Geometry 1B is a one semester course that builds off of definition-based, informal understandings of geometric concepts learned the previous semester (in Geometry 1A) to develop direct proof-writing skills, mathematical communication skills (oral, English and symbolic), and patient, creative problem solving. This course is designed with three different class structures to emulate several facets of the creative, investigative and protracted work of professional mathematicians: investigation of definitions and proving thematic theorems as in-class group work, presentations of individually-completed homework assignments that apply thematic theorems followed by student-centered peer review, and project work-shopping that asks students to extend their knowledge to very demanding and individualized proofs in fluid groups. The primary initiative of this student-centered course is to increase student persistence, confidence, creativity, and autonomy in solving problems that are predominantly non-procedural as well as in critiquing the work of other students.
-Basic properties and postulates (reflexive, symmetric, and transitive properties; multiplication and division postulates, etc.)
-Using addition, subtraction, reflexive, substitution and partition postulates to prove line segments and angles congruent
-Definitions of right, vertical, supplementary and complementary angles and related theorems
-Triangle Congruency Postulates (SSS, SAS, ASA)
-Corresponding Parts of Congruent Triangles are Congruent
-Definition of parallel lines, relationships between angles formed when parallel lines are crossed by a transversal and related theorems
-Definitions of isosceles and equilateral triangles and related theorems
-Definition of median, angle bisector and altitude
-Definition of perpendicular bisector and related theorems
Semester-Long Process and Proof Topics...
-Understanding theorems and definitions as conditional and biconditional statements
-Using a problem-solving process of analyzing given information, making observations, making conjectures, brainstorming, planning, proving and revising
-Progressive increase in student autonomy and difficulty of problems assigned relative to student skills at that time as semester develops
-Giving clear explanations of work and criticism of the work of others
-Progressive increase in comfort level with flawed or incomplete 'early' ideas and errors, understanding them as a natural part of the problem solving process, progressive increase in comfort level with revision and critical feedback from other students and the teacher
Project 1: Triangle Congruency
Students receive a personal collection of three triangle congruency statements to prove with accompanied diagrams and given information. In proving these three statements, students demonstrate mastery of the use of triangle congruency postulates (SSS, SAS and ASA), basic properties and postulates, angle definitions and theorems and adding and subtracting angles and line segments. Students are required to give written evidence of making initial observations based on the assigned givens and conjectures, brainstorming, planning a proof, writing it, and revising it. Students present the rough drafts of their proofs for critique from the class. Then they write a process paper detailing their strategies for proving their triangle congruency statements and how they arrived at these strategies, explaining observations, flawed strategies, revisions and lines of reasoning as in depth as possible.
Project 2: Using Triangle Congruency to Prove Other Conjectures
Students receive a personal collection of three statements to prove, as in the first project. Although the predominant skill to demonstrate mastery of in this project is use of the 'Corresponding Parts of Congruent Triangles are Congruent' theorem, students must devise a wide range of strategies with greater complexity and appeal to a larger selection of thematic theorems to prove their conjectures than in the first project. Students also complete a rough draft presentation with a peer review and then a process paper as in project 1.
Each student is assigned a biweekly presentation of a challenging/ novel proof from their homework for peer review. Classmates give warm and cool feedback and offer ideas to improve the proof/ delineate issues with its validity. There is at least one student presentation almost every day of class. This is intended as a catalyst for class discussion around main topics, a way to clear up common errors, and to give importance to individual assignments completed at home.
PBAT: Prove Your Own Conjecture
Students are given a personal diagram from which they make observations and formulate conjectures. Students endeavor to prove one or more conjectures that "show off" the full range of thematic theorems that they learned during the course and a level of proof complexity, sophistication, and creativity that is their very best work.