Geometry Standard for Grades Pre-K–2

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Geoboards and Polygons
(Part 1)

Teachers must provide materials and structure the environment appropriately to encourage students to explore shapes and their attributes. For example, young students can compare and sort building blocks as they put them away on shelves, identifying their similarities and differences. They can use commonly available materials such as cereal boxes to explore attributes of shapes or folded paper to investigate symmetry and congruence. Students can create shapes on geoboards or dot paper and represent them in drawings, block constructions, and dramatizations. »

Students need to see many examples of shapes that correspond to the same geometrical concept as well as a variety of shapes that are nonexamples of the concept. For example, teachers must ensure that students see collections of triangles in different positions and with different sizes of angles (see fig. 4.12) and shapes that have a resemblance to triangles (see fig. 4.13) but are not triangles. Through class discussions of such examples and nonexamples, geometric concepts are developed and refined.

Fig. 4.12. Examples of triangles

Fig. 4.13. Examples of nontriangles

Students also learn about geometric properties by combining or cutting apart shapes to form new shapes. For example, second-grade students can be challenged to find and record all the different shapes that can be created with the two triangles shown in figure 4.14. Interactive computer programs provide a rich environment for activities in which students put together or take apart (compose and decompose) shapes. Technology can help all students understand mathematics, and interactive computer programs may give students with special instructional needs access to mathematics they might not otherwise experience.

Fig. 4.14. Two triangles can be combined to make different shapes.

Four types of mathematical questions regarding navigation and maps can help students develop a variety of spatial understandings: direction (which way?), distance (how far?), location (where?), and representation (what objects?). In answering these questions, students need to develop a variety of skills that relate to direction, distance, and position in space. Students develop the ability to navigate first by noticing landmarks, then by building knowledge of a route (a connected series of landmarks), and finally by putting many routes and locations into a kind of mental map (Clements 1999b).

Teachers should extend young students' knowledge of relative position in space through conversations, demonstrations, and stories. When students act out the story of the three billy goats and illustrate over and » under, near and far, and between, they are learning about location, space, and shape. Gradually students should distinguish navigation ideas such as left and right along with the concepts of distance and measurement. As they build three-dimensional models and read maps of their own environments, students can discuss which blocks are used to represent various objects like a desk or a file cabinet. They can mark paths on the model, such as from a table to the wastebasket, with masking tape to emphasize the shape of the path. Teachers should help students relate their models to other representations by drawing a map of the same room that includes the path. In similar activities, older students should develop map skills that include making route maps and using simple coordinates to locate their school on a city map (Liben and Downs 1989).

Tangram Challenge (Part 1)

Teachers should choose geometric tasks that are accessible to all students and sufficiently open-ended to engage students with a range of interests. For example, a second-grade teacher might instruct the class to find all the different ways to put five squares together so that one edge of each square coincides with an edge of at least one other square (see fig. 4.15). The task should include keeping a record of the pentominoes that are identified and developing a strategy for recognizing when they are transformations of another pentomino. Teachers can » encourage students to develop strategies for being systematic by asking, "How will you know if each pentomino is different from all the others? Are you certain you have identified all the possibilities?" They can challenge students to predict which of their pentominoes, if cut out of grid paper, would fold into an open box and then verify (or reject) their predictions by cutting them out and trying to fold them into boxes.

Fig. 4.15. Examples of pentominoes and nonpentominoes

Teachers should guide students to recognize, describe, and informally prove the symmetric characteristics of designs through the materials they supply and the questions they ask. Students can use pattern blocks to create designs with line and rotational symmetry (see fig. 4.16) or use paper cutouts, paper folding, and mirrors to investigate lines of symmetry.

Fig. 4.16. Symmetries can be found in designs.

Spatial visualization can be developed by building and manipulating first concrete and then mental representations of shapes, relationships, and transformations. Teachers should plan instruction so that students can explore the relationships of different attributes or change one characteristic of a shape while preserving others. In the activity in figure 4.17, students are holding a long loop of string so that each student's hand serves as a vertex of the triangle. In this arrangement, students experiment with changing a shape by increasing the number of sides while the perimeter is unchanged. Conversations about what they notice and how to change from one shape to another allow students to hear different points of view and at the same time give teachers insight into their students' understanding. Work with concrete shapes, illustrated in this activity, lays a valuable foundation for spatial sense. To further develop students' abilities, teachers might ask them to see in their "mind's eye" the shapes that would result when a shape is flipped or when a square is cut diagonally from corner to corner. Thus, many shape and transformation activities build spatial reasoning if students are asked to imagine, » predict, experiment, and check the results of the work themselves (see fig. 4.18).

Fig. 4.17. Making a string triangle

Fig. 4.18. Paper cutting can aid spatial visualization and reasoning

Classroom activities that enhance visualization, such as asking students to recall the configuration of the dots on dominoes and determine the number of dots without counting, can promote spatial memory. A teacher could place objects (such as scissors, a pen, a leaf, a paper clip, and a block) on the overhead projector, show the objects briefly, and ask students to name the objects they glimpsed. Or the teacher might have students close their eyes; she could then take one object away and ask which one was removed.

In another "quick image" activity, students can be briefly shown a simple configuration such as the one in figure 4.19 projected on a screen and then asked to reproduce it. The configuration is shown again for a couple of seconds, and they are encouraged to modify their drawings. The process may be repeated several times so that they have opportunities to evaluate and self-correct their work (Yackel and Wheatley 1990). Asking, "What did you see? How did you decide what to draw?" is likely to elicit different explanations, such as "three triangles," "a sailboat sinking," "a square with two lines through it," "a y in a box," and "a sandwich that has been cut into three pieces." Students who can see the configuration in several ways may have more mathematical knowledge and power than those who are limited to one perspective.

Fig. 4.19. In a quick-image activity, students try to reproduce this image, which has briefly been projected on a screen.

Spatial visualization and reasoning can be fostered in navigation activities when teachers ask students to visualize the path they just walked from the library and describe it by specifying landmarks along the route or when students talk about how solid geometric shapes look from different perspectives. Teachers should ask students to identify structures from various viewpoints and to match views of the same structure portrayed from different perspectives. Using a variety of magazine photographs, older students might discuss the location of the photographers when they took each one.

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Geoboards and Polygons (Part 2)