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EOC Navigation Geometry EOC: The Proof Is In The Put-in Jacqueline Hailey Erik Hailey CAMT July 19, 2012 JHAILEY/GPISD 2 Logic Proof JHAILEY/GPISD 3 Pre AP High School Mathematics with Geometry Focus (p 2) • The rule of four (analytical, numerical, graphical, verbal) limits, sequences, rate of change, functions, area under a curve, variation, trigonometry, geometric means, construction, areas of plane figures, areas and volumes of solids, coordinate geometry, and transformations . • Using manipulatives to develop geometric concepts before rigorous application . • Assessment in geometry. Connecting geometric proof with the Pre-AP idea of "justifying your answer". JHAILEY/GPISD 4 JHAILEY/GPISD 5 STAAR: Geometry EOC FORMAT Reporting Categories Reporting Category 1: Geometric Structure Reporting Category 2: Geometric Patterns and Representations Number of Standards Number of Questions Readiness Standards G.2B, 3C Supporting Standards G.1B, 1C, 2A, 3A, 3B, 3D, 3E Total Readiness Standards G.5A, 5D 2 Supporting Standards G.4A, 5B, 5C 3 Total 5 7 9 2 10/52 10/47 19-21% Not tested: G.1A: Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. JHAILEY/GPISD 6 Italicized TEKS were not tested on TAKS. G.1 Geometric structure: Basic Elements. The student understands the structure of, and relationships within, an axiomatic system. • (B)recognize the historical development of geometric systems and know mathematics is developed for a variety of purposes; and • (C)compare and contrast the structures and implications of Euclidean and non-Euclidean geometries. JHAILEY/GPISD 7 G.2 Geometric structure: Making Conjectures. The student analyzes geometric relationships in order to make and verify conjectures (P3) • (A) use constructions to explore attributes of geometric figures and make conjectures about geometric relationships; and • (B) make conjectures about angles, lines, polygons, circles, and three-dimensional figures and determine the validity of the conjectures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic JHAILEY/GPISD 8 G.3 Geometric structure: Axiomatic Systems. The student applies logical reasoning to justify and prove mathematical statements. • (A)determine the validity of a conditional statement, its converse, inverse, and contrapositive; • (B) construct and justify statements about geometric figures and their properties; • (C) use logical reasoning to prove statements are true and find counter examples to disprove statements that are false; • (D) use inductive reasoning to formulate a conjecture; and • (E) use deductive reasoning to prove a statement JHAILEY/GPISD 9 G.4 Geometric structure. The student uses a variety of representations to describe geometric relationships and solve problems. • (A) select an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. JHAILEY/GPISD 10 G.5 Geometric patterns: Patterns and Transformations. The student uses a variety of representations to describe geometric relationships and solve problems. • (A) use numeric and geometric patterns to develop algebraic expressions representing geometric properties; • (B) use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles; • (C) use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations; and • (D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 JHAILEY/GPISD 11 and 30-60-90) and triangles whose sides are Pythagorean triples. Essential Questions • What is the relationship between reasoning, justification, and proof in geometry? • What is a truth-value? • How does a truth-value apply to conditional statements? • How do deductive reasoning and Truth Tables help judge the validity of logical arguments? JHAILEY/GPISD 12