Geometry was derived from real world measurements of lines, planes and solids. These developed into concepts that were idealized and defined. A systematic logical approach was then made with the relations these idealized figures have with themselves. Therefore, it is always useful to start with tangible figures and intuitively develop definitions and agreed upon propositions as a basis for study in geometry.

1. Along with numbers, geometry uniquely connects mathematics with the physical world.

2. Geometry uniquely enables ideas from other areas of mathematics to be pictured.

3. Geometry non-uniquely provides an example of a mathematical system.

The direct connection that geometry makes with the physical world takes the form of shapes for buildings, city layouts, and construction of many types. It answers questions like, "How far?", "How big?" or "How long?". Areas, perimeters, volumes and the Pythagorean relations are examples of the usefulness of geometry. Also analysis and classifications of shapes and relationships between figures using congruence or similarity are useful ideas explored in geometry. Geometry also can be used to picture algebraic ideas. Using coordinate geometry, graphs of lines and curves can be generated. Sine, cosine and tangent curves can be pictured. The derivative of a function as a tangent to a curve at a point on that curve, and statistics using bar and circle graphs and curve fitting are uses of geometry. Despite these unique and direct aspects for studying geometry, the non-unique aspect of geometry as a mathematical system, historically has been the most influential focus contained in the content of the geometry course. The emphasis on proofs must be there but not to the exclusion of the other aspects of geometry. In order to do that:

1. Treat obvious statements informally and not as rigorous proofs.

2. Shorten the prolonged periods for proofs using the same two column format.

3. Include topics from coordinate geometry, and transformational geometry and simple uses of statistics.

Geometry has traditionally been taught between Algebra I and Algebra II. Flexibility should be kept so that geometry could be taught after Algebra II.

Critical Components

In order to satisfy the graduation requirement, geometry should deal significantly with:

I.    Geometry as a logical System

A.    Goal:

Students will understand and approach geometric problems using measurement.

Students will

1. Use measurement to derive definitions and assumptions.

2. Use the deductive system to prove basic (important) theorems.

3. Use other than formal methods of demonstration for non-essential problems and theorems.

II. Problem Solving

A.    Goal:

Students will understand problem solving and select strategies generally used in geometry.

Students will

1. Learn and apply the four step problem solving procedure of

a. Identifying and analyzing the problem.
b. Formulating a plan to solve the problem.
c. Solving the problem.
d. looking back for patterns that can be useful for solving other problems.

2. Learn and use the following problem solving strategies

a. Drawing a picture or diagram.
b. Solving part of the problem.
c. Looking for a pattern.
d. Work backwards from conclusion to condition.

III. Lines and Angles

A.    Goal:

Students will understand and demonstrate proficiency in the geometry of lines.

Students will

1. Define and use

a. Parallel and perpendicular lines, transversals and skew-lines.
b. Parts of lines (points, segments, rays).
c. Angles, angle measurement and classification of angles.

IV. Triangles

A Goal:

Students will understand triangles and some of their applications.

Students will

1. Define and use

a. Congruence of triangles.
b. Calculation and application of area of triangles.
c. Similarity and proportionality of triangles.
d. The Pythagorean theorem and apply this principle to meaningful problems.
e. Elementary right triangle trigonometry functions (sine, cosine, tangent) and apply to meaningful problems.

V. Geometric Constructions

A. Goal

Students will understand how to construct certain geometric figures.

Students will

1. Explain methods and construct

a. line segments, angles, triangles
b. The bisection of segments and angles, subdivision of a line into "N" equal segments.
c. Perpendiculars, parallels and simple polygons.

2. Use the concept of locus of points to define a curve.

VI. Polygons

A. Goal:

Students will understand parts and uses of polygons.

Students will

1. Name the components of polygons and their properties.

2. Compute areas of polygons.

3. Measure interior and exterior angles of polygons.

VII. Circles, Arcs

A. Goal:

Students will understand the properties of the circle and arc of a circle.

Students will

1. Define properties and relationships involving circles and arcs or circles and interior angles.

2. Calculate and use areas of circles.

3. Develop equations of circles.

VIII. Coordinate Geometry

A Goal:

Students will understand the use of geometry to picture functions.

Students will

1. Use the coordinate system to define graph functions.

2. Define slope of the line, parallel and perpendicular lines.

3. Relate linear equations to specific lines and vice versa.

IX. Transformations

A. Goal:

Students will understand the unifying nature of transformations concerning congruence, symmetry and similarity in conclusions and deductions about geometry.

Students will

1. Define and demonstrate reflection, rotation, translation and dilation as moves in showing congruence and similarity of geometric figures.

X. Solid Geometry

A. Goal:

Students will understand introductory ideas of three-dimensional geometry.

Students will

1. Study and use ideas from

a. Families of polyhedra: Prisms, pyramids, and regular polyhedra.
b. Cones and cylinders.
c. Geometry of spheres: Areas of great circles, volume of spheres.
d. Perspectives and cross sections.
e. Drawing three-dimensional shapes.
f. Surface areas and volumes of solids.
g. The proportionality of lengths, areas, and volumes.

return_arrow.gif (322 bytes)

back_arrow.gif (305 bytes)

next_arrow.gif (304 bytes)

return to courses of study index

previous course

next course