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 10am11:15am on Tues/Thurs
Tutorials: S Ross 105 or MC 111 from 10:30am11:20am on Wed
office hours: Monday 122pm and Thursday 2:303:30pm 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

$z^2 = 1\sqrt{3}i$, $x^2  5x+7i=0$, $x^36x+13x12=0$, $x^4 + x^2 + 1=0$

Feb 5

Assignment #3


Feb 6

roots of $z^na$ are centered about 0

Prove (for $n$ odd): $sin(0)+sin(2 \pi/n) + sin(4\pi/n)$ $+ \cdots
+ sin(2(n1)\pi/n)=0$

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, 710pm. 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+(mn)/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 122pm 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 5pm10pm 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.