Five Fabulous Activities for Your Math Circle

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Five Fabulous Activities for Your Math Circle is a guidebook for teachers, parents, and math circle leaders. The book is primarily for those working with middle and high school students, with some activities for younger math friends.

Five Fabulous Activities for Your Math Circle

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About the book

A math circle is any group of people gathering to explore mathematics—it could be in your home, in school, or even online. Math circle activities are often interactive, exploratory, flexible, open-ended, and social. This might sound like a really fun idea, but how does one make it happen?

Five Fabulous Activities - Bard 1

Five Fabulous Activities for Your Math Circle will help show you how. This book is a guide and a collection of recipes for anyone who wants to help others discover joyful and challenging math.

Our book will help you guide mathematical learners of a wide variety of ages. We primarily target middle and high school students. We have also included activities and explanations intended for younger math friends. The underlying mathematics can be of interest even to adult learners, including ourselves!

Each chapter of this book contains resources for organizing a single math circle session or short series of sessions. The activities are also excellent for family mathematics or classroom enrichment. The supplies are easy to assemble: paper, pencils, scissors, and your imagination for a science expedition to Mars or a storytelling gathering at a Chokwe village in Africa.

Five Fabulous Activities - easy supplies

In the chapters you’ll find the mathematical background, such as motivation and conclusions, as well as clear explanations. You’ll also see how to divide the material into a series of guided and discovery-based activities, and suggestions on how to present them to students. Each activity includes a “Jump Right In” warm-up for a quick exploration.

Here are the chapters:

Bulgarian Solitaire – A simple one-player game leads to fascinating patterns and a mathematical structure.

Sona – An African storytelling art form connects readers with intermediate counting and number theory concepts.

Five Fabulous Activities - Lusona

Nim & Jim – Nim is a classic game that can be analyzed using binary numbers. Jim is Japheth’s new twist, making the analysis come to life for those who like to discover things for themselves.

Splitting to Mars – A problem of splitting a team of space scientists into two groups leads to a combinatorial puzzle with a satisfying solution.

Hyperbinary Numbers and the Calkin-Wilf Tree – The idea of using coins with binary values has a surprising, accessible connection to the problem of enumerating the rational number system.

Glossary – Last but not least, this chapter has short, fabulous background pieces on key concepts used in the book, plus more quick challenges to share with math friends!

Five Fabulous Activities - layout sketch 1

229 crowdfunders fully supported the book’s production – thank you!

About the Authors

Five Fabulous Activities - Paul

Paul Ellis is an Associate Professor of Mathematics and chair of the Department of Mathematics and Computer Science at Manhattanville College. He has led the Westchester Area Math Circle in Purchase, NY since 2013. He has given math circle demonstrations in New York, New Jersey, Boise, Budapest, and Shanghai. He is the lead mathematics instructor at the Math League Summer Tournament in China.

Five Fabulous Activities - Sam

Samuel Coskey is an Associate Professor of Mathematics at Boise State University. He has led the Boise Math Circle since 2014. He is often a mentor at the Boise Math REU.

Five Fabulous Activities - Japheth

Japheth Wood is a Continuing Associate Professor of Mathematics at Bard College, which includes teaching duties in the Bard Prison initiative. He leads the Bard Math Circle, including their annual summer camp. He previously served as the Executive Director for the New York Math Circle, one of the largest in the country. He also serves as senior staff at the Hampshire College Summer Studies in Mathematics Program.


When the three of us connected at math events, we quickly realized we have a common interest in math circles. Since then we have been sharing our activities with each other, and even presenting in each others’ circles. After more than five years, we decided to share some of our favorite activities with a broader audience. We further saw our collaboration as an opportunity to share some of our methods for overcoming the challenges we faced when we were new to leading activities.

If you are someone who facilitates mathematical exploration, we hope you will use this book to help spread the joy of math to others. If you love to explore math, encourage your mentors to use this book to help guide you in your journey!

Five Fabulous Activities - Bard 3


James Tanton (Mathematician at Large—Mathematical Association of America; more about James).

Three top-notch Math Circle leaders have come together to share with the world their insight and wisdom – and most important – their exuberant joy in math by presenting lesson plans to five of their favorite mathematical wonders for novice and expert math circle leaders alike! Woohoo! Carefully laid out to be inviting and welcoming to all, and carefully and thoughtfully structured to nudge the facilitator deeper and deeper into the delight and awe of running a math circle session (with rest stops and side turns along the way, if desired), this book aptly portrays the beautiful human spirit of the Math Circle community and the genuine process of exploring mathematics. So, come! Play with math! Here are five truly fabulous ways.

Mark Saul (Executive Director—Julia Robinson Math Festival; more about Mark).

This is an exciting book. It brings us the freshness of experience. The mathematics is not laid out in logical order, but in pedagogical order. The writers use stories of their work with students to guide us through the discoveries that a group is likely to make, as a game, puzzle, or activity elicits mathematical thought. The activities start very simply, then soar to the heights, in some cases linking to open research questions.

The authors use these activities to introduce specific mathematical techniques (counting techniques, concepts from number theory, methods of proof) and also strategies (generalizing, considering special cases, re-formulating the problem). The activities are carefully chosen and structured. The exposition is well thought out, both mathematically and pedagogically.

And even emotionally. The writers, because they have worked directly with students, are able to recount their observations of the students’ emotional, as well as mathematical experiences. Emotions—the frustration of puzzlement, the joy of discovery—are the key to motivating the student to learn. And this is the core task of the teacher.

Meet the Publisher

nm_logo_100pxAs a parent, teacher, or math circle leader, you want books that are helpful for your children. Will the book make sense? Will it add rich, beautiful, and kind math to your child’s life? Will it be fun? So that you can say YES to all of these questions, the Natural Math community has developed a unique publishing process.

Our publishing gives authors, artists, and even readers a lot of creative control. A book starts with amazing authors who deeply care about their work in mathematics education. Each author spends many hours improving every aspect of the manuscript by testing the book with children, families, and classes.

Next, Natural Math invites beta readers: brave community volunteers who field-test the draft in their own families and classes, without the authors on-scene to help. Our beta readers are parents, principals and classroom teachers, as well as leaders of math circles, learner coops, museum activities and other extracurricular programs. They come from all learning backgrounds and all continents (except Antarctica). Their input is how Natural Math books become so real.

Crowdfunding comes next. Our crew of professional editors, proofreaders, and designers are paid with the crowdfunding money. They make the paper books a joy to hold, and ensure that ebooks flow well on all devices. Natural Math books have been featured by The Atlantic, The NY Times, The NPR, and many lovely sites dedicated to learning, parenting, and mathematics.

And One More Thing

As all our previous books, Five Fabulous Activities is published under a Creative Commons license. That means that people all over the world will be able to access its content, translate it into different languages – and share their ideas based on the book. The Creative Commons site features an interview with the Natural Math founder in its celebration of innovative projects in education.

If you already are a part of the Natural Math community, thank you for your continuing support. If you are new to Natural Math, thank you for checking out our latest project. Please join the math adventure!

Thank you for your interest! Please help us spread the word about this book to support math circles