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Proofs Without Words II: More Exercises in Visual Thinking

Roger B. Nelsen
Mathematical Association of America
Publication Date: 
Number of Pages: 
Classroom Resource Materials
[Reviewed by
Robert C. Stolz
, on

Proofs Without Words II is a great resource for teachers. The variety of topics addressed makes it valuable at many levels, and is one of its strength. It is organized into chapters dealing with Geometry & Algebra, Trigonometry, Calculus & Analytic Geometry, Inequalities, Integer Sums, Infinite Series, Linear Algebra and other topics. Presented are theorems and statements which are proven mostly through the use of pictures.

The many pictures are presented in a simple format, a style well suited for this book. It reminds many of us of the simplicity and elegance of showing that something is true through the use of pictures. As is probably expected, there are variety (7) of proofs of the Pythagorean Theorem, along with many other geometric proofs. These types of exercises are well-suited for geometry classes, as well as for classes for pre-service teachers. The chapter on integer sums has many proofs concerning figurative numbers, which is a concept that is also quite appropriate for pre-service teachers in both elementary and secondary programs, as well as for students in elementary number theory classes. The pictures presented are delightful and fairly easy to decipher.

There is a beautiful proof about the area under a polygon of arch with a corollary regarding the area of the cycle. Though in the geometry section, this would be quite appropriate for calculus classes as it addresses areas under curves, as well as the idea of limits. Another favorite, and probably the most compact, is "one figure six identities", which proves the sum and difference trigonometric formulas from one figure — a must see for trigonometry classes.

This book is fun for any mathematics enthusiast. As one peruses the book, you're drawn into the "puzzles" as you try to reconstruct the proof. Proofs Without words II is filled with enrichment activities suitable for classes and club activities; it should be on every mathematics teacher's shelf.








Robert C. Stolz ( is Chair of the Division of Science and Mathematics of the University of the Virgin Islands.