### Geometry Standard for Grades 6–8

Expectations
Instructional programs from prekindergarten through grade 12 should enable all students to— In grades 6–8 all students should—
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
 • precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties; • understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects; • create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems
 • use coordinate geometry to represent and examine the properties of geometric shapes; • use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
Apply transformations and use symmetry to analyze mathematical situations
 • describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling; • examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
Use visualization, spatial reasoning, and geometric modeling to solve problems
 • draw geometric objects with specified properties, such as side lengths or angle measures; • use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume; • use visual tools such as networks to represent and solve problems; • use geometric models to represent and explain numerical and algebraic relationships; • recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

Students should come to the study of geometry in the middle grades with informal knowledge about points, lines, planes, and a variety of two-and three-dimensional shapes; with experience in visualizing and drawing lines, angles, triangles, and other polygons; and with intuitive notions about shapes built from years of interacting with objects in their daily lives.

In middle-grades geometry programs based on these recommendations, students investigate relationships by drawing, measuring, visualizing, comparing, transforming, and classifying geometric objects. Geometry provides a rich context for the development of mathematical reasoning, including inductive and deductive reasoning, making and validating conjectures, and classifying and defining geometric objects. Many topics treated in the Measurement Standard for the middle grades are closely connected to students' study of geometry.

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

Middle-grades students should explore a variety of geometric shapes and examine their characteristics. Students can conduct these explorations using materials such as geoboards, dot paper, multiple-length cardboard strips with hinges, and dynamic geometry software to create two-dimensional shapes.

Students must carefully examine the features of shapes in order to precisely define and describe fundamental shapes, such as special types of quadrilaterals, and to identify relationships among the types of shapes. A teacher might ask students to draw several parallelograms on a coordinate grid or with dynamic geometry software. Students should make and record measurements of the sides and angles to observe some of the characteristic features of each type of parallelogram. They should then generate definitions for these shapes that are correct and consistent with the commonly used ones and recognize the principal relationships among elements of these parallelograms. A Venn diagram like the one shown in figure 6.14 might be used to summarize observations that a square is a special case of a rhombus and rectangle, each of which is a special case of a parallelogram.

 Fig. 6.14. A diagram showing the relationship among types of parallelograms

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The teacher might also ask students to draw the diagonals of multiple examples of each shape, as shown in figure 6.15, and then measure the lengths of the diagonals and the angles they form. The results can be summarized in a table like that in figure 6.16. Students should observe that the diagonals of these parallelograms bisect each other, which they might propose as a defining characteristic of a parallelogram. Moreover, they might observe, the diagonals are perpendicular in rhombuses (including squares) but not in other parallelograms and the diagonals are of equal length in rectangles (including squares) but not in other parallelograms. These observations might suggest other defining characteristics of special quadrilaterals, for instance, that a square is a parallelogram with diagonals that are perpendicular and of equal » length. Using dynamic geometry software, students could explore the adequacy of this definition by trying to generate a counterexample.

 Fig. 6.15. Students can draw the diagonals of parallelograms to make further observations.

 Fig. 6.16. A table of students' observations about the properties of the diagonals of special types of quadrilaterals

Middle-grades students also need experience in working with congruent and similar shapes. From their earlier work, students should understand that congruent shapes and angles are identical and can be "matched" by placing one atop the other. Students can begin with an intuitive notion of similarity: similar shapes have congruent angles but not necessarily congruent sides. In the middle grades, they should extend their understanding of similarity to be more precise, noting, for instance, that similar shapes "match exactly when magnified or shrunk" or that their corresponding angles are congruent and their corresponding sides are related by a scale factor.

Students can investigate congruence and similarity in many settings, including art, architecture, and everyday life. For example, observe the overlapping pairs of triangles in the design of the kite in figure 6.17. The overlapping triangles, which have been disassembled in the figure, can be shown to be similar. Students can measure the angles of the triangles in the kite and see that their corresponding angles are congruent. They can measure the lengths of the sides of the triangles and see that the differences are not constant but are instead related by a constant scale factor. With the teacher's guidance, students can thus begin to develop a more formal definition of similarity in terms of relationships among sides and angles.

 Fig. 6.17. Kite formed by overlapping triangles

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Investigations into the properties of, and relationships among, similar shapes can afford students many opportunities to develop and evaluate conjectures inductively and deductively. For example, an investigation of the perimeters, areas, and side lengths of the similar and » congruent triangles in the kite example could reveal relationships and lead to generalizations. Teachers might encourage students to formulate conjectures about the ratios of the side lengths, of the perimeters, and of the areas of the four similar triangles. They might conjecture that the ratio of the perimeters is the same as the scale factor relating the side lengths and that the ratio of the areas is the square of that scale factor. Then students could use dynamic geometry software to test the conjectures with other examples. Students can formulate deductive arguments about their conjectures. Communicating such reasoning accurately and clearly prepares students for creating and understanding more-formal proofs in subsequent grades.

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

Geometric and algebraic representations of problems can be linked using coordinate geometry. Students could draw on the coordinate plane examples of the parallelograms discussed previously, examine their characteristic features using coordinates, and then interpret their properties algebraically. Such an investigation might include finding the slopes of the lines containing the segments that compose the shapes. From many examples of these shapes, students could make important observations about the slopes of parallel lines and perpendicular lines. Figure 6.18 helps illustrate for one specific rhombus what might be observed in general: the slopes of parallel lines (in this instance, the opposite sides of the rhombus) are equal and the slopes of perpendicular lines (in this instance, the diagonals of the rhombus) are negative reciprocals. The slopes of the diagonals are

and

 Fig. 6.18. A rhombus drawn on the coordinate plane

#### Apply transformations and use symmetry to analyze mathematical situations

Transformational geometry offers another lens through which to investigate and interpret geometric objects. To help them form images of shapes through different transformations, students can use physical objects, figures traced on tissue paper, mirrors or other reflective surfaces, figures drawn on graph paper, and dynamic geometry software. They should explore the characteristics of flips, turns, and slides and should investigate relationships among compositions of transformations. These experiences should help students develop a strong understanding of line and rotational symmetry, scaling, and properties of polygons.

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From their experiences in grades 3–5, students should know that rotations, slides, and flips produce congruent shapes. By exploring the positions, side lengths, and angle measures of the original and resulting figures, middle-grades students can gain new insights into congruence. They could, for example, note that the images resulting from transformations have different positions and sometimes different orientations » from those of the original figure (the preimage), although they have the same side lengths and angle measures as the original. Thus congruence does not depend on position and orientation.

Transformations can become an object of study in their own right. Teachers can ask students to visualize and describe the relationship among lines of reflection, centers of rotation, and the positions of preimages and images. Using dynamic geometry software, students might see that each point in a reflection is the same distance from the line of reflection as the corresponding point in the preimage, as shown in figure 6.19a. In a rotation, such as the one shown in figure 6.19b, students might note that the corresponding vertices in the preimage and image are the same distance from the center of rotation and that the angles formed by connecting the center of rotation to corresponding pairs of vertices are congruent.

 Fig. 6.19. Using dynamic geometry software, students can explore the results of reflections (a) and rotations (b).

Teachers can pose additional challenges to develop students' understanding of transformations and congruence. For example, given the three pairs of congruent shapes in figure 6.20, students might be asked to identify a transformation applied to transform one shape into the other. Most students who have had extensive experience with transformations should see that the first pair appears to be related by a reflection, the second pair by a translation, and the third pair by either a reflection or a rotation. As students develop a more sophisticated understanding of transformations, they could be asked to describe the transformation more exactly, using distance, angles, and headings. The transformation for the second pair of shapes, for example, is a translation of 1.5 cm at a 45-degree angle.

 Fig. 6.20. Three pairs of congruent shapes

Teachers may also want students to consider what happens when transformations are composed. For example, the image produced when a figure is reflected through one line and the resulting image is reflected through a different line will be either a translation of the preimage if the lines of reflection are parallel or a rotation if they intersect. To assess students' understanding of transformations, teachers can give students two congruent shapes and have them specify a transformation or a composition of transformations that will map one to the other. This can be done using dynamic geometry software.

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Transformations can also be used to help students understand similarity and symmetry. Work with magnifications and contractions, called dilations, can support students' developing understanding of similarity. For » example, dilation of a shape affects the length of each side by a constant scale factor, but it does not affect the orientation or the magnitude of the angles. In a similar manner, rotations and reflections can help students understand symmetry. Students can observe that when a figure has rotational symmetry, a rotation can be found such that the preimage (original shape) exactly matches the image but its vertices map to different vertices. Looking at line symmetry in certain classes of shapes can also lead to interesting observations. For example, isosceles trapezoids have a line of symmetry containing the midpoints of the parallel opposite sides (often called bases). Students can observe that the pair of sides not intersected by the line of symmetry (often called the legs) are congruent, as are the two corresponding pairs of angles. Students can conclude that the diagonals are the same length, since they can be reflected onto each other, and that several pairs of angles related to those diagonals are also congruent. Further exploration reveals that rectangles and squares also have a line of symmetry containing the midpoints of a pair of opposite sides (and other lines of symmetry as well) and all the resulting properties.

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

Students' skills in visualizing and reasoning about spatial relationships are fundamental in geometry. Some students may have difficulty finding the surface area of three-dimensional shapes using two-dimensional representations because they cannot visualize the unseen faces of the shapes. Experience with models of three-dimensional shapes and their two-dimensional "nets" is useful in such visualization (see fig. 6.25 in the "Measurement" section for an example of a net). Students also need to examine, build, compose, and decompose complex two-and three-dimensional objects, which they can do with a variety of media, including paper-and-pencil sketches, geometric models, and dynamic geometry software. Interpreting or drawing different views of buildings, such as the base floor plan and front and back views, using dot paper can be useful in developing visualization. Students should build three-dimensional objects from two-dimensional representations; draw objects from a geometric description; and write a description, including its geometric properties, for a given object.

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Students can also benefit from experience with other visual models, such as networks, to use in analyzing and solving real problems, such as those concerned with efficiency. To illustrate the utility of networks, students might consider the problem and the networks given in figure 6.21 (adapted from Roberts [1997, pp. 106–7]). The teacher could ask students to determine one or several efficient routes that Caroline might use for the streets on map A, share their solutions with the class, and describe how they found them. Students should note the start-end point of each route and the number of different routes that they find. Students could then find an efficient route for map B. They should eventually conclude that no routes in map B satisfy the conditions of the problem. They should discuss why no such route can be found; the teacher might suggest that students count the number of paths attached to each node and look at where they "get stuck" in order to » understand better why they reach an impasse. To extend this investigation, students could look for efficient paths in other situations or they might change the conditions of the map B problem to find the pathway with the least backtracking. Such an investigation in the middle grades is a precursor of later work with Hamiltonian circuits, a foundation for work with sophisticated networks.

 Fig. 6.21. Networks used to solve efficiency problems (Adapted from Roberts [1991, pp. 101–7])

Visual demonstrations can help students analyze and explain mathematical relationships. Eighth graders should be familiar with one of the many visual demonstrations of the Pythagorean relationship—the diagram showing three squares attached to the sides of a right triangle. Students could replicate some of the other visual demonstrations of the relationship using dynamic geometry software or paper-cutting procedures, and then discuss the associated reasoning.

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Geometric models are also useful in representing other algebraic relationships, such as identities. For example, the visual demonstrations of the identity (a + b)2 = a2 + 2ab + b2 in figure 6.22 makes it easy to remember. A teacher might begin by asking students to draw a square with side lengths (2 + 5). Students could then partition the square as shown in fig. 6.22a, calculate the area of each section, and finally represent the total area. Students could then apply this approach to the » general case of a square with sides of length (a + b), as shown in figure 6.22b, which demonstrates the identity (a + b)2 = a2 + 2ab + b2.

 Fig. 6.22. Geometric representation demonstrating the identity (a + b)2 = a2 + 2ab + b2

Many investigations in middle-grades geometry can be connected to other school subjects. Nature, art, and the sciences provide opportunities for the observation and the subsequent exploration of geometry concepts and patterns as well as for appreciating and understanding the beauty and utility of geometry. For example, the study in nature or art of golden rectangles (i.e., rectangles in which the ratio of the lengths is the golden ratio, (1 + )/2) or the study of the relationship between the rigidity of triangles and their use in construction helps students see and appreciate the importance of geometry in our world.