101 (dates also below).

 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.

### Evaluation:

The evaluation will be based on the following criteria
 Participation based on attendance and in class assignments Assignments roughly one every 4 weeks 20% Tutorial writeups see below 25% Quizzes 6 total, 3 per term, best 2 from each term 25% 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.

Assignments: There will be roughly one assignment 4-5 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.

Tutorial Assignments: 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. Here is a breakdown of some aspects that I plan to evaluate your solutions from the tutorial:
(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.

### Tutorials:

(Sept 19, 2011) First tutorial assignment - due Oct 3 for tutorial 1, October 17 for tutorial 2
(Oct 3, 2011) Tutorial assignment 2 - due Oct 24 for tutorial 1, October 31 for tutorial 2
(Oct 24, 2011) Tutorial assignment 3 - due November 7 for tutorial 1, November 14 for tutorial 2.
(Nov 7, 2011) Tutorial assignment 4 - due November 21 for tutorial 1, November 28 for tutorial 2.
(Nov 21, 2011) Tutorial assignment 5 - due December 5 for tutorial 1, January 9 for tutorial 2.
(Dec 5, 2011) Tutorial assignment 6 - due January 16 for tutorial 1, January 23 for tutorial 2.
(Dec 5, 2011) Some instructions about presenting a tutorial problem starting in January - this is probably not the final version but I am posting this early so you know what to expect.
(Jan 11, 2012) Tutorial assignment 7 - first group will be presenting Jan 16 and 23 - due January 30 for tutorial 1, February 6 for tutorial 2.
(Jan 23, 2012) Tutorial assignment 8 - second group will be presenting Jan 30 and Feb 6 - due February 13 for tutorial 1, February 27 for tutorial 2.
(Feb 6, 2012) Tutorial assignment 9 - third group will be presenting Feb 13 and Feb 27 - due March 5 for tutorial 1, March 12 for tutorial 2.
(Feb 27, 2012) Tutorial assignment 10 - fourth group will be presenting March 5 and 12 - due March 19 for tutorial 1, March 26 for tutorial 2.
(March 12, 2012) Tutorial assignment 11 - fifth group will be presenting March 19 and 26 - due April 2 for tutorial 1, April 9 for tutorial 2.

### Handouts:

(Sept 12, 2011) A copy of the syllabus
(Sept 19, 2011) First homework assignment - due in lecture Oct 17
(Sept 19, 2011) Exerpt from Kranz - Techniques of Problem Solving - useful for HW #1.
(Oct 4, 2011) A few notes about telescoping sums
(Oct 24, 2011) Second homework assignment - due in lecture Nov 14
(Nov 28, 2011) Third homework assignment - due in lecture Jan 16
(Jan 23, 2012) Fourth homework assignment - due in lecture Feb 27
(February 27, 2012) Fifth homework assignment - due in lecture March 19
(March 20, 2012) Sixth homework assignment - not due
(March 25, 2012) A copy of the final exam from a few years ago
(March 25, 2012) A copy of the final exam from last year

### Announcements:

(Sept 1, 2011) 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 1, 2011) Those that would like to review some concepts that you are expected to have from high school, there are a few online references I can recommend to start: (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.

(September 19, 2011) I wasn't able to meet the class on Monday September 12 but class was held by Dorota Mazur that evening. She gave you an in class assignment that I labelled 'assessment quiz.' I wanted to make sure that you had some basic algebra skills that you will be using in almost all of your math courses: exponents, logarithms, trig functions, manipulations of polynomials, functions and pattern recognition. There were 11 questions in total (5 questions but 2 of them had 4 parts each). In total 31 people took the quiz.
 number of problems correct 2 3 4 5 6 7 8 9 10 11 number of papers 2 1 1 6 1 3 7 3 5 2
The reason why I gave this assessment quiz is that I would like you to know some of the things you should be reviewing in order to do well in this class and your other math courses. The question people had the hardest time with was the last one with logarithms. Hint: it is a good idea to review properties of logarithms and the definition.

(September 26, 2011) The link listed above to the few pages of text from Krantz's book is useful for understanding telescoping sums. Make sure you try the following three examples: $1^2 +2^2 + \cdots + n^2 = n(n+1)(2n+1)/6$ $1\cdot 2+ 2\cdot 3 + 3 \cdot 4 + \cdots + n(n+1) = n(n+1)(n+2)/3$ $1\cdot 2\cdot 3\cdot 4 +2\cdot 3\cdot 4\cdot 5 +3\cdot 4\cdot 5\cdot 6 +\cdots + n(n+1)(n+2)(n+3) = n(n+1)(n+2)(n+3)(n+4)/5$ Challenge: What is the general formula? Write it down and show why it is true.

(October 2, 2011) Note you should hand your solutions to your tutorial problems AT THE BEGINNING of your tutorial (usually 2 weeks later).

(October 3, 2011) If you are interested in finishing the logic problem it is here: First past the post. It isn't too hard if you make the right observation. We finished it before we cleared the classroom.

(October 4, 2011) I wrote a few notes on telescoping sums for those of you that have a hard time with them.

(October 17, 2011) The topic of logic that we are discussing in class can be found in Chapter 2 of the book MP:ATAM.

(October 24, 2011) I suggested last week that for practice that you try 2.17, 2.32 and 2.39. Actually, you should do lots of problems. It is the best way of learning mathematics.

(November 21, 2011) This week I took some time to talk about how to succeed in this course (and all your courses for that matter). If you enjoyed the audio that I played, it comes from This American Life. Here is my advice:
(2) Practice for the quizzes.
(3) Do problems. Lots of them. If you don't understand the problems, only then read the book for clarification and ideas.
(4) Stuck? Remember to apply all your methods from Lecture #3 Sept 26, 2011. Never give up.
(5) Never get a 0 on an assignment for not handing it in.
(6) Go to math lab (find out when the TAs for the class are there).
(7) Work on your homework and study for quizzes in groups.
(8) Go to my office hours M5-6 and Th4:30-5:30.
(9) email me or the TAs (this is not ideal because it is often hard to communicate what needs clarification by email, but if you need a hint, help or to make an appointment to see me, this could work).

(November 28, 2011) I posted the next homework assignment that isn't due until January. My advice: start now. For those of you in Tutorial 1, please take a look at the number spirals instructions. There were a bunch of errors and clarifications that didn't get done until after the tutorial on November 21 and that have now been corrected on the pdf that is here.

(December 5, 2011) Starting January 16 there will be 5 tutorials left and we are going to ask you (the students) to get in groups of 2-3 and present the problems. This will require some preparation and the tutorial problems will be made available in advance.

(December 5, 2011) There was a typo on problem 1 (a) of the homework. Please download the new copy and change the word 'even' to 'odd' in two places in the problem. The problem was clearly false before.

(January 3, 2012) Welcome back. Here is an announcement that I received by email today that is relevant for our class. I never remember to make these announcements in class.
Hello All,

Please make an announcement in your classes regarding the upcoming Mathematical Contest in Modeling (MCM).

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 9-13, 2012. 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 Jan 15. There will be an information/training session in late January.

Cheers,

Jane

(January 23, 2012) The class has already been broken into groups for presenting tutorial problems. I'll place your names in the tutorial meeting schedule below for your presentations. You will get the tutorial problem at least one week in advance. I have placed some instructions about how to prepare for your presentation. I would like you to give me a writeup the week after (or the same day if you like) of your answers to the questions.

(February 6, 2012) When we did induction in small groups on January 23 I gave you the following problems. You should figure out for what values of $n$ the statements are true and then prove them by induction:
A. $1^3+2^3+3^3+\cdots+n^3 = \frac{n^2(n+1)^2}{4}$ B. $1^2+3^2+5^2+\cdots+(2n-1)^2=\frac{n(4n^2-1)}{3}$ C. $1^2+4^2+7^2+\cdots+(3n-2)^2=\frac{n(6n^2-3n-1)}{2}$ D. $1 + 2 + 4 + 5 + 7 + \cdots + (3n-1) + (3n+1) = 3n^2+3n+1$ E. $\frac{a^{n}- b^n}{a-b} = a^{n-1} + a^{n-2} b + a^{n-3} b^2 + \cdots + b^{n-1}$ F. If $a_n =$ the number of dots in the $n^{th}$ diagram below (where the next diagram is equal to the previous one plus three extra segments each containing $n$ dots), then $a_n = \frac{3n^2 - n}{2}$.
G. $\begin{bmatrix} 1 & 1\\0&1\\\end{bmatrix}^n = \begin{bmatrix} 1 & n\\0&1\\\end{bmatrix}$
(February 6, 2012) I am modifying the homework assignment and adding new rules. If you got 9, 10, 11 or 12 on the fourth quiz, I don't care if you hand in this assignment and I give you a free pass on it. If you got a 6,7,8 on the quiz I would like you to do the homework as planned. If you got less than 6 on the quiz, then you must do ALL THE PROBLEMS ON THIS PAGE. I am considering making this the rule for future homework assignments where the number of problems you have to do to get full credit is inversely proportional the the score that you get on the quiz.

(February 6, 2012) I gave you a problem for next time. Say that the sequence $F_0 = 0, F_1 = 1, F_2 = 1$, and $F_{n+1} = F_n + F_{n-1}$ for $n\geq 2$. Show by induction that $\begin{bmatrix} 1 & 1\\1&0\\\end{bmatrix}^n = \begin{bmatrix} F_{n+1} & F_{n}\\F_{n}&F_{n-1}\\\end{bmatrix}$
(February 14, 2012) One more example of an induction problem (in case you were sick of them already). Let the sequence $a_n$ be defined recursively as follows: $a_0 = a_1 = 1$ and $a_n = 3 a_{n-1} - 2 a_{n-2} + 2$ for $n\geq 2$. Show that $a_n = 2^{n+1} - (2n+1)$ for $n\geq 0$. This one was an important example because you can't use just the normal inductive assumption and base case. This is 'strong induction' in your book.

(February 14, 2012) For homework, I would like you to look up (and learn) the definition of one-to-one. It is in your book, but you should be able to find it on wikipedia or other online sources as well. There are only a few things that I will ask that you memorize in this class: one of them is the definition of 'divides', another is the definitions we just covered: 'function,' 'onto' and 'one-to-one.'

(March 20, 2012) I've posted homework 6 for practice. Don't worry about handing it in, but be sure to do the questions.

(March 23, 2012) I love XKCD.
(March 31, 2012) I've updated my office hours April 2, 5-6pm; April 5, 2-3pm; April 9 at 3-4:30pm.

(April 5, 2012) The tutorial problem is due April 9 for tutorial 2. You may either bring it to my office hours or you may bring it to the final exam on the 11th and hand it in then. Make sure that if you have any outstanding work that you have not handed in that you also bring that to the final exam. One thing that I did not receive from a number of people is an explanation of the presentation that you did in the tutorial (you should hand in one per group).

## Lecture meeting schedule

 Date Topic Notes Sept 12 Intro to course, sample problem, assessment quiz Sept 19 the difference between conjecture and proof, the technique of telescoping sums hw #1 assigned Sept 26 how to make a conjecture and then explain it, telescoping sum example Oct 3 logic and logic puzzle Quiz 1 Oct 10 reading week, no class Oct 17 truth tables, tautologies, contradictions and fallacies hw #1 due Oct 24 quantified logical statements hw #2 assigned Oct 31 Proving statements true or false, vacuous and trivial return hw 1 Nov 7 definition of divides, direct proof Quiz 2 Nov 14 go over quiz 2, definition of transitive order return quiz 2, hw #2 due Nov 21 pep talk (see remark above), examples of proofs return hw 2 Nov 28 Examples of proof in groups presented on the board HW #3 given Dec 5 Mostly just did the quiz Quiz 3 Jan 9 Went over the quiz, homework was "prove $\sqrt{11}+\sqrt{13}$ is irrational" return quiz 3 Jan 16 irrational proof, Polya strikes out HW #3 due Jan 23 induction HW #4 assigned Jan 30 Quiz 4 Feb 6 Go over the quiz and homework return quiz #4 and homework #3 Feb 13 A bit more induction, definition of a function and onto, played: 'is it onto?' Feb 20 reading week, no class Feb 27 1-1 and onto HW #4 due, HW#5 assigned Mar 5 Quiz 5 Mar 12 relations, transitive, reflexive, symmetric return quiz #5 Mar 19 relations again, binomial coefficients, Pascal's triangle and paths in a rectangle HW #5 due, hw #6 assigned Mar 26 more binomial coefficients and paths Apr 2 Quiz 6

## Tutorial meeting schedule - NOTE: Tutorial 1 is held in Ross S156 and Tutorial 2 is held in Ross S101

 Date Topic Notes Sept 12 won't meet because tutorials don't meet first week of Fall Sept 19 Diagonals of a Rectangle Tut 1 Sept 26 Diagonals of a Rectangle Tut 2 Oct 3 first tutorial problem due, cells of a prism Tut 1 Oct 10 reading week, no class Oct 17 first tutorial problem due, cells of a prism Tut 2 Oct 24 second tutorial problem due, Square Bashing Tut 1 Oct 31 second tutorial problem due, Square Bashing Tut 2 Nov 7 third tutorial problem due, non-transitive dice Tut 1 Nov 14 third tutorial problem due, non-transitive dice Tut 2 Nov 21 fourth tutorial problem due, number spiral Tut 1 Nov 28 fourth tutorial problem due, number spiral Tut 2 Dec 5 fifth tutorial problem due, Archimedes' Regions Tut 1 Jan 9 fifth tutorial problem due, Archimedes' Regions Tut 2 Jan 16 6th due, (Andy Koh, Ajki Lolja, Yan Xu) counting rectangles Tut 1 Jan 23 6th due, (Injy Raad, Michael Rutledge, Dhruv Popli) counting rectangles Tut 2,3 Jan 30 7th due, (Peter Le, JiaHao Yu) escalators Tut 1 Feb 6 7th due, (Raluca Antonescu, Jasdeep Dhaliwal) escalators Tut 2,3 Feb 13 8th due, (Blessing Enitan, Talmai Lamont) number grids Tut 1 Feb 20 reading week, no class Feb 27 8th due, (Ali Hajmanouchehri, Chao Zhang, Nu Chen) number grids Tut 2 Mar 5 9th due (Joelle Chung, Joshviraj Gunesh, Rima Belousova) dividing an inheritance Tut 1 Mar 12 9th due, (Christopher Couto, Henok Weldyes, Stephen Lidderdale) dividing an inheritance Tut 2 Mar 19 10th due, (Jian Wang, Syad Asghar, Yixin Yang) around the table Tut 1 Mar 26 10th due, (Emmaz Rastgar, Giasuddin Ahmed Kazi, Zohaib Ratani) around the table Tut 2 Apr 2 Review MEET IN Ross S156