## Required and Recommended Books

The Berkeley Math Circle and Math Taught the Right Way will be providing some of these required and recommended books to participants, teachers, and instructors at a discounted price (please note that we will not be making the books available to anyone else). Below you will find information, descriptions, and prices for these books, which are available for purchase at our Monday or Wednesday office hours while we have them in stock. Please note that the books at the BMC website are for registered students only and that we do not sell them to people not registered at BMC. All of these books may alternatively be purchased on the web.

**Strongly Recommended Books**

#### A Decade of the Berkeley Math Circle: The American Experience, Volume I

Edited by: Zvezdelina Stankova and Tom Rike

A co-publication of the AMS and Mathematical Sciences Research Institute

Excerpt of book description by the American Mathematical Society:

"Many mathematicians have been drawn to mathematics through their experience with math circles: extracurricular programs exposing teenage students to advanced mathematical topics and a myriad of problem solving techniques and inspiring in them a lifelong love for mathematics. Founded in 1998, the Berkeley Math Circle (BMC) is a pioneering model of a U.S. math circle, aspiring to prepare our best young minds for their future roles as mathematics leaders. Over the last decade, 50 instructors --from university professors to high school teachers to business tycoons--have shared their passion for mathematics by delivering more than 320 BMC sessions full of mathematical challenges and wonders. Based on a dozen of these sessions, this book encompasses a wide variety of enticing mathematical topics: from inversion in the plane to circle geometry; from combinatorics to Rubik's cube and abstract algebra; from number theory to mass point theory; from complex numbers to game theory via invariants and monovariants. The treatments of these subjects encompass every significant method of proof and emphasize ways of thinking and reasoning via 100 problem solving techniques. Also featured are 300 problems, ranging from beginner to intermediate level, with occasional peaks of advanced problems and even some open questions. The book presents possible paths to studying mathematics and inevitably falling in love with it, via teaching two important skills: thinking creatively while still "obeying the rules," and making connections between problems, ideas, and theories. The book encourages you to apply the newly acquired knowledge to problems and guides you along the way, but rarely gives you ready answers. "Learning from our own mistakes" often occurs through discussions of non-proofs and common problem solving pitfalls. The reader has to commit to mastering the new theories and techniques by "getting your hands dirty" with the problems, going back and reviewing necessary problem solving techniques and theory, and persistently moving forward in the book. The mathematical world is huge: you'll never know everything, but you'll learn where to find things, how to connect and use them. The rewards will be substantial."

**Purchase at: **A Decade of the Berkeley Math Circle Vol. I

#### A Decade of the Berkeley Math Circle: The American Experience, Volume II

Edited by: Zvezdelina Stankova and Tom Rike

A co-publication of the AMS and Mathematical Sciences Research Institute

Excerpt of book description by the American Mathematical Society:

"Many mathematicians have been drawn to mathematics through their experience with math circles: extracurricular programs exposing teenage students to advanced mathematical topics and a myriad of problem solving techniques and inspiring in them a lifelong love for mathematics. Founded in 1998, the Berkeley Math Circle (BMC) is a pioneering model of a U.S. math circle, aspiring to prepare our best young minds for their future roles as mathematics leaders. Over the last decade, 50 instructors --from university professors to high school teachers to business tycoons--have shared their passion for mathematics by delivering more than 320 BMC sessions full of mathematical challenges and wonders. Based on a dozen of these sessions, this book encompasses a wide variety of enticing mathematical topics: from inversion in the plane to circle geometry; from combinatorics to Rubik's cube and abstract algebra; from number theory to mass point theory; from complex numbers to game theory via invariants and monovariants. The treatments of these subjects encompass every significant method of proof and emphasize ways of thinking and reasoning via 100 problem solving techniques. Also featured are 300 problems, ranging from beginner to intermediate level, with occasional peaks of advanced problems and even some open questions. The book presents possible paths to studying mathematics and inevitably falling in love with it, via teaching two important skills: thinking creatively while still "obeying the rules," and making connections between problems, ideas, and theories. The book encourages you to apply the newly acquired knowledge to problems and guides you along the way, but rarely gives you ready answers. "Learning from our own mistakes" often occurs through discussions of non-proofs and common problem solving pitfalls. The reader has to commit to mastering the new theories and techniques by "getting your hands dirty" with the problems, going back and reviewing necessary problem solving techniques and theory, and persistently moving forward in the book. The mathematical world is huge: you'll never know everything, but you'll learn where to find things, how to connect and use them. The rewards will be substantial."

**Purchase at: **A Decade of the Berkeley Math Circle Vol. II

#### Textbooks for Middle and High School Students: Adapted from leading Eastern European Programs

Textbooks for grades 6-9: 6A, 6B, 7A, 7B, 8A, 8B, 9A. 9B materials are in preparation. Each grade is split into two semesters (part A and part B), and each semester includes a textbook and workbook.

These books are used at the Math Taught the Right Way and are highly recommended to students at BMC if they feel that math taught at their school is not challenging enough for them.

Example textbook/workbooks:

In 6A, a core goal is to develop basic numeracy — as students study the algebraic operations and their properties, they encounter the natural numbers and decimals. They begin work with variables, setting up a solid base for the work that is to come in later books.

For geometry, students focus on more complicated polyhedral and round solids. In algebra, students will encounter ratios and proportions and a few of their many applications, including more experience with representing and analyzing data and practical problems such as recipe and map scaling. The semester ends with a study of polynomials and familiar factoring formulas.

For 8B algebra, students encounter the basics of algebraic functions and their graphs, learn to solve systems of linear equations and inequalities, solidify their understanding of polynomial equations, and begin working with rational expressions. Their study of advanced geometry continues with a rigorous development of classic theorems on circles and their inscribed angles and polygons. In the final chapter, students get an introduction to combinatorics, a field often used to provide bridges between other branches of mathematics.

In algebra, students continue their study of functions, master systems of quadratic equations, and learn methods for solving inequalities involving polynomials, rational expressions, and/or absolute value. In geometry, students have a unit on similar triangles that makes heavy use of algebra they learned early on and sets the stage for trigonometry, which will be studied in 9B. Finally, students use their combinatorics knowledge from 8B to develop basic probability results.

**Purchase these textbooks at:** ArchiMath Publishing

#### The Art and Craft of Problem Solving, 3rd Edition

Published by: Wiley

"Appealing to everyone from college-level majors to independent learners, The Art and Craft of Problem Solving, 3rd Edition introduces a problem-solving approach to mathematics, as opposed to the traditional exercises approach. The goal of The Art and Craft of Problem Solving is to develop strong problem solving skills, which it achieves by encouraging students to do math rather than just study it. Paul Zeitz draws upon his experience as a coach for the international mathematics Olympiad to give students an enhanced sense of mathematics and the ability to investigate and solve problems."

**Purchase at: **The Art and Craft of Problem Solving, 3rd Edition

#### Kiselev's Geometry: Book 1, Planimetry

Translated from Russian by Alexander Givental

Published by: Sumizdat

This is a wonderful, easy-going introduction to plane geometry, which was used for decades as a regular textbook in Russian middle schooles. It has been translated from its original Russian to English by one of UC Berkeley's very own math instructors, Professor Alexander Givental.

**Purchase at: **Kiselev's Geometry, Book I

#### Kiselev's Geometry: Book 2, Stereometry

Translated from Russian by Alexander Givental

Published by: Sumizdat

This is the second volume of the famous Kiselev's work. A marvelous self-contained exposition on stereometry that proved to be a favorite for generations of students and mathematicians in Russia. Thanks to our UC Berkeley Professor Alexander Givental this book is now available in English.

**Purchase at: **Kiselev's Geometry, Book II

**Highly Recommended for BMC Beginner and Intermediate**

#### Math Circle by the Bay Topics for Grades 1-5 (MSRI Mathematical Circles Library)

Authors: Laura Givental, Maria Nemirovskaya, Ilya Zakharevich

Published by: American Mathematical Society

Excerpt of book description on Amazon.com:

This book is based on selected topics that the authors taught in math circles for elementary school students at the University of California, Berkeley; Stanford University; Dominican University (Marin County, CA); and the University of Oregon (Eugene). It is intended for people who are already running a math circle or who are thinking about organizing one. It can be used by parents to help their motivated, math-loving kids or by elementary school teachers. We also hope that bright fourth or fifth graders will be able to read this book on their own. The main features of this book are the logical sequence of the problems, the description of class reactions, and the hints given to kids when they get stuck. This book tries to keep the balance between two goals: inspire readers to invent their own original approaches while being detailed enough to work as a fallback in case the teacher needs to prepare a lesson on short notice. It introduces kids to combinatorics, Fibonacci numbers, Pascal's triangle, and the notion of area, among other things. The authors chose topics with deep mathematical context. These topics are just as engaging and entertaining to children as typical recreational math problems, but they can be developed deeper and to more advanced levels.

**Purchase at: **Math Circle by the Bay Topics for Grades 1-5

#### Mathematical Circles: Russian Experience

Authors: Dmitri Fomin, Sergey Genkin, and Ilia V. Itenberg

Published by: American Mathematical Society

Excerpt of book description on Amazon.com:

"What kind of book is this? It is a book produced by a remarkable cultural circumstance in the former Soviet Union which fostered the 8creation of groups of students, teachers, and mathematicians called 'mathematical circles'. The work is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport--without necessarily being competitive. This book is intended for both students and teachers who love mathematics and want to study its various branches beyond the limits of school curriculum. It is also a book of mathematical recreations and, at the same time, a book containing vast theoretical and problem material in main areas of what authors consider to be 'extracurricular mathematics'. The book is based on a unique experience gained by several generations of Russian educators and scholars."

**Purchase at: **Mathematical Circles: Russian Experience

**Suggested for BMC Elementary and Beginner**

#### Math Olympiad Contest Problems for Elementary and Middle Schools

Author: Dr. G. Lenchner

Published by: Mathematical Olympiads For Elementary and Middle Schools, Inc.

Book description by the publisher:

The Math Olympiad contests presented these 400 challenging problems and ingenious solutions over a period of 16 years.Aimed at young students, their teachers and parents, the book contains an unusual variety of problems, a section of hints to help the reader get started, and seven unique appendices that inform and enrich, among other features.

**Purchase at:*** *Math Olympiad Contest Problems, Vol. I

#### Math Olympiad Contest Problems, Vol. 2

Editor: Richard Kalman

Published by: Mathematical Olympiads For Elementary and Middle Schools, Inc.

Book description by the publisher:

A continuation of our first volume, Math Olympiad Contest Problems for Elementary and Middle Schools, it is full of useful features for PICO[Person In Charge of Olympiads] and mathlete alike, and can be a valuable addition to your library.

**Purchase at:*** *Math Olympiad Contest Problems, Vol. II

**Recommended for BMC Beginner, Intermediate and Advanced**

#### Art of Problem Solving Books

Published by: the Art of Problem Solving

Book description by the publisher:

The Art of Problem Solving mathematics curriculum is specifically designed for outstanding math students in grades 6-12, and presents a much broader and deeper exploration of challenging mathematics than a typical math curriculum. The Art of Problem Solving texts have been used by tens of thousands of high-performing students, including many winners of major national contests such as MATHCOUNTS and the AMC.

**Purchase at:** Art of Problem Solving

#### Proofs that Really Count: The Art of Combinatorial Proof

Authors: Arthur T. Benjamin and Jennifer J. Quinn

Published by: Mathematical Association of America

Excerpt of book description on Amazon.com:

"Mathematics is the science of patterns, and mathematicians attempt to understand these patterns and discover new ones using a variety of tools. In

Proofs That Really Count, award-winning math professors Arthur Benjamin and Jennifer Quinn demonstrate that many number patterns, even very complex ones, can be understood by simple counting arguments. The arguments primarily take one of two forms:

- A counting question is posed and answered in two different ways. Since both answers solve the same question, they must be equal.
- Two different sets are described, counted, and a correspondence found between them. One-to-one correspondences guarantee sets of the same size. Almost one-to-one correspondences take error terms into account. Even many-to-one correspondences are utilized.
The book explores more than 200 identities throughout the text and exercises, frequently emphasizing numbers not often thought of as numbers that count: Fibonacci Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to name a few. Numerous hints and references are given for all chapter exercises and many chapters end with a list of identities in need of combinatorial proof. The extensive appendix of identities will be a valuable resource. This book should appeal to readers of all levels from high school math students to professional mathematicians."

**Purchase at: **Proofs that Really Count

#### Count Down: Six Kids Vie for Glory at the World's Toughest Math Competition

Author: Steve Olson

Excerpt from the official book description from the publisher (available at Amazon.com):

"Each summer six math whizzes selected from nearly a half-million American teens compete against the world's best problem solvers at the International Mathematical Olympiad. Steve Olson followed the six 2001 contestants from the intense tryouts to the Olympiad's nail-biting final rounds to discover not only what drives these extraordinary kids but what makes them both unique and typical. Beyond the Olympiad, Olson sheds light on many questions, from why Americans feel so queasy about math, to why so few girls compete in the subject, to whether or not talent is innate."

Note: three members of the Berkeley Math Circle were on this team as well as 2009 BMC instructor Ian Le.

**Purchase at:** Count Down

**Recommended for BMC Advanced**

#### Contest Problem Book VIII

Authors: J. Douglas Faires and David Wells

Published by: Mathematical Association of America

Past problems with complete solutions from the American Mathematics Competitions 10 (AMC 10), which is one of the first tests in the series of contests that determines the United States International Math Olympiad team. This book includes all AMC 10 tests from 2000-2007.

**Purchase at:** Contest Problem Book VIII

#### Contest Problem Book IX

Authors: David Wells and J. Douglas Faires

Published by: Mathematical Association of America

Past problems with complete solutions from the American Mathematics Competitions 12 (AMC 12), which is one of the first tests in the series of contests that determines the United States International Math Olympiad team. This book includes all AMC 12 tests from 2001-2007.

**Purchase at:** Contest Problem Book IX

#### Mathematical Omnibus: Thirty Lectures on Classical Mathematics

Authors: Dmitry Fuchs and Serge Tabachnikov

Published by: American Mathematical Society

Dmitry Fuchs, a longtime lecturer at the Berkeley Math Circle, has compiled his notes from BMC Sessions into this wonderful book published by AMS. The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology. The reader's effort will be rewarded in seeing the harmony of each subject. The common thread in the selected subjects is their illustration of the unity and beauty of mathematics. Most lectures contain exercises, and solutions or answers are given to selected exercises. A special feature of the book is an abundance of drawings (more than four hundred), artwork by an award-winning artist, and about a hundred portraits of mathematicians. Almost every lecture contains surprises for even the seasoned researcher.

**Purchase at:** Mathematical Omnibus

#### Mathematical Adventures For Students and Amateurs

Edited by: David F. Hayes and Tatiana Shubin

Published by: Mathematical Association of America (this book contains many lectures given by our own Berkeley Math Circle Instructors at a monthly lecture series in San Jose/Santa Clara State Universities).

Excerpt of book description on Amazon.com:

"How should you encode a message to an extraterrestrial? What do frogs and powers of 2 have in common? How many faces does the Stella Octangula have? Is a plane figure of constant diameter a circle, and what does this have to do with NASA? Is there any such thing as a truly correct map? What patterns are possible in juggling? What do all of these questions have in common? They--and many others--are answered in this book."

"This is a partial record of the Bay Area Mathematical Adventures (BAMA), a lecture series for high school students (and incidentally their teachers, parents, and other interested adults) hosted by San Jose State and Santa Clara Universities in the San Francisco Bay Area of California. These lectures are aimed primarily at bright high school students, the emphasis on 'bright', and as a result, the mathematics in some cases is far from what one would expect to see in talks at this level. There are serious mathematical issues addressed here."

**Purchase at: **Mathematical Adventures For Students and Amateurs