Using the correct materials and manipulatives appropriately allows the richness of geometrical concepts to be fully investigated to enable students to explore and reason about shapes. This article explores some of the key ideas involved in learning about two dimensional shapes and discusses how materials can be used to support learning.

If an object is conveniently located in front of two plane mirrors placed at an angle, an observer can see a superposition of images that results in a face with three eyes, called in this text a "triclops." The conditions of occurrence of such an image may seem trivial, but this is incorrect: rather, the correct interpretation of this…

Several times a week I walk by a metal chair that is fastened to a flat concrete slab at an outdoor bus stop here in Boulder. One day I noticed on the concrete a nice shadow image of the woven metal seat of the chair (Fig. 1). The seat and back of the chair are formed from 3.8-cm wide strips of metal spaced 3.8 cm apart. The seat is about 39 cm…