Defining Representation
 The Representation Standard | Visual Representations | Tables, Graphs, and Variables | Additional Points | Summary | Your Journal
 In the Building Viewpoints video from the Observe part of this session, the students are given a two-dimensional description of a three-dimensional structure. From that description, they build the structure and then closely examine each view. In order to communicate their thinking, they represent each view on grid paper. From that information, they can determine relationships among the views and describe the original structure. The culmination of this activity is learning how to represent three-dimensional objects in a diagram. For example, students can create two-dimensional drawings on grid paper that represent the three-dimensional aspects of their buildings. In the middle grades, we want students to use representations to solve problems, to clarify and extend their mathematical ideas, and to recognize that two different representations might describe the same phenomenon. Drawings and models are especially appropriate as students work with geometry and spatial visualization concepts. However, these representations can also connect what students know in one strand to their experiences in other strands. For example, students who use area models to develop an understanding of multiplication can connect the concept of multiplication of whole numbers to finding the area of two-dimensional figures. Fraction area models help students develop understanding of proportional reasoning. Area models can also be used to develop concepts in probability. This single form of representation, first used with a simple concept, becomes a tool for students to use to help give meaning to more complex ideas. Watch the video segment (duration 0:29) in the viewer box on the upper left to hear a reflection from Pam Hardaway, a middle school mathematics teacher.
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