​                           Notes for Teachers 

                                        

These notes provide an overview of units to help teachers determine the best way to use Geometry for School and Home with their students.

 
General:  Most students have some experience with geometry prior to their present class  However, some may be learning concepts for the first time, and many can benefit from a review (even where a school curriculum might assume otherwise).  This is particularly true for the content of the first two units.

 

Unit 1:  Basic geometry concepts (point, line, and plane), related definitions and postulates (including intersection, collinear, coplanar, parallel and skew, transformations, line segments, congruent line segments, distance, and midpoints)  These topics are presented in a simple logical order: points and lines; planes (with relation to points and lines); points and lines in the coordinate plane; line segments; distance and length; and midpoints.  (Rays are not introduced until Unit 2.)  Constructions with line segments and midpoints are used to reinforce concepts, but are not critical to the development of the unit.  Of particular note is the treatment of distance and measurement.  The ability to measure is too often assumed of students, but that is not always the case.  Lack of skill in measuring can hinder a student throughout the course since the understanding of many geometric relationships depends on seeing relationships of measured elements.

 

Unit 2:   Topics related to angles including definitions such as perpendicular, angle bisectors and many types of angle pairs  As in Unit 1, measurement is essential in learning relationships.  The ability of students to measure angles is not assumed and several assignments develop the concept of measurement and the use of a protractor.  Angle related topics are presented in a simple logical order: naming rays and angles; angle congruence; adjacent angles and angle measure addition; and angle bisectors. Special angle pairs are introduced with emphasis on identification, not the relationship of the measures of those angles (Theorems that relate angle pairs are not developed until Unit 4 where inductive and deductive reasoning are introduced and applied.). Postulates about parallel and perpendicular lines are developed.  As in Unit 1, constructions (congruent angles, angle bisectors, perpendicular lines and parallel lines) are excellent ways to reinforce but are not critical to the development of the concepts in the unit.

 

Unit 3:  Polygons: definitions and classifications; transformations (including transformation functions in the coordinate plane); and symmetry  Polygon definitions and general classifications are introduced. Polygons are then the platfrom for transformational geometry as well as symmetry (since polygons can easily be cut out or traced, and then moved, flipped, or turned).  The remainder of the unit emphasizes the relationship among sets and subsets of polygons and being able to name a polygon in the most descriptive way. The unit also includes many polygon-related concepts: diagonals, medians, altitudes, points of concurrency, and midsegments.  Constructions (regular polygons, medians, altitudes) are available as reinforcement.

 

Unit 4:  Types of reasoning and relationships involving lengths and angle measures.  The unit begins with the introduction of informal Inductive and deductive reasoning. Inductive reasoning is used to develop relationships in angle pairs (linear pairs, vertical angles, alternate interior and corresponding angles with parallel lines); angle measure and side length equality and inequality in triangles; sums of angle measures in triangles and other polygons (including exterior angles); lengths of midsegments in triangles and trapezoids; lengths of sides and diagonals in quadrilaterals.  

 

Unit 5: Congruent triangles, congruence postulates, writing formal congruent triangles proofs using three forms (flow chart, two column, paragraph) The unit consists of three parts: an introduction to congruent triangles with the development of postulates for congruence in triangles (SSS, SAS, ASA, AAS, HL); logic (arguments, conditional statements, converse, inverse, and contrapositive, syllogisms, Law of Detachment); and writing proofs of congruent triangles in three forms: flow charts, two column (T-charts), and paragraph. Proof writing is developed gradually in six steps:

1. The concepts of Given and Prove are introduced and related to the roles of premises and conclusions in the Law of Detachment. Two forms (flow-charts and two column) are used.

2. The first congruent triangle proofs have specifically stated side and angle congruences.  Students learn how to mark the congruences on the triangles and write simple (4 statement) proofs based on the given congruent pairs and the triangle congruence postulates.

3. Proofs requiring congruences that can be determined from the given figures (reflexive property and vertical angles) are introduced.  Students mark all congruences (those that are given as well as those that can be concluded from the figure) and write both flow-chart and two column (T) proofs using the congruence postulates.

4. Some basic definitions (isosceles, angle bisector, midpoint, and perpendicular lines) are restated.  Students complete and write proofs that involve a conclusion based on one of the stated definitions. 

5. Parallel line and angle relationships (corresponding and alternate interior angles) are restated.  Students complete and write proofs that involve a conclusion  using a parallel lines theorem.

6. CPCTC is introduced.  Students prove segments or angles congruent by first proving triangles congruent and then using CPCTC.

7. The remainder of the unit involves applying CPCTC, paragraph proofs, indirect proof, and proving theorems using congruent triangles and CPCTC.

 

Unit 6:  Proportions in geometry, triangle proportion theorems, dilations (including dilations on the coordinate plane), similarity of polygons, triangle similarity postulates and proof, indirect measurement, the geometric mean in right triangles   The first two assignments review ratio and proportions concepts (writing and simplifying ratios, cross-products, and solving proportions) through problems that are based on geometric figures.  Next, geometric proportion theorems are developed through measurement. The remainder of the unit is about similarity and its various applications.  Similarity is introduced through dilations. Students compare the relationship between a dilation, angle congruence, and proportionality with a formal definition of similarity.  The definition is reinforced through problems to name congruent angles and calculate lengths of sides using proportions. The triangle similarity postulates are developed intuitively and used for problems and proofs.  Particular emphasis is on the AA postulate used for indirect measurement.  

 

Unit 7:  Geometry of right triangles, the Pythagorean Theorem and trigonometry   The Pythagorean Theorem is introduced through measurement of several right triangles and is proved using the geometric mean in right triangles.  Answers in assignments evolve from whole numbers to approximations to radicals (and simplifying). The remainder of the first part of the unit includes the Pythagorean converse and its use in classifying triangles, the application of the theorem in finding lengths in triangles and rectangles; and the development of the special right triangle (isosceles and 30-60) theorems.  The second part of the unit builds trigonometry concepts from basic vocabulary (hypotenuse, opposite leg, adjacernt leg) through using the sin, cos, and tan ratios to solve for side lengths and angle measures (using inverses) in right triangles and concludes with applications and the Law of Sines.

 

Unit 8:  Area of polygons and use of formulas; introduction to polyhedrons and their surface area; volume of prisms and pyramids   Students usually but not always have some knowledge of area formulas.  This unit emphasizes the development of the formulas.  Area is introduced by counting square units and approximating areas using the one-half rule (a concept that might be new to all).  The area formulas evolve from comparing the results by counting with calculations using measurement of dimensions (rectangles, squares, parallelogram) and from figures being parts (e.g. triangle as half of a parallelogram) or composites (e.g. trapezoid as two triangles) of others.  The remainder of the unit is about the relationship of area to other concepts: area and the Pythagorean Theorem; area of special triangles, ratio of areas of similar polygons; geometric probability; relation between area and perimeter.  The last part of the unit introduces polyhedrons, surface area, and volume. 

 

Unit 9:  The Distance Formula; using the Distance, Slope, and Midpoint Formulas to classify and determine relationships in triangles and quadrilaterals; algebraic proofs   The Distance Formula is introduced as a consequence of the Pythagorean Theorem and is used to find the distance between points.  In subsequent assignments, the emphasis is on using the Distance Formula as well as slopes and midpoints to determine relationships in triangles and quadrilaterals: the Distance Formula to show congruence of sides and diagonals; slope to show parallel and perpendicular sides and perpendicular diagonals; midpoints to show bisecting. The last part of the unit is a gradual introduction to the use of algebra: writing and identifying algebraic coordinates; finding slope, distance, and midpoints from algebraic coordinates; completing algebraic proofs.

 

Unit 10: Definitions related to circles; relationships among lines, segments, angles, and arcs in circles; pi and circumference and area; surface area and volume of cylinders, cones, and spheres  Basic circle definitions and relationships in circles are introduced including chords and distances from the center of a circle, angles, arcs and arc measurement, tangents, and secants. Inductive reasoning is used to develop rules for angles and intercepted arc measure relationships. The second part of the unit begins with an activity to develop an understanding of pi and then covers formulas for circumference (and arc lengths), area (circles, sectors, segments, and composite figures), surface area and volume (cylinders, cones, and spheres).  The last part of the unit develops the relations of the lengths of segments of chords, secants, and tangents.