Download 9-12 Intro Geometry

GEOMETRY
INTRODUCTION
California Mathematics Framework
The main purpose of the geometry curriculum is to develop geometric skills and concepts as
well as 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 to place 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. This includes giving
counter examples to disprove a statement.
Starting with undefined terms and axioms, students learn to establish the validity of other
assertions by logical deductions, i.e., 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 and not
the literary polish of the writing or the adherence to a particular proof format.
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 as well as amass a wealth of geometric data in the process. Many students come out of a
course in geometry---including many high achieving ones---with so little geometric intuition that,
given three non-collinear points, they cannot visualize in the crudest way what their circumcircle
must be like. One way to develop this geometric sense is to become familiar with the basic
straightedge-compass constructions, as illustrated in the standard:
16.0. Students perform basic constructions with 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.
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.
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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 school geometry texts to start with axioms that incorporate the real
numbers. Although it runs counter to the spirit of Euclid to do geometric proofs with real
numbers, it 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 at most one line parallel to L.
Because of the fundamental role it plays in the development of mathematics up to the nineteenth
century, the significance of this postulate should be discussed. Also, 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. This gives a natural context to show students the key concept of
uniqueness in mathematics. Experience indicates that students usually find this concept difficult.
It is also recommended that the topics of circles and similarity be taught as early as possible.
Once they are available, the course enters a new phase not only because of the interesting
theorems that can now be proved, but also because similarity opens up the applications of
algebra to geometry. These could include the determination of one side of a regular decagon on
the unit circle by use of the quadratic formula as well as the applications of geometry to practical
problems.
It is usually not realized that circles can be introduced very early. 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:
(i) base angles of isosceles triangles are equal, and
(ii) exterior angle of a triangle equals the sum of opposite interior angles.
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 the use of paragraph proofs is the expense of typesetting more elaborate
formats, not that they are intrinsically better or clearer. In point of fact, neither of these claims of
superiority for paragraph proofs is actually valid. Furthermore, it appears that in order for
beginners to learn the precision of argument needed, the two-column format is best. After
students have shown a mastery of the basic logical skills, then 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.
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One of the high points of elementary mathematics, in fact of all of mathematics, is the
Pythagorean theorem:
14. Students prove the Pythagorean Theorem.
It 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, (see standards 8 below), students can
prove the Pythagorean theorem in at least two more ways using the familiar picture of four
congruent right triangles with legs a and b nestled inside a square of side a + b.
8.
Students know, derive, and solve problems involving the perimeter, circumference,
area, volume, lateral area, and surface area of common geometric figures.
10. 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 a particularly simple one since
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 in an
intuitive manner without proofs. If the area of disk and length of circle are
defined as the limit of exhausting by inscribing or circumscribing regular polygons, then p is
either the area of the unit disk or the ratio of circumference to diameter, and heuristic arguments
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 rectilinear figures
(polyhedra) and should be touched on only lightly and entirely intuitively.
8.
Students know, derive, and solve problems involving perimeter, circumference,
area, volume, lateral area, and surface area of common geometric figures.
9.
Students compute the volumes and surface areas of prisms, pyramids, cylinders,
cones, and spheres.
An important aspect of teaching 3-dimensional geometry is to cultivate students’ spatial intuition.
Although most students find spatial visualization difficult, this 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 via the introduction of trigonometric functions.
18.
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, (e.g., tan (x) = sin
(x)/cos (x), (sin2 (x) + cos2 (x) = 1).
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19.
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.
The discussion of the trigonometric functions can be brief because they will be scrutinized at
length in more advanced courses, but students should at least get used to the sine, cosine, and
tangent functions. Here, the emphasis should be on the simple applications of these functions to
practical problems and to geometry itself. For example, finding the
height of a mountain by using the angle of inclination, or yielding a new formula for the area of a
triangle in terms of two sides and the sine of their included angle.
Finally, the Pythagorean theorem leads naturally to the introduction of rectangular coordinates
and coordinate geometry in general. A significant portion of the curriculum could be devoted to
the teaching of topics embodied in the next two standards.
17.
22.
Students prove theorems by using coordinate geometry, including the midpoints of a
line segment, distance formula, and various forms of equations of lines and circles.
Students should know the effect of rigid motion on figures in the coordinate plane
and space, including rotations, translations, and reflections.
These standards lead students to the next level of sophistication: an algebraic and
transformation-oriented approach to geometry. Students begin to see how algebraic concepts
add a new dimension to the understanding of geometry, and conversely, how geometry gives
substance to algebra. Thus straight lines are no longer merely a simple geometric object; they
are also the graphs of linear equations. Conversely, solving simultaneous linear equations now
becomes finding the point of intersection of straight lines. Another example is the interpretation
of the geometric concept of congruence in the Euclidean plane as a correspondence under an
isometry of the coordinate plane. Concrete examples of isometries are studied: rotations,
reflections and translations. It is strongly suggested that the discussion be rounded off with the
proof of the structure theorem: every isometry of the coordinate plane is a translation, or the
composition of a translation and a rotation or the composition of a translation, a rotation, and a
reflection.
Special attention should be given to the fact that a gap left open in Algebra I must be filled here.
Standards 7.0 and 8.0 of Algebra I assert that
(1) the concept of slope of a straight line makes sense,
(2) the graph of a linear equation is a straight line, and
(3) two straight lines are perpendicular if and only if their slopes have product -1.
These facts should now be proved.
Additional Comments and Cautionary Notes.
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(A) An important point to make to students concerning proofs is that while the written proofs
presented in class should serve as models for exposition, they should in no way be a model of
how proofs are discovered. The perfection of the finished product can easily mislead students
into thinking that they must likewise arrive at their proofs with the same
apparent ease. Teachers need to make clear to their students that the actual thought process is
usually full of false starts and many zigzags between promising leads and dead ends. Only trials
and errors can lead to a correct proof.
This awareness of the nature of doing mathematics could lead to a de-emphasis of the rigid
requirements on the writing of two-column proofs in some classrooms..
(B) The first part of the course sets the tone of students' perception of proofs. With this in mind,
it is advisable to discuss, mostly without proofs, those first consequences of the axioms that are
needed for later work. A few proofs should be given, naturally, for illustrative purposes. For
example, the equality of vertical angles, or the equality of base
angles of an isosceles triangle and its converse. There are two reasons for this recommendation.
The foremost is that a complete logical development is neither possible nor desirable. This has
to do with the intrinsic complexity of the structure of Euclidean geometry (cf. M.J. Greenberg,
Euclidean and Non-Euclidean Geometries, 3rd ed., W.H.
Freeman, 1993, pp. 1--146). A second reason is the usual misconception that such elementary
proofs are easy for beginners. Working on the level of axioms is actually more difficult for
beginners than working with theorems that come a little later in the logical development. This is
because, on the one hand, working with axioms requires a heavy
reliance on formal logic without recourse to intuition---in fact often in spite of one's intuition. On
the other hand, doing spade work on the level of axioms does not usually have a clear direction
or goal, and it is difficult to convince students to learn something without a clearly stated goal. If
one so desires, students can always be made to go back to prove the elementary theorems after
they have already developed a firm grasp of proof techniques.
(C) Students’ first attempts at proofs need to be structured with care. At the beginning of the
development of this skill it might be good strategy, instead of asking students to do many trivial
proofs, after showing them the proofs of two or three easy theorems, to proceed as follows:
a) students could be shown as early as possible a generous number of proofs of
substantive theorems so that they can get a well-rounded idea of what a proof is about
before they write any proofs themselves;
b) as a prelude to constructing proofs themselves, students could be to supply reasons for
some of the steps in these sample (substantive) proofs, instead of constructing extremely
easy proofs on their own, and
c) after an extended exposure to non-trivial proofs students could then be asked to give
proofs of simple corollaries of substantive theorems.
The reason for (b) and (c) is to make students associate " proofs" with real mathematics rather
than a formal ritual from the beginning. This goal can be accomplished with the use of local
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axiomatics, i.e., before proving a theorem, state clearly the facts needed for the proof. These
facts need not be previously proven ones, but should ideally be sufficiently
plausible even without a proof. Extensive use of local axiomatics would make possible the
presentation of interesting but perhaps advanced theorems sufficiently early in the course.
In the Appendix " Circumcenter, orthocenter, and centroid", the ideas in (B) and (C) are put to
use as a demonstration how they might work.
(D) For the purpose of developing students' geometric intuition, the following geometric
constructions are recommended ("construction" means "construction with straightedge and
compass"). It is understood that all of them will be proved at sometime during the course of
study.
∗ bisecting an angle
∗ constructing the perpendicular bisector of a line segment
∗ constructing the perpendicular to a line from a point on the line, and from a point not on
the line
∗ duplicating a given angle
∗ constructing the parallel to a line through a point not on the line
∗ constructing the circumcircle of a triangle
∗ dividing a line segment into n equal parts
∗ constructing the tangent to a circle from a point on the circle
∗ constructing the tangents to a circle from a point not on the circle
∗ locating the center of a given circle
∗ constructing a regular n-gon on a given circle for n=3, 5, 6
(E) This is the place to add a word about the use of technology. The availability of good
computer software makes the accurate drawing of geometric figures almost a pleasure. Such
software can enhance the experience of making the drawings in the constructions above. In
addition, the ease of making accurate drawings encourages the formulation and
exploration of geometric conjectures. For example, it is now easy to convince oneself that the
intersections of adjacent angle trisectors of the angles of a triangle are likely the vertices of an
equilateral triangle (Morley’s theorem). If students do have access to such software, the
potential for a more intense mathematical encounter is certainly there. In encouraging them to
make use of the technology, however, one should not lose sight of the fact that the excellent
visual evidence thus provided must never be taken as a replacement for understanding. For
instance, software may give the following heuristic evidence for why the angle sum of a triangle
is 180° . Click any three points on the screen and a triangle with these three points as vertices
appears. Click each angle again and there would be three numbers that give the angle
measurement of each angle. Add these number and 180 would be the answer. Furthermore, no
matter the shape of the triangle, the result would always be the same.
Now this may be taken as a big psychological boost for one’s belief in the validity of the
theorem about the angle sum, and if students find this kind of evidence helpful, so much the
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better. But it must be recognized that these angle measurements have added nothing to one’s
understanding of why this theorem is true. Furthermore, if one really wants to have a hands-on
experience with angle measurements in order to check the validity of this theorem, the best way
is to do it painstakingly by hand on paper. Morley’s theorem mentioned above is another
illustration of the same principle: evidence can not replace proofs, and the computer program
would not reveal the reason why the three points are always the vertices of an equilateral
triangle.
(F) Students should know that the coordinate plane provides a concrete example that satisfies
all the axioms of Euclidean geometry if lines are defined as the graphs of linear equations
ax+by=c with at least one of a and b not equal to zero. Lines a1 x + b1 y =c1 and a2 x + b2 y
=c2 are defined to be parallel if (a1, b1) is proportional to (a2, b2) but (a1, b1, c1) is not
proportional to (a2, b2, c2). The verification of the axioms is straight forward.
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