MATH 1200 B & C  Problems, Conjectures, Proofs

Professor: Mike Zabrocki
email: Office: TEL 2028
Office hours: Monday 46, Tuesday 46pm Textbook: Mathematical Proofs: A transition to advanced mathematics, by Chartrand, Polimeni, Zhang
As a alternate/optional textbook: Thinking Mathematically, by Mason, Burton, Stacey 
Course Description: Students entering a university
level mathematics program often lack the experience to deal with
questions and problems when there is no obvious method
to apply. 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 and proof. Learning how to
present convincing reasoning  or proof  is another
course outcome. This course is about thinking and about
communicating.
To do well in upper division courses at York, students will need to be proficient in these types of skills and Math 1200 is a required first year course to help students succeed in their later courses. Class and tutorial attendance is mandatory and active participation is expected of all students. The course textbook will be Mathematical Proofs: A Transition to Advanced Mathematics. The text is useful because it has lots of examples and problems. We will be covering Chapters 27 and occasionally digress in to subjects that appear in the other chapters. We will also be working with the most recent edition of J. Mason, L. Burton, and K. Stacey, Thinking Mathematically (Prentice Hall). The problems in this book are easily accessible while at the same time allowing for rich and varied investigations. 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. Prerequisite: 12U Advanced Functions and Introductory Calculus or equivalent. 
Participation 
based on attendence and in class assignments 

Assignments 
roughly one every 4 weeks 
25% 
Journal/Investigation Projects 
see below 
20% 
Quizzes 
6 total, 3 per term, best 2 from each term 
25% 
Final Examination 
Winter exam period 
30% 
Tutorial 1  CC 335  Tutorial 2 CC 318  Tutuorial 3  CC 109 
Varvara Nika  Varvara Nika  Natasha May 
Mark Beider  Mark Beider  Seyed Mohammad Tavalla 
Mihai Alboiu Kyle Ali Jeremiah Bolante Kevin Chui Jennifer Daechsel Brittany Duarte Kamal Fadlia Matthew Goodman Nikolay Karpenko Doyun Kim Hoi Lun Rachael Milwid Anusiga Nandakumar Matthew Noon Chinedum Opara 
Alexander Ashbourne Mayooran Balakrishnan Karena Cooper Christopher D'Alonzo Alexander Keen Derry Largey Liting Liang Kristeen Marshall Mitesh Mistry Dhanraj Oomajee Kratima Shukla Michelle Swampillai Diana Talvan Justin Tong Qian Wu 
Rokhaya Fall Alfred Ferwerda Jodie Gonzalez Yehoshua Komarovsky Shiyam Pillai Lavanya Ramanathas Avrohom Rosenberg David Shabudin Pavel Shuldiner Kent Tam Shabneez Toorabally Boyin Wang Meijin Zhu Yi Zou 
Tutorial 1  BC 225  Tutorial 2  CC 335 
Elissa Ross  Elissa Ross 
Seyed Mohammad Tavalla  Seyed Mohammad Tavalla 
Craig Fernandes James Fu Nan Jiang Jeff Lee Hongjun Li Sajeda Mamun Matthew Mendes Anna Miadzvedzeva Cheshta Narula Hetal Patel Shashi Ramkeesoon Abirami Sivalingam Alyssa Strassler Sarathambika Sundaralingam Aditi Tandon Luca Tarea Cong Wang Christina Zakko Zhexin Zhao Khrystyna Zhdan 
Laura An Joel Bakole Kalamba Daniel Booker Yi Chen Hinsviraj Gunesh Sophia Han Bolong He Anopan Jeyabalan Yee Lai Richard Lax Yi Liu Thomas Norman Narae Park Adam Podstawka Ioana Popa Zekeria Qassem Wincy Wong Tony Youbi Shuaiqi Zhang 
 \[ 1+3+5+\cdots+(2k+1) = (k+1)^2 (1) \]
 \[ 1^3+2^3+3^3+\cdots+n^3 = \frac{n^2(n+1)^2}{4} \]
 \[ 1^4+2^4+3^4+\cdots+k^4 = \frac{k(k+1)(2k+1)(3k^2+3k1)}{30} \]
 \[ 1^2+3^2+5^2+\cdots+(2k1)^2=\frac{k(4k^21)}{3} \]
 \[ 1^2+4^2+7^2+\cdots+(3k2)^2=\frac{k(6k^23k1)}{2} \]
 \[ (0a+1)^2+(1a+1)^2+(2a+1)^2+\cdots+(ka+1)^2=\frac{(ak+1)(k+1)+a^2(2k+1)(k+1)k}{6} \]
 \[ 1 + 2 + 4 + 5 + 7 + \cdots + (3n1) + (3n+1) = 3n^2+3n+1 \]
 \[ 1 + 3 + 4 + 6 + \cdots + (3n2) + (3n) = 3n^2+n \]
 \[ {{n} \choose {0}} + {{n+1} \choose {1}} + {{n+2}\choose {2}} + \cdots +{{n+r}\choose {r}} = {{n+r+1} \choose {r}} \]
 \[ {{r}\choose {r}} + {{r+1} \choose {r}} + {{r+2}\choose {r}} + \cdots + {{n}\choose {r}} = {{n+1}\choose {r+1}} \]
 \[ (11/\sqrt{2})(11/\sqrt{3})\cdots(11/\sqrt{n})<2/n^2 \]
 \[ \frac{1}{2} \frac{3}{4} \frac{5}{6} \cdots \frac{2n1}{2n} \leq \frac{1}{\sqrt{3n+1}} \]
If \(x \geq 1\), then \((1 + x)^n \geq 1 + nx\) for all \(n \geq 1\) $7^{2n}  48n  1$ is divisible by $2304$  \[ n^2 < 2^n < n! \]
 \[ 2(\sqrt{n+1}1) < 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots + \frac{1}{\sqrt{n}} \]
 \[ 1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\cdots + \frac{1}{\sqrt{n}} < 2 \sqrt{n} \]
Date 
Topic 
Notes 
Sept 13 or 16 
Intro to course, techniques of problem solving 
HW #1 given 
Sept 20 or 23 
more problem solving, sequences and sums 

Sept 27 or 30 
sequences and sums, conjecture does not equal theorem 

Oct 4 or 7 
Quiz 1 
HW #1 due, HW #2 
Oct 11 or 14 
Thanksgiving/reading week 

Oct 18 or 21 
Go over quiz, logic and English 
quiz and HW #1 returned 
Oct 25 or 28 
Logic, implications, shapes, tautologies 

Nov 1 or 4 
Quiz 2 
HW #2 due 
Nov 8 or 11 
review quiz, proof by cases 
HW #3 given, quiz #2 returned 
Nov 15 or 18 
properties of real numbers, contrapositive, vacuously true statements 
HW #2 returned 
Nov 22 or 25 
more properties of real numbers 

Nov 29 or Dec 2 
properties of integers and defintion of \(a \equiv b~(mod~n)\) 
HW #3 due 
Dec 6 or 9 
Quiz 3 

Jan 10 or 6 
review Quiz #3, review \(ab\) and \(a \equiv b~(mod~m)\) 
return quiz #3, hw #3 
Jan 17 or 13 
induction 

Jan 24 or 20 
more induction 
HW #4 given 
Jan 31, 27 
Quiz 4 

Feb 7 or 3 
review quiz 4, binomial coefficients 
return quiz #4, HW #4 due 
Feb 14 or 10 
binomial coefficients 
return HW#4 (Feb 14) 
Feb 21 or 24 
Reading week 

Feb 28 or 17 
more binomial coefficients 
return HW#4 (Feb 17), HW #5 given 
Mar 7 or 3 
Quiz 5 

Mar 14 or 10 
functions, domain, range, 11, onto 
HW #5 due, return quiz 5 
Mar 21 or 17 
functions, domain, range, 11, onto 
HW #6 given 
Mar 28 or 24 
functions, domain, range, 11, onto, relations 

Apr 4 or Mar 31 
relations 
Quiz 6 
Date 
Topic 
Notes 
Sept 13 or 17 

Sept 20 or 24 
honey bee ancestors 
Tut 1 
Sept 27 or Oct 1 
honey bee ancestors 
Tut 2,3 
Oct 4 or 8 
matchsticks 
Tut 1 
Oct 11 or 15 
Thanksgiving 

Oct 18 or 22 
matchsticks 
Tut 2,3 
Oct 25 or 29 
shirts 
Tut 1 
Nov 1 or 5 
shirts 
Tut 2,3 
Nov 8 or 12 
triangle of bits 
Tut 1 
Nov 15 or 19 
triangle of bits 
Tut 2,3 
Nov 22 or 26 
circular sequence 
Tut 1 
Nov 29 or Dec 3 
circular sequence 
Tut 2,3 
Dec 6 or 10 
Polya Strikes Out 
Tut 1 
Jan 10 or 7 
Polya Strikes Out 
Tut 2,3 
Jan 17 or 14 
Postage denomination 
Tut 1 
Jan 24 or 21 
Postage denomination 
Tut 2,3 
Jan 31 or 28 
agonal numbers 
Tut 1 
Feb 7 or 4 
agonal numbers 
Tut 2,3 
Feb 14 or 11 
review 
Tut 1 
Feb 21 or 25 
Reading week 

Feb 28 or 18 
review 
Tut 2,3 
Mar 7 or 4 
tilings of a rectangle 
Tut 1 
Mar 14 or 11 
tilings of a rectangle 
Tut 2,3 
Mar 21 or 18 
practice for final, triangle of numbers 
Tut 1 
Mar 28 or 25 
practice for final, triangle of numbers 
Tut 2,3 
Apr 4 (CC 335) or 1 (BC 225) 
review session 
Tut 1,2,3 
Apr 7 
no class, but journals due 5th floor Ross drop box 