(something close to) Penrose tiles

MATH 1200
section B  Problems, Conjectures, Proofs
2012  2013 
Professor:
Mike Zabrocki
email: Office: TEL 2028
Office hours: Monday 12:302:30pm, Thursday 45pm 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. 
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% 
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, 11 and onto 
Assignment for tutorial #1 due, HW #5 assigned 
Feb
18 
reading week, no class 

Feb
25 
11, onto, functions 
Assignment for tutorial #2 due 
Mar
4 
11 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 
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 