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11 652 Extending Geometry Extending Geometry ISBN: 0-536-08809-8 Mathematics for Elementary School Teachers , Fourth Edition, by Phares O'Daffer, Randall Charles, Thomas Cooney, John Dossey, and Jane Schielack. Published by Addison Wesley. Copyright 2008 by Pearson Education, Inc. 653 Chapter Perspective The beauty of mathematics is accentuated through the lens of geometry. Geometric designs in buildings, such as the Alhambra in Granada, Spain shown in this photo, reveal symmetry and give insight into this important mathematical idea. The intriguing world of patterns, designs, and tessellations brings to life some fascinating relationships among mathematics, nature, and art. Also, whether you are thinking of an astronomer describing the changes in the positions of stars in the sky, or a physicist describing the movement of particles in an accelerator, you notice that motion plays an integral part in natural events. The idea of transformations helps us give useful mathematical descriptions of motions in our world. In this chapter, we focus on fig- ures, relationships, and patterns in space, including geometric descrip- tions of motion, tessellations, and special polygons to extend the development of your spatial sense. Big Ideas Proportionality: If two quantities vary proportionally, that relationship can be represented as a linear function. Patterns: Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways. Orientation and Location: Objects in space can be oriented in an infinite number of ways, and an objects location in space can be described quantitatively. Shapes and Solids: Two- and three-dimensional objects with or without curved surfaces can be described, classified, and analyzed by their attributes. Transformations: Objects in space can be transformed in an infinite number of ways, and those transformations can be described and analyzed mathematically. Connection to the NCTM Principles and Standards The NCTM Principles and Standards for School Mathematics (2000) indicate that the mathematics curriculum in geometry for grades PreK8 should prepare students to apply transformations and use symmetry to analyze mathematical situations; use visualization, spatial reasoning, and geometric modeling to solve problems (p. 41) . Connection to the PreK8 Classroom In grades PreK2, students use their own physical experiences with shapes, such as fitting pieces into a puzzle, to learn about slides, turns, ips, and symmetry. In grades 35, students are ready to mentally manipulate shapes to make predictions, learn the mathematical language to describe their predictions, then verify their predictions physically.... View Full Document