### Geometry Standard for Grades 3–5

Expectations
Instructional programs from prekindergarten through grade 12 should enable all students to— In grades 3–5 all students should—
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
 • identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes; • classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids; • investigate, describe, and reason about the results of subdividing, combining, and transforming shapes; • explore congruence and similarity; • make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems
 • describe location and movement using common language and geometric vocabulary; • make and use coordinate systems to specify locations and to describe paths; • find the distance between points along horizontal and vertical lines of a coordinate system.
Apply transformations and use symmetry to analyze mathematical situations
 • predict and describe the results of sliding, flipping, and turning two-dimensional shapes; • describe a motion or a series of motions that will show that two shapes are congruent; • identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
Use visualization, spatial reasoning, and geometric modeling to solve problems
 • build and draw geometric objects; • create and describe mental images of objects, patterns, and paths; • identify and build a three-dimensional object from two-dimensional representations of that object; • identify and draw a two-dimensional representation of a three-dimensional object; • use geometric models to solve problems in other areas of mathematics, such as number and measurement; • recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

The reasoning skills that students develop in grades 3–5 allow them to investigate geometric problems of increasing complexity and to study geometric properties. As they move from grade 3 to grade 5, they should develop clarity and precision in describing the properties of geometric objects and then classifying them by these properties into categories such as rectangle, triangle, pyramid, or prism. They can develop knowledge about how geometric shapes are related to one another and begin to articulate geometric arguments about the properties of these shapes. They should also explore motion, location, and orientation by, for example, creating paths on a coordinate grid or defining a series of flips and turns to demonstrate that two shapes are congruent. As students investigate geometric properties and relationships, their work can be closely connected with other mathematical topics, especially measurement and number.

The study of geometry in grades 3–5 requires thinking and doing. As students sort, build, draw, model, trace, measure, and construct, their capacity to visualize geometric relationships will develop. At the same time they are learning to reason and to make, test, and justify conjectures about these relationships. This exploration requires access to a variety of tools, such as graph paper, rulers, pattern blocks, geoboards, and geometric solids, and is greatly enhanced by electronic tools that support exploration, such as dynamic geometry software.

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

In the early grades, students will have classified and sorted geometric objects such as triangles or cylinders by noting general characteristics. In grades 3–5, they should develop more-precise ways to describe shapes, focusing on identifying and describing the shape's properties and learning specialized vocabulary associated with these shapes and properties. To consolidate their ideas, students should draw and construct shapes, compare and discuss their attributes, classify them, and develop and consider definitions on the basis of a shape's properties, such as that a rectangle has four straight sides and four square corners. For example, many students in these grades will easily name the first two shapes in figure 5.10 as rectangles but will need to spend more time discussing why the third one is also a rectangle—indeed, a special kind of rectangle.

 Fig. 5.10. Examples of rectangles

p. 165 In grades 3–5, teachers should emphasize the development of mathematical arguments. As students' ideas about shapes evolve, they should » formulate conjectures about geometric properties and relationships. Using drawings, concrete materials, and geometry software to develop and test their ideas, they can articulate clear mathematical arguments about why geometric relationships are true. For example: "You can't possibly make a triangle with two right angles because if you start with one side of the triangle across the bottom, the other two sides go straight up. They're parallel, so they can't possibly ever meet, so you can't get it to be a triangle."

When students subdivide, combine, and transform shapes, they are investigating relationships among shapes. For example, a fourth-grade class might investigate the relationship between a rectangle and a nonrectangular parallelogram with equal bases and heights (see fig. 5.11) by asking, "Does one of these shapes have a larger area than the other?" One student might cut the region formed by the parallelogram as shown in figure 5.11 and then rearrange the pieces so that the parallelogram visually matches the rectangle. This work can lead to developing a general conjecture about the relationship between the areas of rectangles and parallelograms with the same base and height. The notion that shapes that look different can have equal areas is a powerful one that leads eventually to the development of general methods (formulas) for finding the area of a particular shape, such as a parallelogram. In this investigation, students are building their ideas about the properties of classes of shapes, formulating conjectures about geometric relationships, exploring how geometry and measurement are related, and investigating the shapes with equal area.

 Fig. 5.11. The relationship between the areas of a rectangle and a nonrectangular parallelogram with equal bases and heights

An understanding of congruence and similarity will develop as students explore shapes that in some way look alike. They should come to understand congruent shapes as those that exactly match and similar shapes as those that are related by "magnifying" or "shrinking." For example, consider the following problem involving the creation of shapes with a particular set of properties:

Make a triangle with one right angle and two sides of equal length. Can you make more than one triangle with this set of properties? If so, what is the relationship of the triangles to one another?

As students make triangles with the stipulated properties (see fig. 5.12), they will see that although these triangles share a common set of characteristics (one right angle and a pair of sides of equal length), they are not all the same size. However, they are all related in that they look alike; that is, one is just a smaller or larger version of the other. The triangles are similar. Although students will not develop a full understanding of similarity until the middle grades, when they focus on proportionality, in grades 3–5 they can begin to think about similarity in terms of figures that are related by the transformations of magnifying or shrinking.

 Fig. 5.12. Right triangles with two sides of equal length

p. 166

When discussing shapes, students in grades 3–5 should be expanding their mathematical vocabulary by hearing terms used repeatedly in context. As they describe shapes, they should hear, understand, and use mathematical terms such as parallel, perpendicular, face, edge, vertex, angle, trapezoid, prism, and so forth, to communicate geometric ideas with greater precision. For example, as students develop a more sophisticated understanding of how geometric shapes can be the same or different, the everyday meaning of same is no longer sufficient, and they begin to need words such as congruent and similar to explain their thinking. »

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

In grades 3–5, the ideas about location, direction, and distance that were introduced in prekindergarten through grade 2 can be developed further. For instance, students can give directions for moving from one location to another in their classroom, school, or neighborhood; use maps and grids; and learn to locate points, create paths, and measure distances within a coordinate system. Students can first navigate on grids by using landmarks. For example, the map in figure 5.13 can be used to explore questions like these: What is the shortest possible route from the school to the park along the streets (horizontal and vertical lines of the grid)? How do you know? Can there be several different "shortest paths," each of which is equal in length? If so, how many different "shortest paths" are there? What if you need to start at the school, go to the park to pick up your little sister, stop at the store, and visit the library—in what order should you visit these locations to minimize the distance traveled? In this activity, students are using grids and developing fundamental ideas and strategies for navigating them, an important component of discrete mathematics.

Students at this level also should learn how to use two numbers to name points on a coordinate grid and should realize that a pair of numbers corresponds to a particular point on the grid. Using coordinates, they can specify paths between locations and examine the symmetry, congruence, and similarity of shapes drawn on the grid. They can also explore methods for measuring the distance between locations on the grid. As students' ideas about the number system expand to include negative numbers, they can work in all four quadrants of the Cartesian plane.

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

p. 167

Students in grades 3–5 should consider three important kinds of transformations: reflections, translations, and rotations (flips, slides, and turns). Younger students generally "prove" (convince themselves) that two shapes are congruent by physically fitting one on top of the other, but students in grades 3–5 can develop greater precision as they describe the motions needed to show congruence ("turn it 90°" or "flip it vertically, then rotate it 180°"). They should also be able to visualize » what will happen when a shape is rotated or reflected and predict the result.

Students in grades 3–5 can explore shapes with more than one line of symmetry. For example:

In how many ways can you place a mirror on a square so that what you see in the mirror looks exactly like the original square? Is this true for all squares?

Can you make a quadrilateral with exactly two lines of symmetry? One line of symmetry? No lines of symmetry? If so, in each case, what kind of quadrilateral is it?

Although younger students often create figures with rotational symmetry with, for example, pattern blocks, they have difficulty describing the regularity they see. In grades 3–5, they should be using language about turns and angles to describe designs such as the one in figure 5.14: "If you turn it 180 degrees about the center, it's exactly the same" or "It would take six equal small turns to get back to where you started, but you can't tell where you started unless you mark it because it looks the same after each small turn."

 Fig. 5.14. Pattern with rotational symmetry

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

Students in grades 3–5 should examine the properties of two-and three-dimensional shapes and the relationships among shapes. They should be encouraged to reason about these properties by using spatial relationships. For instance, they might reason about the area of a triangle by visualizing its relationship to a corresponding rectangle or other corresponding parallelogram. In addition to studying physical models of these geometric shapes, they should also develop and use mental images. Students at this age are ready to mentally manipulate shapes, and they can benefit from experiences that challenge them and that can also be verified physically. For example, "Draw a star in the upper right-hand corner of a piece of paper. If you flip the paper horizontally and then turn it 180°, where will the star be?"

p. 168

Much of the work students do with three-dimensional shapes involves visualization. By representing three-dimensional shapes in two dimensions and constructing three-dimensional shapes from two-dimensional » representations, students learn about the characteristics of shapes. For example, in order to determine if the two-dimensional shape in figure 5.15 is a net that can be folded into a cube, students need to pay attention to the number, shape, and relative positions of its faces.

 Fig. 5.15. A task relating a two-dimensional shape to a three-dimensional shape

Students should become experienced in using a variety of representations for three-dimensional shapes, for example, making a freehand drawing of a cylinder or cone or constructing a building out of cubes from a set of views (i.e., front, top, and side) like those shown in figure 5.16.

 Fig. 5.16. Views of a three-dimensional object (Adapted from Battista and Clements 1995, p. 61)

Exploring Rectangles and Parallelograms

Technology affords additional opportunities for students to expand their spatial reasoning ability. Software such as Logo enables students to draw objects with specified attributes and to test and modify the results. Computer games such as Tetris (Pajitnov 1996) can help develop spatial orientation and eye-hand coordination. Dynamic geometry software provides an environment in which students can explore relationships and make and test conjectures.

Students should have the opportunity to apply geometric ideas and relationships to other areas of mathematics, to other disciplines, and to problems that arise from their everyday experiences. There are many ways to make these connections. For example, measurement and geometry are closely linked, as illustrated in the problem in figure 5.11, where geometric properties are used to relate the areas of two figures of different shapes. Geometric models are also important in investigating number relationships. Number lines, arrays, and many manipulatives used for modeling number concepts are geometric realizations of arithmetic relationships. In algebra, students in grades 3–5 often work with geometric problems to explore patterns and functions (see, for example, the "tower of cubes" problem in fig. 5.5).

In addition to its utility in exploring and understanding other areas of mathematics, geometry is closely associated with other subjects, such as art, science, and social studies. For example, students' work on symmetry can enhance their creation and appreciation of art, and their work on coordinate geometry is related to the maps they create or use in their study of the world. The study of geometry promotes a deeper understanding of many aspects of mathematics, improves students' abstract reasoning, and highlights relationships between mathematics and the sciences.