### Geometry

 (b) Geometric structure: knowledge and skills and performance descriptions. (1) The student understands the structure of, and relationships within, an axiomatic system. Following are performance descriptions. (A) The student develops an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. (B) Through the historical development of geometric systems, the student recognizes that mathematics is developed for a variety of purposes. (C) The student compares and contrasts the structures and implications of Euclidean and non-Euclidean geometries. Interactive Student Interactive Classroom (C) Dan Pedoe's Observation (2) The student analyzes geometric relationships in order to make and verify conjectures. Following are performance descriptions. (A) The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. (B) The student makes and verifies conjectures about angles, lines, polygons, circles, and three-dimensional figures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. Interactive Student (B) Cutting Corners (B) Circle Ratio Interactive Classroom (3) The student understands the importance of logical reasoning, justification, and proof in mathematics. Following are performance descriptions. (A) The student determines if the converse of a conditional statement is true or false. (B) The student constructs and justifies statements about geometric figures and their properties. (C) The student demonstrates what it means to prove mathematically that statements are true. (D) The student uses inductive reasoning to formulate a conjecture. (E) The student uses deductive reasoning to prove a statement. Interactive Student (B) Light Bounce Interactive Classroom (4) The student uses a variety of representations to describe geometric relationships and solve problems. Following is a performance description. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. Interactive Student Interactive Classroom (c) Geometric patterns: knowledge and skills and performance descriptions. The student identifies, analyzes, and describes patterns that emerge from two- and three-dimensional geometric figures. Following are performance descriptions. (B) The student uses 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. () The student uses properties of transformations and their compositions to make connections between mathematics and the real world in applications such as tessellations or fractals. (3) The student identifies and applies patterns from right triangles to solve problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. Interactive Student Interactive Classroom (d) Dimensionality and the geometry of location: knowledge and skills and performance descriptions. (1) The student analyzes the relationship between three-dimensional objects and related two-dimensional representations and uses these representations to solve problems. Following are performance descriptions. (A) The student describes, and draws cross sections and other slices of three-dimensional objects. (B) The student uses nets to represent and construct three-dimensional objects. (C) The student uses top, front, side, and corner views of three-dimensional objects to create accurate and complete representations and solve problems. Interactive Student Teacher (2) The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. Following are performance descriptions. (A) The student uses one- and two-dimensional coordinate systems to represent points, lines, line segments, and figures. (B) The student uses slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. (C) The student develops and uses formulas including distance and midpoint. Interactive Student (e) Congruence and the geometry of size: knowledge and skills and performance descriptions. (8) The student extends measurement concepts to find area, perimeter, and volume in problem situations. Following are performance descriptions. (A) The student finds areas of regular polygons and composite figures. (B) The student finds areas of sectors and arc lengths of circles using proportional reasoning. (C) The student develops, extends, and uses the Pythagorean Theorem. (D) The student finds surface areas and volumes of prisms, pyramids, spheres, cones, and cylinders in problem situations. Interactive Student (C) Pythagorean Puzzles (2) The student analyzes properties and describes relationships in geometric figures. Following are performance descriptions. (A) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties of parallel and perpendicular lines. (B) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of polygons and their component parts. (C) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of circles and the lines that intersect them. (D) The student analyzes the characteristics of three-dimensional figures and their component parts. Interactive Student (3) The student applies the concept of congruence to justify properties of figures and solve problems. Following are performance descriptions. (A) The student uses congruence transformations to make conjectures and justify properties of geometric figures. (B) The student justifies and applies triangle congruence relationships. Interactive Student (B) Congruence Theorems (f) Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems. Following are performance descriptions. (1) The student uses similarity properties and transformations to explore and justify conjectures about geometric figures. (2) The student uses ratios to solve problems involving similar figures. (3) In a variety of ways, the student develops, applies, and justifies triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples. (4) The student describes the effect on perimeter, area, and volume when length, width, or height of a three-dimensional solid is changed and applies this idea in solving problems. (3) Square in a Right Triangle