A Decade of the Berkeley Mathematical Circle: The American Experience
Authors
Description
This book is based on materials from a dozen of the 320 Berkeley Mathematical Circle (CMB) sessions over the past 10 years. In recent discussions, parents of students have described the CMB as a mathematical circle of excellence.
Topics covered in CMB include combinatorics, graph theory, linear algebra, geometric transformations, recursive sequences, series, set theory, groups, number theory, elliptic curves, algebraic geometry, applications to computing, natural sciences, economics, and more.
Each topic is taught by an expert in the field who has the ability to challenge students and support them when they face such challenges. All problems require students to present mathematical proofs.
Target audience
Middle school
Name: A Decade of the Berkeley Mathematical Circle: The American Experience
Author(s): Zvezdelina Stankova e Thomas Rike
Pages: 362
Publication: AMS/IMPA
ISBN: 978-85-244-0441-2
Edition: 1
Foreword
Introduction
1. Mathematical Circles of Excellence
2. Why, What, and For whom?
3. Notation and Technicalities
4. The Art of Being a Mathematician and Problem Solving
5. Acknowledgments
Session 1. Inversion in the Plan. Part I
1. Why Inversion? Motivation
2. Inversion as a Transformation
3. Definition of Inversion
4. Basic Properties of Inversion
5. Troubleshooting Techniques with Inversion
6. Solving the First Problems with Inversion
7. How does inversion affect distances?
8. Proof of Ptolemy’s Theorem
9. How Are Inversion Problems Created?
10. Selected Problem Tips and Solutions
Session 2. Combinatorial. Part I
1. Two Counting Riddles
2. Multiplication, Menus, and Encoding
3. Addition and Partition
4. Division: A Cure for Uniform Overcounting
5. Balls in Urns and other Applications
6. Brotherhoods of Numbers: A Promise Fulfilled
Session 3. Rubik’s cube. Part I
1. Presentation and a Little Notation
2. Mathematically Encoding the Rubik’s Cube
3. Some Basic Characteristics of Movements
4. Visualizing Permutations
5. Structure of Permutations in Cycles
6. Applications of Cyclic Structure to the Cube
7. Conclusions
Session 4. Number Theory. Part I
1. Using the Math Crown
2. Remains: Where It All Began
3. Z-congruences
4. Properties of Congruence
5. Restos Learning to Ride a Bike
6. Brute Force Approach Adjustments
7. Pairs and Severability
8. Tips and Solutions for Selected Problems
Session 5. A Few Words About Proofs. Part I
1. Why Prove Things?
2. Evidence versus Non-Evidence
3. Proof by Contradiction
4. Proofs of Possibility and Impossibility
5. Some Problems Need Two Proofs!
6. Tips and Solutions for Selected Exercises
Session 6. Mathematical Induction
1. Examples and Conjectures
2. Mathematical Induction and Proof
3. Mathematical Induction in Action
4. Strong induction
5. Mathematical Induction in other Areas
6. A Little Caution 136
7. Selected Exercise Tips and Solutions
Session 7. Geometry of Points of Mass
1. Introduction
2. Definition and Properties of Mass Points
3. Key Examples
4. Biseterrics and Heights
5. Areas, Space and Separation of Masses
6. Ceva, Menelaus, and Associativity of Addition
7. Examples of Competition Problems
8. History and References
9. Selected Problem Tips and Solutions
Session 8. More about Exams. Part II
1. Induction Proofs Again
2. Extremes Are Naturally Laborious
3. The Pigeon House Principle
4. Selected Problem Tips and Solutions
Session 9. Complex numbers. Part I
1. A Geometry Problem
2. A Little History
3. Complex Numbers via Geometry
4. Basic Operations with Complex Numbers
5. Complex Multiplication
6. Another Form of Complex Numbers
7. Summary: What Have We Learned?
8. Selected Problem Tips and Solutions
Session 10. Footsteps. Games with Invariants
1. Classic Warm-Ups
2. Invariants with Numbers
3. Footsteps
4. Coatings and More Invariants
5. The Escape of the Clones
6. Tips and Solutions for Selected Problems
Session 11. CMB Favorite Issues. Part I
1. In Search of the Hidden Circle
2. Inscribed and Central Angles
3. “Walking in Circles”
4. “Go Off on a Tangent” Goes Straight to the Point!
5. When Ghost Circles Join Tangents
6. Building Cyclic Trapezoids “From Nothing”
7. Tips and Solutions for Selected Problems
Session 12. Monovariants. Part I
1. Porcelain and Chocolate Bar Shops
2. Strolling through a Mansion
3. Finite vs. Infinite
4. The Monovariant Working Group
5. Women and Men Walking Around the Mansion
6. Non-Numerical Monovariants
7. Mansions – Advanced Reader Appendix
8. Tips and Solutions for Selected Problems
Epilogue
1. What comes of it all
2. The Culture of Circles
3. Eastern Europe vs. Math Circles in the USA
4. History and Power
5. Does USA Need Mathematical Circles of Excellence?
Symbols and Notations
Abbreviations
Biographical Data
Bibliography
Credits
Index