In grades 9-12, the mathematics curriculum should include the study of the geometry of two and three dimensions from an algebraic point of view so that all students can-

• translate between synthetic and coordinate representations;
• deduce properties of figures using transformations and using coordinates;
• identify congruent and similar figures using transformations;
• analyze properties of Euclidean transformations and relate translations to vectors;

and so that, in addition, college-intending students can-

• deduce properties of figures using vectors;
• apply transformations, coordinates, and vectors in problem solving.

Focus

One of the most important connections in all of mathematics is that between geometry and algebra. Historically, mathematics took a great stride forward in the seventeenth century when the geometric ideas of the ancients were expressed in the language of coordinate geometry, thus providing new tools for the solution of a wide range of problems.

More recently, the study of geometry through the use of transformations-the geometric counterpart of functions--has changed the subject from static to dynamic, providing in the process great additional power that can be used, for example, to describe and produce moving figures on a video screen. Viewed as an algebraic system, transformations also provide college-intending students with valuable experiences with properties of function composition and group structure.

The interplay between geometry and algebra strengthens students' ability to formulate and analyze problems from situations both within and outside mathematics. Although students will at times work separately in synthetic, coordinate, and transformation geometry, they should have as many opportunities as possible to compare, contrast, and translate among these systems. A fundamental idea students should come to understand is that specific problems are often more easily solved in one or another of these systems.

Discussion

Objects and relations in geometry correspond directly to objects and relations in algebra. For example, a point in geometry corresponds to an ordered pair (x, y) of numbers in algebra, a line to a set of ordered pairs satisfying an equation of the form ax + by = c, and the intersection of two lines to the set of ordered pairs that satisfy the corresponding equations. It is correspondences like these that allow translation between the two "languages" and permit concepts in one to clarify and reinforce concepts in the other. In fact, deducing properties of geometric figures using their coordinate representations is often easier for students than synthetic proofs are. For example, all students can calculate the coordinates of the midpoints of two sides of a triangle with given numerical coordinates and use the results to deduce that the segment joining them is parallel to the third side of the triangle and equals half its length (fig. 8.1). College-intending students should be able to extend this technique to prove the result in general.

Fig. 8.1. A coordinate argument

Although students will continue to work with two dimensions, every opportunity should be taken to explore the third dimension as well. Algebraic formulations in three-dimensional coordinate geometry should focus on figures that are simple to represent, such as points, planes perpendicular to an axis, and spheres. The coordinate representation of general planes and lines is more difficult and would best be treated as an enrichment project.

Transformations serve as a powerful problem-solving tool and permit students to develop a broad concept of congruence and similarity that applies to all figures. The derivation of congruence properties through isometries (distance-preserving transformations) and of similarity properties through composites of dilations (ratio-preserving transformations) and isometries provides a connection with, and a reinforcement of, synthetic methods. Transformation geometry in three dimensions is straightforward: most three-dimensional transformations are simply direct extensions of their two-dimensional counterparts.

Transformations often are used to represent physical motions, such as slides, flips, turns, and stretches. Students should use computer software based on this dynamic view of transformations to explore properties of translations, line reflections, rotations, and dilations, as well as compositions of these transformations (fig. 8.2). These graphics experiences not only help students develop an understanding of the effects of various transformations but also contribute to the development of their skills in visualizing congruent and similar figures.

Fig. 8.2. Effects on the plane of the composites of two transformations

The midpoint theorem discussed previously in terms of a coordinate approach also can be proved using a dilation with scale factor 2 (or 1/2) and with center at the triangle vertex that is common to the two sides whose midpoints are connected (see fig. 8.3).

Fig. 8.3. A transformation proof

Although this theorem can also be proved by synthetic methods, students should appreciate the economy offered by coordinate, transformation, and even vector techniques (see fig. 8.4).

Fig. 8.4. A vector proof

It is important that all students come to understand how vectors can be used to represent such physical phenomena as velocity and force. They should investigate vector addition and scalar multiplication, both geometrically and algebraically, and be introduced to applications of those ideas such as in quantifying the effect of wind on the course of an airplane. College-intending students should develop facility with the use of vectors to solve problems as well as to prove geometric theorems (as exemplified in fig. 8.4).

Taken together, the two geometry standards advocate a more eclectic approach to the subject, one based on informal explorations and short local axiomatic sequences. The study of geometry should provide students with the ability to recognize and apply effectively the geometric concepts and methods (synthetic, coordinate, transformation, or vector) most appropriate to a given problem situation.