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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.
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.
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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.
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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.
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p. 234
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.
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
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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.
p. 235
Congruence,
Similarity, and Symmetry through Transformations
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.
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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.
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.
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 Euler circuits, a foundation for work with sophisticated networks.
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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.
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.
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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.
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