In grades 5-8, the mathematics curriculum should include the study of the geometry of one, two, and three dimensions in a variety of situations so that students can--

• identify, describe, compare, and classify geometric figures;
• visualize and represent geometric figures with special attention to developing spatial sense;
• explore transformations of geometric figures;
• represent and solve problems using geometric models;
• understand and apply geometric properties and relationships;
• develop an appreciation of geometry as a means of describing the physical world.

Focus

Geometry is grasping space . . . that space in which the child lives, breathes and moves. The space that the child must learn to know, explore, conquer, in order to live, breathe and move better in it. (Freudenthal 1973, p. 403).

The study of geometry helps students represent and make sense of the world. Geometric models provide a perspective from which students can analyze and solve problems, and geometric interpretations can help make an abstract (symbolic) representation more easily understood. Many ideas about number and measurement arise from attempts to quantify real-world objects that can be viewed geometrically. For example, the use of area models provides an interpretation for much of the arithmetic of decimals, fractions, ratios, proportions, and percents.

Students discover relationships and develop spatial sense by constructing, drawing, measuring, visualizing, comparing, transforming, and classifying geometric figures. Discussing ideas, conjecturing, and testing hypotheses precede the development of more formal summary statements. In the process, definitions become meaningful, relationships among figures are understood, and students are prepared to use these ideas to develop informal arguments. The informal exploration of geometry can be exciting and mathematically productive for middle school students. At this level, geometry should focus on investigating and using geometric ideas and relationships rather than on memorizing definitions and formulas.

The study of geometry in grades 5-8 links the informal explorations begun in grades K-4 to the more formalized processes studied in grades 9-12. The expanding logical capabilities of students in grades 5-8 allow them to draw inferences and make logical deductions from geometric problem situations. This does not imply that the study of geometry in grades 5-8 should be a formalized endeavor; rather, it should simply provide increased opportunities for students to engage in more systematic explorations.

Discussion

A teacher's questioning techniques and language in directing students' thinking are critical to the students' development of an understanding of geometric relationships. Students should be challenged to analyze their thought processes and explanations. They should be allowed sufficient time to discuss the quality of their answers and to ponder such questions as, Could it be another way? What would happen if ....? Students should learn to use correct vocabulary, including such common terms as and, or, all, some, always, never, and if .... then, to reason, as well as such words as parallel, perpendicular, and similar to describe. Geometry also has a vocabulary of its own, including terms like rhombus, trapezoid, and dodecahedron, and students need ample time to develop confidence in their use of this new and unique language. Definitions should evolve from experiences in constructing, visualizing, drawing, and measuring two- and three-dimensional figures, relating properties to figures, and contrasting and classifying figures according to their properties. Students who are asked to memorize a definition and a textbook example or two are unlikely to remember the term or its application.

Triangles are a subject of study in all grades, K-12. At the middle school level, most of the basic properties of triangles can be developed through investigations such as the following.

You are given a pile of toothpicks all the same size. First, take three toothpicks. Can you form a triangle using all three toothpicks placed end to end in the same plane? Can a different triangle be formed? What kinds of triangles are possible?

Now take four toothpicks and repeat the questions. Then repeat with five toothpicks, six toothpicks, and so on.

A table such as that in figure 12.1 helps students to organize their data in a systematic manner.

Fig. 12.1. Triangles

In this activity, students find that the sum of the measures of two sides of a triangle must be greater than the measure of the third side. This activity also reinforces the classification of triangles by sides and angles.

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.

Fig. 12.2. Pythagorean theorem

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. Their discovery that a parallelogram is formed can prompt them to ask such questions as, How does the area of the new figure compare to that of the quadrilateral? What quadrilateral would you start with so that the new figure is a rectangle? A square? 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.

Fig. 12.3. Quadrilaterals

Computer software allows students to construct two- and three-dimensional shapes on a screen and then flip, turn, or slide them to view them from a new perspective. Explorations of flips, slides, turns, stretchers, and shrinkers will illuminate the concepts of congruence and similarity. Observing and learning to represent two- and three-dimensional figures in various positions by drawing and construction also helps students develop spatial sense.

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. Most students in grades 5-8 incorrectly believe that if the sides of a figure are doubled to produce a similar figure, the area and volume also will be doubled. See figure 12.4.

Fig. 12.4. Area and volume

Investigations of two- and three-dimensional models fosters an understanding of the different growth rates for linear measures, areas, and volumes of similar figures. These ideas are fundamental to measurement and critical to scientific applications.

Students' understanding of the angle properties of polygons and the concept of area can be enhanced through explorations of tessellations with regular polygons. Which polygons will cover the plane and which ones will not? Why? This exercise can be extended to combining regular polygons and investigating solids constructed from regular polygons. In such a discussion students can also consider why the square is used as a unit of area and the cube as a unit of volume.

Symmetry in two and three dimensions provides rich opportunities for students to see geometry in the world of art, nature, construction, and so on. Butterflies, faces, flowers, arrangements of windows, reflections in water, and some pottery designs involve symmetry. Turning symmetry is illustrated by bicycle gears. Pattern symmetry can be observed in the multiplication table, in numbers arrayed in charts, and in Pascal's triangle.