Math 1200: Problems, Conjectures and Proofs - Fall 2019 - Section A - Zabrocki
This is NOT the current course web page.
It is from Fall 2019 and
it is available for reference only.
Contact information:
Mike Zabrocki
Office: DB (TEL) 2026
Course: FC 202 from 10am-11:15am on Tues/Thurs
Tutorials: Ross S105 and MC 111 from 10:30am-11:20am on Wed
office hours: Monday 12-2pm and Thursday 1-2pm and by appointment
NOTE: For the week of Dec 2-6 office hours will be as scheduled.
For the week of Dec 9-13 I will hold office hours Dec 11, from 10-noon.
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
Martin Liebeck, A Concise Introduction to Pure Mathematics, Third Edition.
It is recommended, but not required. We will plan to cover the following
topics from that text:
4. Inequalities
5. $n^{th}$ roots and rational powers
6. complex numbers
8. induction
10. the integers
11. prime factoriziation
13. congruences of integers
16. counting and choosing
21. Infinity
Other useful references are
Mathematical Reasoning: Writing and Proof by Ted Sundstrom.
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
|
10%
|
Assignments
|
assigned throughout the term
|
35%
|
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. 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.
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 December exam period
and a midterm around midway through the term. The time and date of these exams
will be announced.
Schedule:
Date
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Topic
|
Remarks
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Sept 5
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About the class, introductory problem
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first assignment - due Sept 11
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Sept 10
|
telescoping sums (youtube), notes
|
|
Sept 11
|
tutorial - discuss first and second assignments
|
second assignment - due Sept 25
|
Sept 12
|
numbers, inequalities, AND, OR, IF __ THEN __
|
|
Sept 17
|
inequalities, ___ IFF ___
|
|
Sept 16
|
tutorial - discuss second assignment
|
|
Sept 19
|
quantifiers (forall, exists), direct proof, proof by contradiction
|
Prove $\forall x,y \in {\mathbb Q}, \exists z \in {\mathbb Q}$, $x< z < y$
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Sept 24
|
a little more quantifiers, $x^{a/b}$, $\sqrt{2}$ is irrational
|
|
Sept 25
|
tutorial - Assignment 2 and Assignment 3
|
assignment 3 is due Oct 9
|
Sept 26
|
complex numbers: $e^{i \theta} = cos(\theta) + i sin(\theta)$
|
|
Oct 1
|
complex numbers, trig identities
|
|
Oct 2
|
Assignment 3
|
|
Oct 3
|
trig identity, polynomials
|
|
Oct 8
|
induction
|
|
Oct 9
|
Assignment 4
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Assignment #4 due Oct 23
|
Oct 10
|
induction
|
|
Oct 15,16 & 17
|
Reading Week
|
|
Oct 22
|
Practice problems for the midterm
|
|
Oct 23
|
More midterm practice + assignment #4
|
|
Oct 24
|
Definition of division
|
|
Oct 29
|
Midterm!
|
|
Oct 30
|
Assignment 5 - due Nov 13
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Assignment 5 is due Nov 20
|
Oct 31
|
definition of division
|
|
Nov 5
|
Euclidean algorithm
|
find $r,s$ s.t. $r 1417 + s 1024 = gcd(1417,1024)=1$
|
Nov 6
|
Assignment 5
|
|
Nov 7
|
properties of equivalence mod n
|
|
Nov 12
|
properties of equivalence mod n, solving equations
|
$7x \equiv 2~(mod 12)$
|
Nov 13
|
Assignment 6
|
Assignment 6 is due Nov 27
|
Nov 14
|
modular arithmetic
|
|
Nov 19
|
relations, reflexive, symmetric, transitive, equivalence, equivalence classes
|
$\{(x,y) : x=ay, a \in {\mathbb R}_{>0}, x,y \in {\mathbb C}\}$
|
Nov 20
|
Assignment 6
|
|
Nov 21
|
example equivalence classes, beginning counting
|
Generalize proof that $1+2+\cdots+n = n(n+1)/2$
|
Nov 26
|
Counting proofs
|
Watch this video
|
Nov 27
|
Practice for the final, Assignment 6
|
|
Nov 28
|
Counting proofs
|
More practice for the final (this might have overlap)
|
Dec 3
|
Mostly review
|
|
Announcements:
(September 4, 2019) Welcome. Tutorials for this class will meet for the first
time on Wednesday September 11. For the second class I would like you to watch
this
youtube video on telescoping sums.
(September 10, 2019) I have had a number of students ask to add this course
because it is currently full. Next week (Tuesday September 17) I will give
to math majors in their first year permission to enroll. If you are in
second year you should be taking Math 2200 (offered Winter term).
(September 19, 2019) I've poseted a copy of this website as
the syllabus for this course (
pdf).
(September 24, 2019) I have office hours scheduled on Mondays, but I won't be
available on Monday, October 21, 2019.