One of the most important properties in geometry, the Pythagorean theorem, is introduced in the middle grades. Students can discover this relationship through explorations, such as the one suggested in figure 12.2 . Another interesting problem in which students at different levels can investigate geometric properties and relationships of quadrilaterals is shown in figure 12.3 . Students can explore what happens when they connect the midpoints of the sides of several quadrilaterals. Computer software that allows students to construct geometric figures and determine the measures of arcs, angles, and segments creates a rich environment for the investigation of geometric properties and relationships.
Students can make conjectures and explore other figures to verify their reasoning.
Explorations of flips, slides, turns, stretchers, and shrinkers will illuminate the concepts of congruence and similarity. Measuring and comparing the sides and angles of similar polygons help students develop and understand the mathematical concept of similar figures. The relationship between the angles and the sides of similar triangles is the foundation of trigonometry. Similarity also can be related to such real-world contexts as photographs, models, projections of pictures, and photocopy machines. Students should explore the relationships among the lengths, areas, and volumes of similar solids.