Math Made Visual: Creating Images for Understanding Mathematics (Classroom Resource Material)

Often, the best way to teach a proof is by drawing an image that illustrates the ideas. While mathematics is based on abstract concepts, humans are creatures whose intuition is largely based on images. This book describes many ways in which concepts can be visually represented and these images truly are worth a thousand symbols. Part I consists of twenty chapters, which are called: \*) Representing numbers by graphical elements \*) Representing numbers by lengths of segments \*) Representing numbers by areas of plane figures \*) Representing numbers by volumes of objects \*) Identifying key elements \*) Employing isometry \*) Employing similarity \*) Area-preserving transformations \*) Escaping from the plane \*) Overlaying tiles \*) Playing with several copies \*) Sequential frames \*) Geometric dissections \*) Moving frames \*) Iterative procedures \*) Introducing colors \*) Visualization by inclusion \*) Ingenuity in 3D \*) Using 3D models \*) Combining techniques Each chapter has several sections where a specific type of problem is examined in each section. For example, the three sections of chapter 1 are: \*) Sums of odd integers \*) Sums of integers \*) Alternating sums of squares Each chapter terminates with a section called "challenges", which is a set of problems that are to be solved by creating images similar to those demonstrated in the chapter. Hints and solutions to these problems are provided in an appendix. Part II of the book is called "Visualization in the classroom" and is a short history of visualization and a demonstration of how physical objects can be used to illustrate the concepts. This is an outstanding math book, demonstrating some of the very best strategies that can be used to develop and explain a proof. It is not possible to use a diagram to present every mathematical proof that you may explain to your students. However, it can be done more often than you may realize, so I recommend that all teachers of mathematics examine this book, you most certainly will find a gem that you can use. Published in Journal of Recreational Mathematics, reprinted with permission. The Object Of This Book Is To Show How Visualization Techniques May Be Employed To Produce Pictures That Have Interest For The Creation, Communication, And Teaching Of Mathematics. Introduction -- Pt. 1. Visualizing Mathematics By Creating Pictures -- 1. Representing Numbers By Graphical Elements -- 1.1. Sums Of Odd Integers -- 1.2. Sums Of Integers -- 1.3. Alternating Sums Of Squares -- 1.4. Challenges -- 2. Representing Numbers By Lengths Of Segments -- 2.1. Inequalities Among Means -- 2.2. The Mediant Property -- 2.3. A Pythagorean Inequality -- 2.4. Trigonometric Functions -- 2.5. Numbers As Function Values -- 2.6. Challenges -- 3. Representing Numbers By Areas Of Plane Figures -- 3.1. Sums Of Integers Revisited -- 3.2. The Sum Of Terms In Arithmetic Progression -- 3.3. Fibonacci Numbers -- 3.4. Some Inequalities -- 3.4. Some Inequalities -- 3.5. Sums Of Squares -- 3.6. Sums Of Cubes -- 3.7. Challenges -- 4. Representing Numbers By Volumes Of Objects -- 4.1. From Two Dimensions To Three -- 4.2. Sums Of Squares Of Integers Revisited -- 4.3. Sums Of Triangular Numbers -- 4.4. A Double Sum -- 4.5. Challenges. 5. Identifying Key Elements -- 5.1. On The Angle Bisectors Of A Convex Quadrilateral -- 5.2. Cyclic Quadrilaterals With Perpendicular Diagonals -- 5.3. A Property Of The Rectangular Hyperbola -- 5.4. Challenges -- 6. Employing Isometry -- 6.1. The Chou Pei Suan Ching Proof Of The Pythagorean Theorem -- 6.2. A Theorem Of Thales -- 6.3. Leonardo Da Vinci's Proof Of The Pythagorean Theorem -- 6.4. The Fermat Point Of A Triangle -- 6.5. Viviani's Theorem -- 6.6. Challenges -- 7. Employing Similarity -- 7.1. Ptolemy's Theorem -- 7.2. The Golden Ratio In The Regular Pentagon -- 7.3. The Pythagorean Theorem -- Again -- 7.4. Area Between Sides And Cevians Of A Triangle -- 7.5. Challenges -- 8. Area-preserving Transformations -- 8.1. Pappus And Pythagoras -- 8.2. Squaring Polygons -- 8.3. Equal Areas In A Partition Of A Parallelogram -- 8.4. The Cauchy-schwarz Inequality -- 8.5. A Theorem Of Gaspard Monge -- 8.6. Challenges. 9. Escaping From The Plane -- 9.1. Three Circles And Six Tangents -- 9.2. Fair Division Of A Cake -- 9.3. Inscribing The Regular Heptagon In A Circle -- 9.4. The Spider And The Fly -- 9.5. Challenges -- 10. Overlaying Tiles -- 10.1. Pythagorean Tilings -- 10.2. Cartesian Tilings -- 10.3. Quadrilateral Tilings -- 10.4. Triangular Tilings -- 10.5. Tiling With Squares And Parallelograms -- 10.6. Challenges -- 11. Playing With Several Copies -- 11.1. From Pythagoras To Trigonometry -- 11.2. Sums Of Odd Integers Revisited -- 11.3 Sums Of Squares Again -- 11.4. The Volume Of A Square Pyramid -- 11.5. Challenges -- 12. Sequential Frames -- 12.1. The Parallelogram Law -- 12.2. An Unknown Angle -- 12.3. Determinants -- 12.4. Challenges -- 13. Geometric Dissections -- 13.1. Cutting With Ingenuity -- 13.2. The Smart Alec Puzzle -- 13.3. The Area Of A Regular Dodecagon -- 13.4. Challenges -- 14. Moving Frames -- 14.1. Functional Composition -- 14.2. The Lipschitz Condition -- 14.3. Uniform Continuity -- 14.4. Challenges. 15. Iterative Procedures -- 15.1. Geometric Series -- 15.2. Growing A Figure Iteratively -- 15.3. A Curve Without Tangents -- 15.4. Challenges -- 16. Introducing Colors -- 16.1. Domino Tilings -- 16.2. L-tetromino Tilings -- 16.3. Alternating Sums Of Triangular Numbers -- 16.4. In Space, Four Colors Are Not Enough -- 16.5. Challenges -- 17. Visualization By Inclusion -- 17.1. The Genuine Triangle Inequality -- 17.2. The Mean Of The Squares Exceeds The Square Of The Mean -- 17.3. The Arithmetic Mean-geometric Mean Inequality For Three Numbers -- 17.4. Challenges -- 18. Ingenuity In 3 D -- 18.1. From 3d With Love -- 18.2. Folding And Cutting Paper -- 18.3. Unfolding Polyhedra -- 18.4. Challenges -- 19. Using 3d Models -- 19.1. Platonic Secrets -- 19.2. The Rhombic Dodecahedron -- 19.3. The Fermat Point Again -- 19.4. Challenges -- 20. Combining Techniques -- 20.1. Heron's Formula -- 20.2. The Quadrilateral Law -- 20.3. Ptolemy's Inequality -- 20.4. Another Minimal Path -- 20.5. Slicing Cubes -- 20.6. Vertices, Faces, And Polyhedra -- 20.7. Challenges. Pt. 2. Visualization In The Classroom -- Mathematical Drawings : A Short Historical Perspective -- On Visual Thinking -- Visualization In The Classroom -- On The Role Of Hands-on Materials -- Everyday Life Objects As Resources -- Making Models Of Polyhedra -- Using Soap Bubbles -- Lighting Results -- Mirror Images -- Towards Creativity -- Pt. 3. Hints And Solutions To The Challenges -- References -- Index -- About The Authors. Claudia Alsina And Roger B. Nelsen. Includes Bibliographical References (p. 161-168) And Index. Is it possible to make mathematical drawings that help to understand mathematical ideas, proofs and arguments? The authors of this book are convinced that the answer is yes and the objective of this book is to show how some visualization techniques may be employed to produce pictures that have both mathematical and pedagogical interest. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called "proofs without words". Hundreds of these have been published in Mathematics Magazine and The College Mathematics Journal, as well as in other journals, books, and on the Internet. Often times, a person encountering a "proof without words" may have the feeling that the pictures involved are the result of a serendipitous discovery or the consequence of an exceptional ingenuity on the part of the picture's creator. In this book the authors show that behind most of the pictures "proving" mathematical relations are some well-understood methods. As the reader shall see, a given mathematical idea or relation may have many different images that justify it, so that depending on the teaching level or the objectives for producing the pictures, one can choose the best alternative The object of this book is to show how visualization techniques may be employed to produce pictures that have interest for the creation, communication and teaching of mathematics. Mathematical drawings related to proofs have been produced since antiquity in China, Arabia, Greece and India but only in the last thirty years has there been a growing interest in so-called 'proofs without words.' In this book the authors show that behind most of the pictures 'proving' mathematical relations are some well-understood methods. The first part of the book consists of twenty short chapters, each one describing a method to visualize some mathematical idea (a proof, a concept, an operation,...) and several applications to concrete cases. Following this the book examines general pedagogical considerations concerning the development of visual thinking, practical approaches for making visualizations in the classroom and a discussion of the role that hands-on material plays in this process.