# Math 1200: Problems, Conjectures and Proofs - Fall 2017/Winter 2018 - Section AY - Zabrocki

## Contact information:

Mike Zabrocki
zabrocki@mathstat.yorku.ca
Life Sciences Building (LSB) 107, Tuesdays 10-11:30am
Office: DB (formerly TEL) 2026
office hours: Mondays 1-2pm, Wednesdays 12-1pm 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 inductions, 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
Martin Liebeck, A Concise Introduction to Pure Mathematics, Third Edition.
It is recommended, but not required.

Other useful references are
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
 Participation based on attendance and in class assignments Assignments roughly one every 3-4 weeks 25% Quizzes 6 total, 3 per term, best 2 from each term 30% Final Examination Winter exam period 45%

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 and you will lose 2 points from your course grade for each class or tutorial that you miss each term. Sometimes attendance at the weekly classes will be measured by a short in class assignment. Non participation in these assignments will result in a lowering of your participation grade. Note that participation is not a percentage of your grade, but non-participation and attendance can lower your overall score.

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.

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.

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.

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.

## Announcements:

(September 5, 2017) Welcome! The first day of classes is Tuesday, September 12 and the class will be held in the Life Sciences Building room 107. Instructions about the tutorials (which are scheduled for Wednesdays) will be given on the first day of class. They will meet for the first time on Wednesday, September 20.

(Sept 12, 2017) For next time I would like you to:
(1) read carefully and be aware of the online academic integrity tutorial
(2) learn to use LaTeX and write a summary of the problem we discussed today in class. I will ask you to hand it in on Moodle.
(3) watch the following video on telescoping sums.

(Sept 12, 2017) While I was in class I was hoping to upload a picture and show you how to insert it in your document, but I couldn't find how to upload a picture onto ShareLaTeX. It turns out that I was missing it because it is subtle and it looks like this:
You might want then to include that image in your file with the LaTeX command like \includegraphics[width=3in]{image.jpg}.

(Sept 15, 2017) I adjusted the schedule a bit and corrected the dates for the tutorials. They will meet every other week and both tutorials will meet the same day. There will be 12 meetings of the tutorials and during 6 of those we will have the quizzes.

(Oct 2, 2017) You will have a quiz tomorrow in the tutorial. You will need to be able to prove a sum formula using telescoping sums. You will also need to justify a some simple logical statements or provide a counter-example.

(Oct 2, 2017) I read your homework assignments and gave you some feedback. Everyone who handed in a latex assignment got full credit, but future assignments will be evaluated more on content. Some general comments about what I saw:
• Reread your sentences and make sure they are as simple and clear to another reader as you can make them
• Don't ramble on. Your final solution should be short and direct. We don't need to know your whole process about how you discovered the answer, just a clear and short explanation of why it is true
• Make sure that you define all notation and abbreviations that you use
• Don't use shorthand notation when you write unless it makes the text more readable
• Do not begin a sentence with a symbol

(Oct 29, 2017) I posted a pdf file with some problems to act as practice for the quiz this week.

(Jan 8, 2018) Welcome back. I am going to have to adjust my office hours for this term. I will not be available this week, but I will start them the week of the 15th.

(Jan 23, 2018) Just an FYI, a version of the worksheet on binomial identities that I had on the website earlier today had a few typos. They have been corrected and a version was added that indicates the date and time it was updated (at the bottom left). You should use this worksheet to practice writing expressions in summation notation as well as proving binomial identities by induction.

(Jan 23, 2018) Do you want to see the Sierpinski triangle in Pascal's triangle? Try the command that I used in class in the Sage cell server:
sum([point2d((n,k)) for n in range(128) for k in range(n+1) if mod(binomial(n,k),2)==1])
The command draws all of the points $(n,k)$ for $0 \leq n \leq 127$ and $0 \leq k \leq n$ such that ${n \choose k}$ is odd.

(March 12, 2018) The class will go on as scheduled and I will record the lectures. Tutorials are cancelled until the strike is over.

(March 20, 2018) The quiz for next week is cancelled and will be held on the last day of class (April 3) instead. This will be the case even if the strike is resolved and tutorials begin again next week. Homework #7 is an optional assignment. Hand it in on the moodle before April 11.

(March 27, 2018) I will be holding a quiz next week. I'll weight it so that it is to your advantage to take it (either I transfer the weight to the final if the strike continues, or I will count it if it improves your grade).

We had a guest come into class just before it started to make a short announcement about an information session about internships. This meeting is to be held tomorrow at 4pm in Founders 152. If you are interested please email actsciyorku@gmail.com

One of the problems on the practice had two parts to it. The first part we were able to argue that there must be at least 3 points in a square with area 1, but the second part we needed to explain why the largest triangle that fits inside of a square with area 1 has area 1/2. None of this made it into the video.

(April 16, 2018) I graded the quizzes, but not the two homework assignments yet. Since I won't be able to cover the quiz in class let me make a few comments here:
1. The first question was to find and prove a formula for $f(n) = \sum_{r=1}^n \frac{1}{(4r+1)(4r-3)}$. It really separated the people who understood summation notation from those who didn't. That means you first have to understand what that expression represents. It says that the variable $r$ varies from 1 to $n$ and you add up all of the expressions when you plug in the values of $r$. That means if $n=1$, then $r$ varies from $1$ to $1$ (that means there is one term) and $f(1) = \frac{1}{(4\cdot 1+1)(4\cdot1-3)} = 1/5$. If $n=2$ then there are two terms in the sum, $f(2) = \frac{1}{(4\cdot 1+1)(4\cdot1-3)} + \frac{1}{(4\cdot 2+1)(4\cdot2-3)} = 2/9$. If $n=3$, $f(3) = \frac{1}{(4\cdot 1+1)(4\cdot1-3)} + \frac{1}{(4\cdot 2+1)(4\cdot2-3)} + \frac{1}{(4\cdot 3+1)(4\cdot3-3)}= \frac{3}{13}$ and $f(4) = \frac{1}{(4\cdot 1+1)(4\cdot1-3)} + \frac{1}{(4\cdot 2+1)(4\cdot2-3)} + \frac{1}{(4\cdot 3+1)(4\cdot3-3)} + + \frac{1}{(4\cdot 4+1)(4\cdot4-3)} = \frac{4}{17}$ . What was really great was it seemed that all the people that were able to guess at the formula for $f(n)$ almost all also gave a clear and complete proof by induction.
2. The second problem no one gave a complete answer (although there were many that were close). It was almost the same as the homework question so I was expecting more to get this correct. You were asked to find a polynomial $p(x)$ such that $p(1)=2, p(2)=11$ and $p(3)=28$ and $p(k)$ for $k\geq 1$ is the value on the ray of the diagram traveling East from the 2 in the picture
The hard way to do this is to set up system of linear equations and solve. Go back and watch the video from March 20 to see the easy way of writing down a polynomial. You were then asked to tell me what $p(0), p(-1), p(-2),$ etc. was equal to. Almost everyone assumes that $p(0)=1$, $p(-1) = 6$ and $p(-2)=19$ (BUT IT IS NOT THE CASE!).

## Schedule (Tuesdays during the Fall and Winter term):

 Date Topic Remarks Sept 12 About the class, introductory problem Hw #1 Sept 19 On writing math and solving problems (latex) academic tutorial, HW assignment, video: Telescoping sums Sept 26 Discussion about logic, set notation Oct 3 More logic discussion, worksheet on logic Video: if...then... statements Oct 10 Examples of direct proof, proof by contrapositive and contradiction worksheet on logic Oct 17 More direct proof worksheet on proofs for divides Oct 24 Review of arithmetic (exponents, logs, trig, polynomials), complex numbers Oct 31 Induction Nov 7 Practice induction induction proofs Nov 14 More practice on induction induction proofs Nov 21 Good and bad proofs by induction bad induction proofs Nov 28 induction Jan 9 Summation notation Jan 16 binomial coefficients, multiplication principle Jan 23 binomial identities and summation notation binomial identities Jan 30 even 3 digit numbers with all digits different, go over quiz, quotients and remainders, gcd Feb 6 Euclidean algorithm, linear equations over integers linear diophantine equations Feb 13 Homework, modular arithmetic modular arithmetic Feb 20 Reading week, no class Feb 27 modular arithmetic and Fermat's Little Theorem Mar 6 review of quiz, proofs that $a^{p-1} \equiv 1~(mod~p)$ Mar 13 some practice proofs problems Mar 20 polynomial equations problems that are now homework 7 Mar 27 more problems as practice for the final more problems Apr 3 Quiz!!!

## Schedule (Tutorials every other Wednesday during the Fall and Winter term):

Tutorial 01 is in DB 0013 on Wednesdays at 10:30-11:30am
Tutorial 02 is in CB 122 on Wednesdays at 10:30-11:30am
 Date Topic Remarks Sept 20 First tutorial second hw assignment Oct 4 Quiz #1 Oct 18 third hw assignment Nov 1 Quiz #2 Nov 15 fourth hw assignment Nov 29 Quiz #3 Jan 10 fifth hw assignment Jan 24 Quiz #4 Feb 7 sixth hw assignment Feb 28 Quiz #5 Mar 14 tutorial cancelled due to strike Mar 28 tutorial cancelled due to strike