In grades 9-12, the mathematics curriculum should include the continued study of the geometry of two and three dimensions so that all students can--

• interpret and draw three-dimensional objects;
• represent problem situations with geometric models and apply properties of figures;
• classify figures in terms of congruence and similarity and apply these relationships;
• deduce properties of, and relationships between, figures from given assumptions;

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

• develop an understanding of an axiomatic system through investigating and comparing various geometries.

Focus

This component of the 9-12 geometry strand should provide experiences that deepen students' understanding of shapes and their properties, with an emphasis on their wide applicability in human activity. The curriculum should be infused with examples of how geometry is used in recreations (as in billiards or sailing); in practical tasks (as in purchasing paint for a room); in the sciences (as in the description and analysis of mineral crystals); and in the arts (as in perspective drawing).

High school geometry should build on the strong conceptual foundation students develop in the new K-8 programs. Students should have opportunities to visualize and work with three-dimensional figures in order to develop spatial skills fundamental to everyday life and to many careers. Physical models and other real-world objects should be used to provide a strong base for the development of students' geometric intuition so that they can draw on these experiences in their work with abstract ideas.

Discussion

Prior to the work of the ancient Greeks (e.g., Thales and Pythagoras), geometric ideas were tied directly to the solution of real-world problems. Hence, the subsequent abstraction and formalization of these ideas, which evolved into the subject of geometry as we know it, has always had many applications in the real world. More recently, fractal geometry, which originated in the mid-1970s with the pioneering work of Benoit Mandelbrot, has provided useful models for analyzing a wide variety of phenomena, from changes in coastlines to chaotic fluctuations in commodity prices. It is the intent of this standard that, whenever possible, real-world situations will provide a context for both introducing and applying geometric topics.

The construction trades use geometric ideas on a daily basis. When a builder is laying the foundation for a rectangular garage, opposite sides are measured to be the same length. However, it is also essential that corner angles be right angles. Students could be encouraged to find ways to check whether this is so using only a tape measure. The two parallelograms on the top in figure 7.1 show two possible methods for solution; the second method, measuring the diagonals, also can be used to "square up" the frame of the garage if it is to be in the shape of a rectangular solid and it is known that opposite edges of the faces are the same length. If the diagonals are of equal length, the angles are right angles. Students may have the opportunity to apply these methods in actual construction in vocational classes or at home. Some students, especially the college intending, should be expected to cite or prove theorems to justify these methods.

Fig. 7.1. Squaring up a rectangular foundation and building

Geometry also is used extensively in the sciences. In the biological sciences, for example, scaling (the geometric concept of similarity) is used to identify limiting factors on the growth of various organisms. Students who have learned some basic ideas of similarity, including the relationship between ratios of sides, areas, and volumes of similar figures, may find such practical, real-world applications of their geometry studies fascinating. Through discussion, students could agree that a very tall human being is roughly similar to one of more normal height, that weight is a function of volume, and that the ability to support weight is a function of the area of a cross-section of leg bones. Similarity concepts are then sufficient for the students to show that to enlarge a six-foot, 175-pound person by a scale factor of 2 would result in a twelve-foot giant, weighing 8 x 175 or 1400 pounds. The cross-sectional area of the giant's bones would be increased by a factor of 4. Thus, the pressure on the leg bones of a twelve-foot person who weighed 1400 pounds would be the same as that of a six-foot person who weighed 700 pounds. This explains why the size of human beings and other organisms is limited by their structural characteristics; the giant flies, spiders, and apes in science-fiction films could not exist in the real world. As an extension of this geometric method, students could read about and discuss practical size limits for inanimate objects such as trees, buildings, and mountains, which can be drawn from a knowledge of the strength of the supporting material such as wood, steel, and granite.

Geometry also provides an opportunity for students to experience the creative interplay between mathematics and art. For example, by repeated tracing of a regular polygon about a point so that the tracings coincide only along edges and do not overlap, and then by extending their tracings outward on the paper (plane), students can discover whether the polygon might be used to form a tiling (tessellation) of the plane. Out of this very informal experience arises a fundamental question: Which regular polygons can be used to tile a plane? This question, of course, leads to student investigation of how to determine the angle measure of a regular polygon, an opportunity for reasoning both deductively (from a knowledge of the angle-sum property of a triangle) and inductively. Once students have determined that only the equilateral triangle, square, and hexagon can be used singly to form a tiling pattern, project work for groups of students could include exploring (a) the number and nature of semiregular tilings using a combination of two or more regular polygons and where these patterns appear in their environment; (b) the existence of nonregular polygons that would serve as a fundamental tiling unit; or (c) the graphics work of M. C. Escher and the creation of Escher-type tessellations (fig. 7.2). This last activity is appealing to many high school students and provides an excellent setting for creative expression.

Fig. 7.2. Creating an Escher-type tessellation

Instruction should focus increased attention on the analysis of three-dimensional figures. Such work is especially important to students who may pursue careers in art, architecture, drafting, and engineering. Appropriate use of three-dimensional representation and CAD (computer-assisted design) software is of particular value in such exercises. It is also important to understand that visualization includes plane figures as well. For example, computer graphics software that allows students to create and manipulate shapes provides an exciting environment in which they can make conjectures and test their attempts at two-dimensional visualization. There are, of course, ample opportunities for visualization within standard activities that do not use a computer. In particular, exercises that require the student to represent given information by drawing a diagram provide an excellent setting for facilitating the reading of mathematics, as well as a special opportunity for problem translation.

Computer microworlds such as Logo turtle graphics and the topics of constructions and loci provide opportunities for a great deal of student involvement. In particular, the first two contexts serve as excellent vehicles for students to develop, compare, and apply algorithms.

Although the hypothetical deductive nature of geometry first developed by the Greeks should not be overlooked, this standard proposes that the organization of geometric facts from a deductive perspective should receive less emphasis, whereas the interplay between inductive and deductive experiences should be strengthened. For example, students should first use an interactive computer software package that allows experimentation with figures and relations to observe across several trials that the length of the median to the hypotenuse of any right triangle appears to be equal to the lengths of the segments it cuts off on the hypotenuse. In the second phase, they would provide a deductive argument verifying their discovery.

Both inductive and deductive reasoning are required as students begin to develop short sequences of theorems. For example, students could be provided the necessary definitions and a set of postulates consisting of three triangle-congruence statements (SSS, SAS, ASA) and a parallel-line statement (if two parallel lines are cut by a transversal, the alternate interior angles are congruent). After class discussion has established a definition of a parallelogram, students could be assigned the task of formulating and then proving or disproving their own conjectures about properties of parallelograms. Following agreement on definitions of a rectangle, square, and rhombus, students could discover and verify properties of these figures using the deduced properties of parallelograms. Students should perform these exercises without reference to similar proofs in a textbook. Figure 7.3 shows a sequence of theorems that could be deduced from these postulates.

Fig. 7.3. An example of local axiomatics

Other topics amenable to organization by local axiomatics include theorems related to parallelism in a plane (in space), similarity of polygons, right-triangle relationships including the Pythagorean theorem, and areas of polygons.

Logical reasoning also is conveyed through carefully designed numerical exercises. Consider, for example, the following exercise (for which the diagram in fig. 7.4 would not be provided):

Fig. 7.4. Numerical context for logical reasoning

In right triangle ABC with hypotenuse AB = 32, M, N, P, Q, and R are midpoints of segments AB, AC, CB, BM, and AM, respectively. Find the perimeter of NPQR.

In this single exercise, students would need to apply logical reasoning in translating the information into a diagram, to derive useful information from appropriate theorems, and to organize that information to lead to the correct calculations. The connection between the problem solution and the sequence of steps in a proof should be emphasized.

College-intending students also should gain an appreciation of Euclidean geometry as one of many axiomatic systems. This goal may be achieved by directing students to investigate properties of other geometries to see how the basic axioms and definitions lead to quite different--and often contradictory--results. For example, great circles, which play the role of lines in spherical geometry, always meet. Thus, in spherical geometry, instead of having exactly one line parallel to a given line through a point not on the line, there are no such lines. Figure 7.5 shows another interesting difference between Euclidean geometry and the geometry of a sphere. Students also could examine some of the history associated with attempts to prove Euclid's famous fifth postulate from both a mathematical and a cultural perspective.

Fig. 7.5. Geometry of a sphere

In summary, synthetic geometry at the high school level should focus on more than deductive reasoning and proof. Equally important is the continued development of students' skills in visualization, pictorial representation, and the application of geometric ideas to describe and answer questions about natural, physical, and social phenomena.