The reasoning skills that students develop in grades 35 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 35 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.
In the early grades, students will have classified and sorted geometric objects
such as triangles or cylinders by noting general characteristics. In grades
35, 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 rectangleindeed, a special kind of rectangle.
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.
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 35 they can begin to think about similarity in terms of figures that are related by the transformations of magnifying or shrinking.
When discussing shapes, students in grades 35 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. »
In grades 35, 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 libraryin 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 in grades 35 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 35 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 35 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 35, 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."
Students in grades 35 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
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.
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 35 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.
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