# Math 1200: Problems, Conjectures and Proofs - Winter 2020 - Section A - Zabrocki

## This is NOT the current course web page. It is from Winter 2020 and it is available for reference only.

Perhaps you are looking for the web page from Fall 2021?

## Contact information:

Mike Zabrocki
My last name at mathstat.yorku.ca
Office: DB (TEL) 2026
Course: VH 3006 from 10am-11:15am on Tues/Thurs
Tutorials: S Ross 105 or MC 111 from 10:30am-11:20am on Wed
office hours: Monday 12-2pm and Thursday 2:30-3:30pm and by appointment

## Course description:

Extended exploration of elementary problems leading to conjectures, partial solutions, revisions, and convincing reasoning, and hence to proofs. Emphasis on problem solving, reasoning, and proving. Regular participation is required. Prerequisite: 12U Advanced Functions (MHF4U) or Advanced Functions and Introductory Calculus (MCB4U). NCR note: Not open to any student who is taking or has passed a MATH course at the 3000 level or higher.

Most High School mathematics problems are solved using algorithmic methods or via reference to model solutions. 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. Learning how to present convincing reasoning — or proof — is one of the course outcomes.

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.

The main goal of this course is to develop skills that lead to understanding and communicating a convincing argument. Support will be given for proof presentation, especially for the kinds of proofs that students are expected to produce in their second year and higher level courses. This includes induction, and arguments with counting and with inequalities. Formal proof writing exercises will be introduced in the second half of the course, once problem solving and informal justification skills reach an acceptable level.

Class and tutorial attendance is mandatory and active participation is expected of all students.

## Course references:

The course textbook is available free online
Mathematical Reasoning: Writing and Proof by Ted Sundstrom.
Other useful references are

Martin Liebeck, A Concise Introduction to Pure Mathematics, Third Edition.
John Mason, Leone Burton, Kaye Stacey, Thinking Mathematically, Second Edition. This book gives an approach to problem solving and the problem solving experience. It is also a source for rich and varied problems.
G. Polya, How to Solve It: A New Aspect of Mathematical Method.

## Course components:

The evaluation will be based on the following criteria
 Tutorial presentations based on attendance and in class assignments 15% Assignments assigned throughout the term 30% Midterm During December exam period 20% Final Examination During April exam period 35%

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. I recommend that you look carefully at 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 of the material. Attendance at the weekly classes and the tutorials is obligatory. The TAs will be calling on people at random from their class list to show solutions to the problems that you have been working on and they will be grading the presentation and solution of those problems. Non participation in these assignments will result in a lowering of your tutorial presentation grade.

Assignments: There will be roughly one assignment every 2 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. Your assignment should include complete sentences and explanations and not just a few equations or numbers. You should follow the writing guidelines that are given in Appendix A of your book. 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.

You should prepare your assignments in LaTeX and hand them in on the online Moodle. LaTeX is a program that was designed for writing mathematics. Information about how to do this is provided on this page and we will discuss it more in class.

Note: Late assignments will be penalized by 10% 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.

There are typically two types of assignments that I will ask you do work on for the homework in this class. Sometimes they will consist of smaller problems related to the discussion that we have in class. Other times the assignments will ask you to write and explain a problem that will require careful analysis and understanding by dividing a long solution into smaller, more manageable steps.

Here is a breakdown of some aspects that I plan to evaluate your solutions. Before you hand in your assignment, I recommend that you read it though carefully and try to address the points from this list.:

(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

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.

Midterm and Final Examination: There will a final exam during the April exam period and a midterm around midway through the term. The time and date of these exams will be announced.

## Schedule:

 Date Topic Remarks Jan 7 LaTeX, introductory problem first assignment (src) - due January 16 Jan 8 First assignment Jan 9 More LaTeX, telescoping sums (text, youtube) $1^3+2^3+3^3+\cdots+n^3$ Jan 14 telescoping sums, logical connectives - AND, OR, IF - THEN - [Sundstrom, Ch 1.1] Jan 15 LaTeX, telescoping sums, assignment #1,#2 Assignment #2 (src) - due January 29 Jan 16 IFF, quantifiers, contrapositive, how to prove IF - THEN - statements [Sundstrom, Ch 2.1, 2.2, 2.4] Jan 21 definition of divides, proving if then statements [Sundstrom, Ch 1.2, 3.1] Jan 22 Assignment #2 Jan 23 axioms of inequalities, proofs, solving equations [Sundstrom, Ch 3.2, 3.3, 3.5] Jan 28 the number line, rational numbers, $\sqrt{2}$ is irrational, complex numbers [Sundstrom, Ch 3.3, 3.4] Jan 29 assignment due and next one introduced Assignment #3 (src) - due Feb 12 Jan 30 More complex numbers Feb 4 polar coordinates, polynomial long division, quadratic formula $z^2 = 1-\sqrt{3}i$, $x^2 - 5x+7-i=0$, $x^3-6x+13x-12=0$, $x^4 + x^2 + 1=0$ Feb 5 Assignment #3 Feb 6 roots of $z^n-a$ are centered about 0 Prove (for $n$ odd): $sin(0)+sin(2 \pi/n) + sin(4\pi/n)$$+ \cdots + sin(2(n-1)\pi/n)=0$ Feb 11 Induction [Sundstrom, Ch 4] Feb 12 also midterm practice and midterm practice and Assignment #3 Assignment #4 (src) - due Mar 4 Feb 13 More induction [Sundstrom, Ch 4] Feb 18,19,20 Reading Week - More practice for midterm Feb 25 Practice for midterm - one, two, three Feb 26 practice for midterm, Assignment #4 Feb 27 Midterm exam Mar 3 induction, binomial coefficients ${n}\choose{k}$, def mod, Euclidean algorithm compute $gcd(1024, 1417)$, [Sunstrom, Ch 3.5, 8.1] Mar 4 Assignment #4 due Assignment #5 (src) - due Mar 4 Mar 5 Go over in detail things that can go wrong with proofs (by induction, etc.), $a \equiv b~(mod~n)$ 9 statements about equivalence, [Sunstrom, Ch 3.5] Mar 10 Equivalence mod n, Euclidean algorithm and applications Find integers $s$ and $t$ such that $1024 s + 1417 t = 1$, [Sundstrom, Ch 8.1, 8.2] Mar 11 Assignment #5 Mar 12 $17x \equiv 14~(mod~30)$, relation, reflexive, symmetric, transitive, equivalence [Sundstrom, Ch 7.1], determine if $R = \{ (x,x) : x \in {\mathbb Z}\}$ is reflexive, symmetric and/or transitive Mar 17 Relations, reflexive, symmetric, transitive slides, [Sundstrom 7.1, 7.2] Mar 18 Assignment #5 due Assignment #6 (src) - due April 1 Mar 19 Equivalence relations and equivalence classes slides, [Sundstrom 7.2] Mar 24 Relations, summation notation slides Mar 25 Assignment #6 Mar 26 summation notation, practice for final slides Mar 31 practice for final slides Apr 1 Assignment #6 due Apr 2 practice for final slides

## Announcements:

(January 6, 2020) Welcome. Tutorials for this class will meet for the first time on January 7. For the second class I would like you to watch this youtube video on telescoping sums.

(January 15, 2020) In Fall 2019 the second assignment was about how many regions were in a particular picture: HERE IS THE ASSIGNMENT.
It might be helpful to see what is expected for a solution to this assignment: HERE IS THE SOLUTION.
Notice that my solution is not long and part of the solution explains how the picture relates to the equations.

(January 29, 2020) Our book does not cover complex numbers, but this text from UBC covers details that we will use about the subject.

(February 8, 2020) A tentative date has been assigned for the final exam of Tuesday, April 14, 7-10pm. Normally, it takes a few weeks for the registrar's office to confirm this date while they make adjustments, but I figure it is better to know earlier and update the information than to wait until the schedule is completely fixed.

(February 10, 2020) Someone asked me what the clock looks like in problem #3. It is an "artist's" rendering and is not necessarily to scale.

(March 3, 2020) We computed the first 8 or 9 rows of Pascal's triangle in class and I colored in the odd values. I pointed out that there seemed to be a pattern. If you paste the command:
m=128;sum(point((k+(m-n)/2,-n),axes=False) for n in range(m) for k in range(n+1) if binomial(n,k)%2==1)
into Sage it will draw the picture of the first 128 rows of Pascal's triangle where there is a blue dot on the odd values (it plots a point *if* the code binomial(n,k)%2==1 is true and this is where binomial(n,k) is equivalent to 1 mod 2). Click here to see what that looks like.

(March 11, 2020) At the beginning of class on Tuesday I spoke about the possibility of moving the class online. This will become necessary in the case that the risk of infection on campus increases. I will be trying to make options for "social distancing" more available. The online platform that I am looking into right now is Microsoft Teams. This platform may require you to log in with your York University email address so please try to log in just in case.

(March 12, 2020) I've decided that Microsoft Teams did not work so I will try out Zoom next. More details to come. We are being warned that we need to be prepared to go online. I posted lecture notes on relations from today's class. I will continue to try to do this.

(March 14, 2020) I will be sending you announcements about how to access the course online and I will be posting information about the course on the website and on the moodle. I will not be using the Microsoft Teams setup because I was not impressed with the outcome (which is why I tried the experiment on Thursday).
Monday's office hours 12-2pm will be held online through Zoom. I will post a list of links for the class meetings as well as material such as prepared slides on the course Moodle.
I will send you links for future classes through moodle so make sure you are receiving messages.
If you have any questions please email me.

(April 6, 2020) The final exam will take place 5pm-10pm on April 14, 2020. An announcement was sent out through moodle with some instructions. A further announcement will be made before the date of the exam.