5.8. GjeometriaGrade 8-12
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The geometry skills and concepts developed in this discipline are useful to all students. Aside from learning these skills and concepts, students will develop their ability to construct formal, logical arguments and proofs in geometric settings and problems. GEO.1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
GEO.2.0 Students write geometric proofs, including proofs by contradiction.
GEO.4.0 Students prove basic theorems involving congruence and similarity.
GEO.6.0 Students know and are able to use the triangle inequality theorem.
GEO.8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures. GEO.9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders. GEO.10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
GEO.11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
GEO.13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
GEO.14.0 Students prove the Pythagorean theorem.
GEO.16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. GEO.17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles. GEO.18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan( x ) = sin( x )/cos( x ), (sin( x )) 2 + (cos( x )) 2 = 1. GEO.19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side. GEO.20.0 Students know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles. GEO.21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. GEO.22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections. |
Comments for Basic Skills of Geometry
The main purpose of the geometry curriculum is to develop geometric skills and concepts and the ability to construct formal logical arguments and proofs in a geometric setting. Although the curriculum is weighted heavily in favor of plane (synthetic) Euclidean geometry, there is room for placing special emphasis on coordinated geometry and its transformations. The first standards introduce students to the basic nature of logical reasoning in mathematics:
1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Starting with undefined terms and axioms, students learn to establish the validity of other assertions through logical deductions; that is, they learn to prove theorems. This is their first encounter with an axiomatic system, and experience shows that they do not easily adjust to the demand of total precision needed for the task. In general, it is important to impress on students from the beginning that the main point of a proof is the mathematical correctness of the argument, not the literary polish of the writing or the adherence to a particular proof format.
Inductive Reasoning Standard 1.0 also calls for an understanding of inductive reasoning. Students are expected not only to recognize inductive reasoning in a formal sense but also to demonstrate how to put it to use. To this end students should be encouraged to draw many pictures to develop a geometric sense and to amass a wealth of geometric data in the process. Many students—including high-achieving ones—complete a course in geometry with so little geometric intuition that, given three non-collinear points, they cannot even begin to visualize what the circumcircle of these points must be like. One way to develop this geometric sense is to have the students become familiar with the basic straightedge-compass constructions, as illustrated in the following standard:
16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line. It would be desirable to introduce students to these constructions early in the course and leave the proofs of their validity to the appropriate place of the logical development later.
Geometric Proofs The subject then turns to geometric proofs in earnest. The foundational results of plane geometry are embodied in the following standards:
2.0 Students write geometric proofs, including proofs by contradiction.
4.0 Students prove basic theorems involving congruence and similarity.
7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
It has become customary in high school geometry textbooks to start with axioms that incorporate real numbers. Although doing geometric proofs with real numbers runs counter to the spirit of Euclid, this approach is a good mathematical compromise in the context of school mathematics. However, the parallel postulate occupies a special place in geometry and should be clearly stated in the traditional form: Through a point not on a given line L, there is exactly one line parallel to L. Because this postulate played a fundamental role in the development of mathematics up to the nineteenth century, the significance of the postulate should be discussed. And because there always exists at least one parallel line through a point to a given line, the import of this postulate lies in the uniqueness of the parallel line. A discussion of this postulate provides a natural context to show students the key concept of uniqueness in mathematics—a concept that experience indicates students usually find difficult. It is also recommended that the topics of circles and similarity be taught as early as possible. Once those topics have been presented, the course enters a new phase not only because of the interesting theorems that can now be proved but also because the concept of similarity expands the applications of algebra to geometry. These applications might include determining one side of a regular decagon on the unit circle through the use of the quadratic formula as well as the applications of geometry to practical problems. It is often not realized that theorems for circles can be introduced very early in a geometry course. For instance, the remarkable theorem that inscribed angles on a circle which intercept equal arcs must be equal can in fact be presented within three weeks after the introduction of axioms. All it takes is to prove the following two theorems:
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1. Base angles of isosceles triangles are equal.
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At this point it is necessary to deal with one of the controversies in mathematics education concerning the format of proofs. It has been argued that the traditional two-column format is stultifying for students and that the format for proofs in the mathematics literature is always paragraph proofs. While the latter observation is true, teachers should be aware that a large part of the reason for using paragraph proofs is the expense of typesetting more elaborate formats, not that paragraph proofs are intrinsically better or clearer. In fact, neither of these claims of superiority for paragraph proofs is actually valid. Furthermore, it appears that for beginners to learn the precision of argument needed, the two-column format is best. After the students have shown a mastery of the basic logical skills, it would be appropriate to relax the requirements on form. But the teacher should never relax the requirement that all arguments presented by the students be precise and correct.
Pythagorean Theorem One of the high points of elementary mathematics, in fact of all of mathematics, is the Pythagorean theorem:
14.0 Students prove the Pythagorean theorem. This theorem can be proved initially by using similar triangles formed by the altitude on the hypotenuse of a right triangle. Once the concept of area is introduced (Standard 8.0), students can prove the Pythagorean theorem in at least two more ways by using the familiar picture of four congruent right triangles with legs a and b nestled inside a square of side a + b.
8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
For rectilinear figures in the plane, the concept of area is simple because everything reduces to a union of triangles. However, the course must deal with circles, and here limits must be used and the number p defined. The concept of limit can be employed intuitively without proofs. If the area or length of a circle is defined as the limit of approximating, inscribing, or circumscribing regular polygons, then p is either the area of a disk of unit radius or the ratio of circumference to diameter, and heuristic arguments (see the glossary) for the equivalence of these two definitions would be given. The concept of volume, in contrast with that of area, is not simple even for polyhedra and should be touched on only lightly and intuitively. However, the formulas for volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres (Standard 9.0) should be memorized. An important aspect of teaching three-dimensional geometry is to cultivate students’ spatial intuition. Most students find spatial visualization difficult, which is all the more reason to make the teaching of this topic a high priority. The basic mensuration formulas for area and volume are among the main applications of geometry. However, the Pythagorean theorem and the concept of similarity give rise to even more applications through the introduction of trigonometric functions. The basic trigonometric functions in the following standards should be presented in a geometry course:
18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan (x) = sin (x)/cos (x), (sin (x))2 + (cos (x))2 = 1.
19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
Finally, the Pythagorean theorem leads naturally to the introduction of rectangular coordinates and coordinate geometry in general. A significant portion of the curriculum can be devoted to the teaching of topics embodied in the next two standards:
17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
