(something close to) Penrose tiles MATH 1200 section B - Problems, Conjectures, Proofs 2012 - 2013 Professor: Mike Zabrocki email: Office: TEL 2028 Office hours: Monday 12:30-2:30pm, Thursday 4-5pm Textbook: Mathematical Proofs: A transition to advanced mathematics, by Chartrand, Polimeni, Zhang As a alternate/optional textbook: Thinking Mathematically, by Mason, Burton, Stacey

 Calendar copy: 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.

(Sept 6, 2012) URGENT! This course has currently too many students. We are looking for volunteers to move to a new section which has just opened up for Thursdays from 8:30-10am. The first meeting of this section will be on September 13 so please enquire if you would be willing. Interested students should contact Janice Grant at the mathematics undergraduate office (janiceg@mathstat.yorku.ca).

### 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.

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 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.

### Handouts:

(Sept 17, 2012) A copy of the syllabus
(Sept 17, 2012) Homework assignment #1 - due Oct 1
(Sept 17, 2012) First assignment for tutorial - hexagonal segments - due Oct 15 for Tutorial 1, Oct 22 for Tutorial 2
(Sept 17, 2012) An excerpt about telescoping sums from Steven Krantz's book
(March 18, 2013) A few page writeup about telescoping sums that I wrote a few years ago
(October 2, 2012) The second homework assignment - due Oct 29
(October 22, 2012) Second assignment for tutorial - inheritance - due Nov 12 for Tutorial 1, Nov 19 for Tutorial 2
(November 2, 2012) Solutions to the second homework assignment
(November 13, 2012) The third homework assignment - due Dec 3
(January 14, 2013) The fourth homework assignment - note there are special instructions (due Feb 4)
(January 28, 2013) The third tutorial assignment
(January 28, 2013) The fourth tutorial assignment
(January 28, 2013) The fifth tutorial assignment - due Feb 11 and Feb 25
(February 11, 2013) The sixth tutorial assignment - due March 4 and March 11
(February 11, 2013) The fifth homework assignment - note there are special instructions (due March 18)
(March 4, 2013) Old exam questions to do in tutorial
(March 18, 2013) The sixth homework assignment - not due (see note below)

### Announcements:

(Sept 6, 2012) URGENT! This course has currently too many students. We are looking for volunteers to move to a new section which has just opened up for Thursdays from 8:30-10am. The first meeting of this section will be on September 13 so please enquire if you would be willing. Interested students should contact Janice Grant at the mathematics undergraduate office (janiceg@mathstat.yorku.ca).

(Sept 10, 2012) There may be several editions of the textbook floating around. Any of them should suffice. We will primarily be following along with Mathematical Proofs: A transition to advanced mathematics. Having the textbook is not critically necessary but is recommended. You may also want to find Thinking Mathematically as an optional textbook. It gives techniques that are helpful for solving problems as well as sample problems.

(Sept 10, 2012) 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.

(Sept 17, 2012) There has not been significant movement on the enrolment numbers for this class. At the moment the only people who are getting into 1200 are those that are signing up for the Thursday morning section.

(October 1, 2012) Just to give you an idea of what we are aiming towards in this class, here are some copies of old exams (2011) (2010).

(October 10, 2012) I will have to cancel my office hours October 11, 2012 because I will be out of town. If you need to meet with me I will be available Monday 12:30-2:30 or it is possible to make an appointment to meet with me.

(October 15, 2012) I have opened up a moodle for this course which will allow you to track your grades. This is an internet tool which will allow me to post interactive course material. You should log in with your passport York account. Currently your quiz grade as well as your homework grade should be online.

(October 22, 2012) I said at one point in one of the classes that I would try to put up a better explanation of why the number of subsets of an $n$ element set is equal to $2 \times$ the number of subsets of an $n-1$ element set because my first one was awkward when I wrote it on the board in class.
It doesn't matter what the $n$ element set or the $n-1$ set that we are working with is, so let us consider the set $\{1,2,3,\ldots, n-1\}$ as our set with $n-1$ elements. Every subset of this set is also a subset $\{1,2,3,\ldots, n\}$. Moreover, every subset of $\{1,2,3,\ldots, n-1\}$ with the element $n$ added in the set is also a subset of $\{1,2,3,\ldots, n\}$. Therefore, we have produced $2 \times$ the number of subsets of $\{1,2,\ldots,n-1\}$ which are subsets of $\{1,2,3,\ldots, n\}$. Since every subset of $\{1,2,3,\ldots, n\}$ either does not contain $n$ (and hence is one of the first type of subsets) or it contains $n$ (and is one of the second type of subsets) then we have accounted for all of them.

(October 26, 2012) I was reading a newsletter and the following advice was written by a mathematician Audrey Terras (who was one of my instructors when I went to school). "1. Don't (EVER) give up. Mathematics is hard to do. Progress can be slow. Patience is necessary. It is important to keep working. When stuck on a problem, start writing up the proofs of the things you have already done. Heisenberg said: 'You just have to be able to drill in very hard wood ... and keep thinking beyond the point where thinking begins to hurt.'" If you would like to read the rest of her comments, I found them here.

(October 29, 2012) Since classes aren't being held on Thursday, I won't be holding my office hours Thursday afternoon either. If you need to see me then, please either schedule an appointment for another time (I can make Tuesday at 4pm this week).

(November 4, 2012) I ran across an article Why Writers Should Learn Math. His case says that a writer should understand the beauty and ideas that one finds in mathematics. I would go much further than this author does because I often see how mathematical logic (as an example of a recent topic of discussion in this class) is imbedded deep within our language and how this logic is then lost by many who are in a position to exposit political arguments, philosophical questions and descriptions of scientific reasoning. Sigh.

(November 17, 2012) I found a video that I really liked. Starting off in math and want to tell people why?

(November 19, 2012) I ran over a little tonight. I showed that if $2|m$ and $3|m$, then $6|m$. Next, I asked you to show that for $a$ and $b$ are positive integers: if $a|m$ and $b|m$, then $(ab)|m$. Try and come up with an explanation of this for next time.

(March 3, 2013) Here is an application of symbolic logic.

This is from the comic Saturday Morning Breakfast Cereal

(March 9, 2013) I finished grading the quizzes only yesterday. They took a long time to read. The solution to the quiz is posted on the moodle under quiz #5. Please read it carefully and make sure that you understand it. I posted the following instructions at the end of the description.
If you had a score $\leq$ 24 on this quiz I would like you to do homework #5 problem 1 and hand it in for Monday, March 18

If you had a score $\leq$ 16 on this quiz I would like you to do all of homework #5 and hand it in for Monday March 18

(I am extending the due date because it took me several days to read these assignments)

(March 9, 2013) As discussed in class, you should do the following 80 problems as practice for the final. Find some finite sets $A, B, C$ (each with just a few elements) and functions $f : B \rightarrow C$ and $g : A \rightarrow B$ such that
1-8. $f$ is $X$, $g$ is $Y$ and $f \circ g$ is $Z$
where $X, Y, Z \in \{$ onto, not onto $\}$ or show that such functions do not exist.
9-16. $f$ is $X$, $g$ is $Y$, $f \circ g$ is $Z$
where $X, Y, Z \in \{$ 1-1, not 1-1 $\}$ or show that such functions do not exist.
17-80. $f$ is $X_0$ and $X_1$, $g$ is $Y_0$ and $Y_1$, $f \circ g$ is $Z_0$ and $Z_1$
where $X_0, Y_0, Z_0 \in \{$ onto, not onto $\}$ and $X_1, Y_1, Z_1 \in \{$ 1-1, not 1-1 $\}$ or show that such functions do not exist.

You should also do the practice for the final that I posted and be prepared to ask questions in tutorial.

(March 9, 2013) You should be practicing for the final exam now. It will take place April 18 at 7pm.

(March 18, 2013) Here is a link to that comic that I showed you on the 11th.

(March 18, 2013) I am not going to collect the 6th homework assignment since I can't really use it to give you feedback before the last quiz and final. But like all of the homework assignments it is necessary practice for your last quiz (compare the quizzes with the homeworks to see how they are related, the same will be true of this one) and for the final exam.

(March 18, 2013) I received the following message about course evaluations:
The Winter 2013 and Fall/Winter 2012-2013 Course evaluations will be available on-line tomorrow (March 19, 2013). The URL link is http://courseevaluations.yorku.ca.
Students should go on-line and complete the course evaluations. They have from March 19, 2013 to April 4, 2013 to submit the evaluations. Please encourage and remind your students to go on-line and complete the course evaluations.

(March 18, 2013) Here are pictures of working and after the work is over.
I tried, but the photo doesn't quite seem readable.

(April 1, 2013) I wasn't joking about my bucket list for university:
1. Learn a second (or third) language
2. Learn to program
3. Don't read - do problems, problems problems
4. Get help from professors, TAs, mathlab, Club Infinity, friends, classmates
5. Do a research project, study abroad, or do something else that will make you happy while you are here (join clubs, sports, hobbys...)

(April 1, 2013) I will not have office hours the week of April 8, but I will have office hours April 15, 3-5pm and April 17, 10am-12pm. The exam is April 18 at 7pm in LAS C (Lassonde Building LSA (formerly Computer Science & Engineering Building)). NB: I had to change my office hours from 2-4 to 3-5 on April 15, because I had a meeting scheduled 12-3

## Lecture meeting schedule

 Date Topic Notes Sept 10 Intro, example problem to course Sept 17 telescoping sums HW #1 assigned Sept 24 problem solving, set notation Quiz 1 Oct 1 sets and logic HW#1 due, HW#2 announced return first quiz Oct 8 Thanksgiving, no class Oct 15 Logic, tautology and fallacy Assignment for tutorial 1 due, return first HW assignment Oct 22 Logic and proof Assignment for tutorial 2 due Oct 29 Some more proof Quiz 2, HW#2 due, returned 1st tutorial assignment Nov 5 definition of divides, even, odd returned 2nd quiz Nov 12 Direct proof, divides, even, odd Assignment for tutorial 1 due, returned 2nd hw assignment Nov 19 contrapositive, examples of proof, proof by cases Assignment for tutorial 2 due Nov 26 Examples of proof/disproof, Euclidean algorithm Dec 3 just the ... Quiz 3, HW#3 due Jan 7 review of proof techniques, rational/irrational Assignment for tutorial 1 due Jan 14 proof by induction Assignment for tutorial 2 due, HW#4 announced Jan 21 Induction practice Jan 28 Induction Quiz 4, Assignment for tutorial #1 due Feb 4 functions and onto Assignment for tutorial #2 due, HW#4 due Feb 11 functions, 1-1 and onto Assignment for tutorial #1 due, HW #5 assigned Feb 18 reading week, no class Feb 25 1-1, onto, functions Assignment for tutorial #2 due Mar 4 1-1 and onto Quiz 5, Assignment for tutorial #1 due Mar 11 complex numbers, relations Assignment for tutorial #2 due, Mar 18 relations and binomial coefficients(?) Homework #5 due Mar 25 properties of $a\equiv b~(mod~n)$ Apr 1 Quiz 6

## Tutorial meeting schedule

 Date Topic Notes Sept 10 won't meet because tutorials don't meet first week of Fall Sept 17 Tutorial assignment expectations + Hexagonal segments Tut 1 Sept 24 Tutorial assignment expectations + Hexagonal segments Tut 2 Oct 1 Hexagonal segments Tut 1 Oct 8 Thanksgiving, no class Oct 15 Hexagonal segments Tut 2 Oct 22 Inheritance Tut 1 Oct 29 Inheritance Tut 2 Nov 5 Inheritance Tut 1 Nov 12 Inheritance Tut 2 Nov 19 ${\mathbb N}$ is isomorphic to ${\mathbb N}^2$ Tut 1 Nov 26 ${\mathbb N}$ is isomorphic to ${\mathbb N}^2$ Tut 2 Dec 3 ${\mathbb N}$ is isomorphic to ${\mathbb N}^2$ Tut 1 Jan 7 ${\mathbb N}$ is isomorphic to ${\mathbb N}^2$ Tut 2 Jan 14 The dating game Tut 1 Jan 21 The dating game Tut 2 Jan 28 Tilings Tut 1 Feb 4 Tilings Tut 2 Feb 11 Square Bashing Tut 1 Feb 18 reading week, no class Feb 25 Square Bashing Tut 2 Mar 4 practice for the final Tut 1 Mar 11 the practice for the final Tut 2 Mar 18 the practice for the final + 80 questions above Tut 1 Mar 25 the practice for the final + 80 questions above Tut 2 Apr 1 the practice for the final + 80 questions above TBA