See also the course materials in THE SAKAI SYSTEM: https://online.deu.edu.tr/
The documents, homeworks etc. for the course that we have given/assigned are the below files ordered by date.
First lecture: Main aim is problem solving! Discussion on problem solving.
Some Problems in the High School Level to Develop The Joy of Problem Solving
(and its Turkish translation: Problem Çözme Zevki̇ni̇ Geli̇şti̇rmek İçi̇n Li̇se Sevi̇yesi̇nde Bazı Problemler)
First lecture: Main aim is problem solving! The above problems are from the following wonderful book which you shall read to develop your problem solving skills:
Polya, G. How to Solve It. A new aspect of mathematical method. Expanded version of the 1988 edition, with a new foreword by John H. Conway. Princeton University Press, 2004.
Turkish translation: George Polya. Nasıl Çözmeli. Tübitak Yayınları, 2019.
From the foreword to this book by John H. Conway: "How to Solve It is a wonderful book! This I realized when I first read right through it as a student many years ago, but it has taken me a long time to appreciate just how wonderful it is. Why is that? One part of the answer is that the book is unique. In all my years as a student and teacher, I have never seen another that lives up to George Polya's title by teaching you how to go about solving problems."
In this book, Polya gives the following list to solve problems and explores this to explain how to develop the heuristic to solve problems:
Polya's "How to Solve It" List
See the following youtube video about Polya's 4 Steps To Solve Any Problem.
Better see the youtube video Polya explains the problem solving technique.
HOMEWORK 1 - Algebraic Properties of the Number Sytems: N, Z, Q, R, C
STUDY and SOLVE THE EXERCISES from the following parts of your textbook for the next weeks; it will be useful to read also its Turkish translation for those who are native speakers of Turkish.
Part I: Study skills for mathematicians (pages 1–49):
Chapter 1 Sets and functions
Chapter 2 Reading mathematics
Chapter 3 Writing mathematics I
Chapter 4 Writing mathematics II
Chapter 5 How to solve problems
Part II: How to think logically (pages 51-96)
Chapter 6 Making a statement
Chapter 7 Implications
Chapter 8 Finer points concerning implications
Chapter 9 Converse and equivalence
Chapter 10 Quantifiers – For all and There exists
Chapter 11 Complexity and negation of quantifiers
Chapter 12 Examples and counterexamples
Chapter 13 Summary of logic
You shall present your answers to the exercises in these parts on the blackboard in the next weeks.
On our lecture on 4th November Friday, we shall firstly complete discussing your answers to all remaining exercises on logic in the remaining Chapters 6-12. We shall then discuss your written answers to the below questions, so come to lectures with fully written answers with complete sentences by explaining every detail, notation, letters, figures etc. to the following questions:
1) Write the proofs of the Cosine Rule and Sine Rule following the advice in Chapters 3 and 4 of your textbook.
2) Write your answers to all the six exercises at the end of Chapter 5 of your textbook (pages 48-49).
3) Write three different proofs of the Pythagoras' Theorem. What about the converse of the Pythagoras' Theorem? Is it true, can you prove it? Reading Chapter 19 (pages 126-135) from your textbook will also be helpful in learning to write these proofs.
We shall then continue with discussing your answers to the some of the many problems in Homework 1 on Number Systems.
Topics for the midterm examination: all the things discussed in the lectures, the Chapters 1-13, 19 from your textbook, the above Homework 1 on Number Systems and Some Problems in the High School Level to Develop The Joy of Problem Solving (and Polya's ''How to Solve It" List).
We shall discuss the answers to the midterm. To learn how you shall write your answers, see the detailed answers:
STUDY and SOLVE THE EXERCISES from the remaining chapters of your textbook (Chapters 14-26, 30-35 and Appendix C), pages 97-184 and 218-262, for the next weeks till the end of the term; you shall present your answers to the exercises in these parts on the blackboard in the next weeks.
Part III
Definitions, theorems and proofs
14 Definitions, theorems and proofs
15 How to read a definition
16 How to read a theorem
17 Proof
18 How to read a proof
19 A study of Pythagoras’ Theorem
Part IV
Techniques of proof
20 Techniques of proof I: Direct method
21 Some common mistakes
22 Techniques of proof II: Proof by cases
23 Techniques of proof III: Contradiction
24 Techniques of proof IV: Induction
25 More sophisticated induction techniques
26 Techniques of proof V: Contrapositive method
Part V
Mathematics that all good mathematicians need
27 Divisors (we shall study that chapter next term)
28 The Euclidean Algorithm (we shall study that chapter next term)
29 Modular arithmetic (we shall study that chapter next term)
30 Injective, surjective, bijective – and a bit about infinity
31 Equivalence relations
Part VI
Closing remarks
32 Putting it all together
33 Generalization and specialization
34 True understanding
35 The biggest secret
Appendix C How to prove that ...
At the same time, you are supposed to solve the problems in the below HOMEWORKS 2, 3 AND 4. In the lectures, we shall discuss your answers to these problems. They contain many problems for real numbers and induction, a fundamental topic that you must learn well.
Homework 2 - Quadratic Polynomials, Completing the Square and Conic Sections: Parabola, Ellipse, Hyperbola
Axioms of the Real Number System
For the axiomatic construction of the real number system and mathematical induction, see pages 17--47 from the book
Apostol, T. M. Calculus, Volume I: One-variable calculus, with an introduction to linear algebra. Second edition. John Wiley & Sons, 1967.
Homework 3 - Real Numbers and Induction
Answers to Homework 3 - To learn how you shall write clearly and precisely, look at these answers only after you have solved and written your answers to Homework 3.
Positive Integers, Integers and Rational Numbers Obtained from the Axioms of the Real Number System
Arithmetic-Geometric Mean Inequality
Homework 4 - Miscellaneous Exercises for Real Numbers Involving Induction