Departmental Course Outline for Geometry

Courses: 231 Lv2 Geometry, 232 Lv1 Geometry, 233 Hon Geometry

Note: This working document describes our coverage plans, but may be revised somewhat during the year. We hope you will find it useful as an overview of the course and as an approximation of what topics will be covered.

Minimum Topic Coverage

General remarks about the following outline:

The first two headings, Ways of Knowing and Basic Objects and Notions, describe general ideas to be covered during the course, not necessarily separate units of the curriculum.
Algebraic skills and concepts should be reviewed during the course as needed, and as appropriate for the course level. Relevant topics may include solving linear and quadratic equations, solving systems of equations, using ratios and proportions, and square roots and their properties.

Topic outline:

Ways of Knowing
- Exploring and conjecturing
  - using physical models
  - using Geometer's Sketchpad
  - inductive reasoning and generalizing
- Logic and reasoning
  - if... then... statements
  - converses, inverses, and contrapositives
  - deduction (modus ponens and modus tollens though not necessarily using those terms)
  - syllogisms
  - Venn diagrams
- Justification and proof
  - deductive proof within an axiomatic system
  - proofs by example or counterexample
  - indirect proof, use of the contrapositive
  - proofs using coordinates
  - (Optional) proofs using vectors
  - (Optional) proofs by mathematical induction
- Explanation and communication
  - written explanation
  - oral explanation
  - 2- and 3-dimensional drawing
  - diagrams and models
Basic Objects and Ideas
- Definitions of geometric objects
  - process of writing mathematical definitions
  - points, lines, planes, etc.
  - polygons and other shapes
  - polyhedra and other solids
- Concepts of measurement and dimensions
  - length, area, and volume
  - angle and arc measures
  - visualization in 2 and 3 dimensions
- Construction
  - compass and straight edge
  - using Geometer's Sketchpad
  - (Optional) using miras
Parallelism and Perpendicularity
- Euclid's parallel postulate
- Transversal theorems and their applications
  - proofs of angle sum for triangles and other polygons
  - parallelograms and their properties
- Parallel and perpendicular objects in 2-dimensional space
- (Optional for Lv2) Parallel and perpendicular objects in 3-dimensional space
Congruence
- Concept of congruence
- Congruence theorems
  - Triangle theorems: SSS, SAS, etc.
  - (Optional) Quadrilateral theorems
- Applications of congruence
  - corresponding parts
  - theorems about equilateral and isosceles triangles
Similarity
- Concept of similarity
- Similarity of triangles and other polygons
- Applications of similarity
  - proportions of lengths
  - scaling of areas and volumes
Transformations and Tessellations
- Transformations of 2-dimensional space
  - translations
  - reflections
  - rotations
  - glide reflections
  - dilations
- Symmetry: reflective, rotational, and glide symmetry
- Tessellations of 2-dimensional space
- (Optional) 3-dimensional transformations, symmetry, and space-filling patterns
Right Triangles
- Pythagorean Theorem
  - proofs of the theorem
  - use of the theorem
  - (Optional?) simplifying radicals
  - application problems
- Special triangles: 30-60-90 and 45-45-90
- Right triangle trigonometry
  - finding lengths using sin, cos, and tan
  - finding angles using inverse sin, cos, and tan
  - application problems
Coordinate Geometry
- Midpoints in 2-dimensional space
- Distance between points in 2-dimensional space
- (Optional) Midpoints and distances in 3-dimensional space
- Parallel and perpendicular slopes
- Proofs using coordinates
Circles
- Circumference and arcs
  - angle and arc measures
  - inscribed angles
  - calculating circumference and arc length
- perpendicular radii and tangents
- perpendicular radii and chords
- (Optional for Lv1 and Lv2) theorems about lengths of chords, tangents, and secants
Area and Volume
- Areas of planar shapes
  - triangles and quadrilaterals
  - regular polygons
  - circles and sectors
- Volumes of solids
  - Prisms and cylinders
  - Pyramids and cones
  - (Optional for Lv2) Spheres
  - (Optional) Polyhedra
- Surface areas of solids
  - Cubes and rectangular solids
  - Cylinders
  - (Optional for Lv2) Spheres
Other Polygon Topics
- Special segments of triangles
  - medians
  - altitudes
  - perpendicular bisectors
  - angle bisectors
- Triangle concurrency points and their properties
  - centroid
  - (Optional for Lv1 and Lv2) orthocenter, circumcenter, and incenter
  - (Optional for Lv1 and Lv2) Euler line
  - (Optional) Nine-point circle
- (Optional) Inequalities involving triangles
  - ("") Pythagorean inequalities
  - ("") Hinge theorems
- (Optional) Solving general triangles
  - ("") Law of Sines
  - ("") Law of Cosines
  - ("") Hero's formula
- Properties of quadrilaterals
  - Special types of quadrilaterals (rectangles, rhombuses, etc.)
  - (Optional) Classifying quadrilaterals by properties of diagonals

Technology Use

Geometer's Sketchpad exploratory software