224 Honors Geometry

The Course

Honors Geometry is a 1-credit course (one semester in the high school block) that serves as the third in a sequence of college preparatory mathematics courses. The key content for this third course includes the measurement of lengths, angles, areas and volumes and the geometry of lines, polygons and circles. Students should also become familiar with formal logic and mathematical proof, including both inductive and deductive reasoning.

Textbook

Schultz, et.al, Geometry (2003), Holt, Rinehart and Winston.

Prerequisites

Before studying geometry, all students must be competent in basic algebra, including solving both linear and quadratic equations and systems of linear equations. Students should also be competent in evaluating roots of real numbers and simplifying radical expressions.

The prerequisites for Honors Geometry can be satisfied by successfully completing Algebra II.

Philosophy

In Honors Geometry, broad concepts and widely applicable methods are emphasized. The focus of the course is neither manipulation nor memorization of an extensive taxonomy of computational methods, properties, theorems, or problem types. Although facility with manipulation and computational competence are important outcomes of the course, they are not the core of the course. Through the unifying themes of reasoning, measurement, and application, the course becomes a cohesive whole, rather than a collection of unrelated topics. These themes are developed using lines, segments, angles, polygons, circles and solids.

Formal logic and proof are central in all mathematics but are traditionally introduced in the geometry course, leading to the common misconception that proof is only necessary in geometry and must be accomplished in the common two-column format. The goal, however, is to lay the groundwork for logical argument throughout the students mathematical career. To this end, it is imperative that the student begins to view mathematics as an axiomatic science, one built on postulates (which are assumptions), definitions (which are choices) and theorems (which follow logically from the postulates and definitions). Thus, it is necessary to introduce the student to different forms of logical reasoning (both inductive and deductive), different forms of proof (both direct and indirect) and different formats for writing proofs (both two-column and paragraph), while continually emphasizing the reason for doing proofs.

The cohesive nature of the course must be emphasized through incremental development of the techniques and skills of geometry with continuous distributed review and frequent cumulative testing. Unifying themes must be revisited over and over again, with continual applications to increasingly complicated problems.

Objectives

1. Students generate and analyze patterns found in geometric situations. They draw diagrams, gather data, make conjectures, judge the validity of a logical argument and give counterexamples to disprove a statement.

2. Students are able to distinguish between postulates, definitions and theorems and can give examples of each. They write formal geometric proofs in both two-column and paragraph format, including proof by contradiction.

3. Students can measure the length of a line segment using a ruler, can measure an angle using a protractor and can construct line segments, angles and bisectors using a compass and a straight edge.

4. Students compute the perimeter and area of plane figures, the circumference and area of circles, and the volume and surface areas of solids. Students use comparison of lengths and areas to compute probabilities geometrically.

5. Students know the effect of rigid and non-rigid transformations on figures in the coordinate plane, including rotations, translations, reflections and dilations.

6. Students use the relationships between complementary, supplementary, vertical and exterior angles and use them to solve problems and prove theorems, including theorems involving parallel lines, perpendicular lines and parallel lines cut by a transversal.

7. Students classify polygons by their number of sides and know the properties of regular polygons. Students classify special quadrilaterals and know their properties and can prove theorems involving quadrilaterals.

8. Students prove theorems involving the congruence and similarity of triangles and use the concept of corresponding parts of both congruent and similar figures.

9. Students can classify triangles based on the measure of their angles and the lengths of their sides and can apply the isosceles triangle theorem and the triangle inequality theorem. They can find the lengths of the sides in a right triangle using the Pythagorean theorem and know the relationship between the sides and angles of both 30?-60?-90? and 45?-45?-90? triangles.

10. Students can identify the radius and diameter of a circle and can compute the measures of central angles, inscribed angles, intercepted arcs, the lengths of tangents, secants and chords and the area of sectors.

11. Students use algebra to solve problems in geometry, including solving linear equations and systems of linear equations and solving quadratic equations by factoring and by the quadratic formula.

12. Students can apply geometric concepts to solve problems in a variety of contexts, including construction, surveying, navigation and scale modeling.

Topical Outline

The outline of topics is intended to indicate the scope of the course, but not necessarily the order in which topics are taught. Although the Final Exam for Geometry will be based on the topics listed in this topical outline, teachers may enrich the course with additional topics.

I. Postulates, Definitions, Theorems and Proof

II. Measurement in Geometry

III. Congruency and Similarity

IV. Points, Lines, Planes and Angles

V. Triangles

VI. Polygons

VII. Circles

VIII. Surface Area and Volume