Geometry

Through the study of geometry, students will learn about geometric shapes and structures and how to analyze their characteristics and relationships. Spatial visualization—building and manipulating mental representations of two-and three-dimensional objects and perceiving an object from different perspectives—is an important aspect of geometric thinking. Geometry is a natural place for the development of students' reasoning and justification skills, culminating in work with proof in the secondary grades. Geometric modeling and spatial reasoning offer ways to interpret and describe physical environments and can be important tools in problem solving.

Geometric ideas are useful in representing and solving problems in other areas of mathematics and in real-world situations, so geometry should be integrated when possible with other areas. Geometric representations can help students make sense of area and fractions, histograms and scatterplots can give insights about data, and coordinate graphs can serve to connect geometry and algebra. Spatial reasoning is helpful in using maps, planning routes, designing floor plans, and creating art. Students can learn to see the structure and symmetry around them. Using concrete models, drawings, and dynamic geometry software, students can engage actively with geometric ideas. With well-designed activities, appropriate tools, and teachers' support, students can make and explore conjectures about geometry and can learn to reason carefully about geometric ideas from the earliest years of schooling. Geometry is more than definitions; it is about describing relationships and reasoning. The notion of building understanding in geometry across the grades, from informal to more formal thinking, is consistent with the thinking of theorists and researchers (Burger and Shaughnessy 1986; Fuys, Geddes, and Tischler 1988; Senk 1989; van Hiele 1986).

Geometry has long been regarded as the place in the school mathematics curriculum where students learn to reason and to see the axiomatic structure of mathematics. The Geometry Standard includes a strong focus on the development of careful reasoning and proof, using definitions and established facts. Technology also has an important role in the teaching and learning of geometry. Tools such as dynamic geometry software enable students to model, and have an interactive experience with, a large variety of two-dimensional shapes. Using technology, students can generate many examples as a way of forming and exploring conjectures, but it is important for them to recognize that generating many examples of a particular phenomenon does not constitute a proof. Visualization and spatial reasoning are also improved by interaction with computer animations and in other technological settings (Clements et al. 1997; Yates 1988).

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Young students are inclined naturally to observe and describe a variety of shapes and to begin to notice their properties. Identifying shapes is » important, too, but the focus on properties and their relationships should be strong. For example, students in prekindergarten through grade 2 may observe that rectangles work well for tiling because they have four right angles. At this level, students can learn about geometric shapes using objects that can be seen, held, and manipulated. Later, the study of the attributes of shapes and of their properties becomes more abstract. In higher grades, students can learn to focus on and discuss components of shapes, such as sides and angles, and the properties of classes of shapes. For example, using objects or dynamic geometric software to experiment with a variety of rectangles, students in grades 3–5 should be able to conjecture that rectangles always have congruent diagonals that bisect each other.

Through the middle grades and into high school, as they study such topics as similarity and congruence, students should learn to use deductive reasoning and more-formal proof techniques to solve problems and to prove conjectures. At all levels, students should learn to formulate convincing explanations for their conjectures and solutions. Eventually, they should be able to describe, represent, and investigate relationships within a geometric system and to express and justify them in logical chains. They should also be able to understand the role of definitions, axioms, and theorems and be able to construct their own proofs.

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

At first, young children learn concepts of relative position, such as above, behind, near, and between. Later they can make and use rectangular grids to locate objects and measure the distance between points along vertical or horizontal lines. Experiences with the rectangular coordinate plane will be useful as they solve a wider array of problems in geometry and algebra. In the middle and secondary grades, the coordinate plane can be helpful as students work on discovering and analyzing properties of shapes. Finding distances between points in the plane by using scales on maps or the Pythagorean relationship is important in the middle grades. Geometric figures, such as lines in the middle grades or triangles and circles in high school, can be represented analytically, thus establishing a fundamental connection between algebra and geometry.

Students should gain experience in using a variety of visual and coordinate representations to analyze problems and study mathematics. In the elementary grades, for example, an interpretation of whole-number addition can be demonstrated on the number line. In later years, students can use the number line to represent operations on other types of numbers. In grades 3–5, grids and arrays can help students understand multiplication. Later, more-complex problems can be considered. For example, in trying to minimize the distance an ambulance would have to travel to reach a new hospital from any location in the community, students in the middle grades might use distances measured along streets. In high school, students can be asked to find the shortest airplane route between two cities and compare the results using a map to the results using a globe. If students were trying to minimize the distances of a car trip to several cities, they might use vertex-edge graphs. High school » students should use Cartesian coordinates as a means both to solve problems and to prove their results.

Apply transformations and use symmetry to analyze mathematical situations

Young children come to school with intuitions about how shapes can be moved. Students can explore motions such as slides, flips, and turns by using mirrors, paper folding, and tracing. Later, their knowledge about transformations should become more formal and systematic. In grades 3–5 students can investigate the effects of transformations and begin to describe them in mathematical terms. Using dynamic geometry software, they can begin to learn the attributes needed to define a transformation. For example, to transform a figure using a rotation, students need to define the center of rotation, the direction of the rotation, and the angle of rotation, as illustrated in figure 3.3. In the middle grades, students should learn to understand what it means for a transformation to preserve distance, as translations, rotations, and reflections do. High school students should learn multiple ways of expressing transformations, including using matrices to show how figures are transformed on the coordinate plane, as well as function notation. They should also begin to understand the effects of compositions of transformations. At all grade levels, appropriate consideration of symmetry provides insights into mathematics and into art and aesthetics.

Fig. 3.3. A clockwise rotation of 120°

Use visualization, spatial reasoning, and geometric modeling to solve problems

Beginning in the early years of schooling, students should develop visualization skills through hands-on experiences with a variety of geometric objects and through the use of technology that allows them to turn, shrink, and deform two-and three-dimensional objects. Later, they should become comfortable analyzing and drawing perspective views, counting component parts, and describing attributes that cannot be seen but can be inferred. Students need to learn to physically and mentally change the position, orientation, and size of objects in systematic ways as they develop their understandings about congruence, similarity, and transformations.

One aspect of spatial visualization involves moving between two-and three-dimensional shapes and their representations. Elementary school students can wrap blocks in nets—two-dimensional figures, usually made of paper, that can be folded to form three-dimensional objects—as a step toward learning to predict whether certain nets match certain solids. By the middle grades, they should be able to interpret and create top or side views of objects. This skill can be developed by challenging them to build a structure given only the side view and the front view, as in figure 3.4. In grades 3–5, students can determine if it is possible to build more than one structure satisfying both conditions. Middle-grades and secondary school students can be asked to find the minimum number of blocks needed to build the structure. High school students should be able to visualize and draw other cross-sections of the structures and of a range of geometric solids.

Fig. 3.4. A block structure (from a presentation by J. de Lange)