Math Made Visual: Creating Images for Understanding Mathematics
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.
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Visualizing Mathematics by Creating Pictures
Representing Numbers by Lengths of Segments
Representing Numbers by Areas of Plane Figures
Representing Numbers by Volumes of Objects
Identifying Key Elements
Escaping from the Plane
Playing with Several Copies
Ingenuity in 3D
Using 3D Models
Visualization in the Classroom 117
Hints and Solutions to the Challenges
angle catenoid Challenges Chapter circle classroom colors compass and straightedge cone congruent construct copies cube curve cylinder dark gray diagonals dimensions dissection dots draw edges equilateral triangles example Fermat point function geometric golden ratio graph gray squares gray triangle hands-on materials hence hyperbola hypotenuse icosahedron illustrated in Figure images inscribed integers intersection isometries isosceles L-tetrominoes line segments mathematical mirrors models monohedral tiling Nelsen o o o o o o O O O OjO objects octahedron ooooooooo ooooooooo ooooooooo original triangle overlay paper parallelogram pentagonal pieces planar plane Platonic solids polygons polyhedra positive numbers problem puzzles pyramids Pythagorean theorem rectangle rectangular representation result rhombic dodecahedron right triangle rotation Section segments of lengths semiperimeter shape shown in Figure side length similar soap film solution spatial straightedge tangent tangram tetrahedron three-dimensional triangle with sides triangular numbers trominoes vertex vertices visual thinking