A Cornucopia of Quadrilaterals collects and organizes hundreds of beautiful and surprising results about four-sided figures—for example, that the midpoints of the sides of any quadrilateral are the vertices of a parallelogram, or that in a convex quadrilateral (not a parallelogram) the line through the midpoints of the diagonals (the Newton line) is equidistant from opposite vertices, or that, if your quadrilateral has an inscribed circle, its center lies on the Newton line. There are results dating back to Euclid: the side-lengths of a pentagon, a hexagon, and a decagon inscribed in a circle can be assembled into a right triangle (the proof uses a quadrilateral and circumscribing circle); and results dating to Erdős: from any point in a triangle the sum of the distances to the vertices is at least twice as large as the sum of the distances to the sides. The book is suitable for serious study, but it equally rewards the reader who dips in randomly. It contains hundreds of challenging four-sided problems. Instructors of number theory, combinatorics, analysis, and geometry will find examples and problems to enrich their courses. The authors have carefully and skillfully organized the presentation into a variety of themes so the chapters flow seamlessly in a coherent narrative journey through the landscape of quadrilaterals. The authors' exposition is beautifully clear and compelling and is accessible to anyone with a high school background in geometry.

A Cornucopia of Quadrilaterals collects and organizes hundreds of beautiful and surprising results about four-sided figures--for example, that the midpoints of the sides of any quadrilateral are the vertices of a parallelogram, or that in a convex quadrilateral (not a parallelogram) the line through the midpoints of the diagonals (the Newton line) is equidistant from opposite vertices, or that, if your quadrilateral has an inscribed circle, its center lies on the Newton line. There are results dating back to Euclid: the side-lengths of a pentagon, a hexagon, and a decagon inscribed in a circle.

A Panoply of Polygons presents and organizes hundreds of beautiful, surprising and intriguing results about polygons with more than four sides. (A Cornucopia of Quadrilaterals, a previous volume by the same authors, thoroughly explored the properties of four-sided polygons.) This panoply consists of eight chapters, one dedicated to polygonal basics, the next ones dedicated to pentagons, hexagons, heptagons, octagons and many-sided polygons. Then miscellaneous classes of polygons are explored (e.g., lattice, rectilinear, zonogons, cyclic, tangential) and the final chapter presents polygonal numbers (figurate numbers based on polygons). Applications, real-life examples, and uses in art and architecture complement the presentation where many proofs with a visual nature are included. A Panoply of Polygons can be used as a supplement to a high school or college geometry course. It can also be used as a source for group projects or extra-credit assignments. It will appeal, and be accessible to, anyone with an interest in plane geometry. Claudi Alsina and Roger Nelsen are, jointly and individually, the authors of thirteen previous MAA/AMS books. Those books, and this one, celebrate and illuminate the power of visualization in learning, teaching, and creating mathematics.

*THIS BOOK IS AVAILABLE AS OPEN ACCESS BOOK ON SPRINGERLINK* One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.

Take an apple and cut it into five pieces. Would you believe that these five pieces can be reassembled in such a fashion so as to create two apples equal in shape and size to the original? Would you believe that you could make something as large as the sun by breaking a pea into a finite number of pieces and putting it back together again? Neither did Leonard Wapner, author of The Pea and the Sun, when he was first introduced to the Banach-Tarski paradox, which asserts exactly such a notion. Written in an engaging style, The Pea and the Sun catalogues the people, events, and mathematics that contributed to the discovery of Banach and Tarski's magical paradox. Wapner makes one of the most interesting problems of advanced mathematics accessible to the non-mathematician.

2019 Update - We have made revisions to correct for minor errata. For a complete list of updates made, please visit us at www.thetutorverse.com. --- If you're taking the Middle Level ISEE, the last thing you need is an all-in-one book that includes materials for other tests. This book is dedicated solely to helping students prepare for the challenges and rigors of the Middle Level ISEE. Access to high-quality and relevant practice questions is the foundation of any study plan. That's why this book contains more Middle Level ISEE practice questions than 6 full-length tests - including 2 full-length practice exams! This book contains: * 2 full-length tests - one full-length diagnostic test, and one full-length practice test. * Detailed answer explanations available online at no additional cost (visit www.thetutorverse.com). * Practice questions organized by topic and content-area, so students can focus on key areas for improvement. * Questions that progress in difficulty, to help students prepare for tough questions. * Helpful tips and suggestions, to complement subject-matter expertise. This book can be used for independent practice or for study with a professional educator. For best results, we recommend students use this book with a tutor or teacher who can help them learn more about new or particularly challenging topics.