# List of Benchmarks for Mathematics |

**Standard 5.** | Understands and applies basic and advanced properties of the concepts of geometry |

| ** Level Pre-K (Grade Pre-K)** |

| 1. | Knows basic geometric language for naming shapes (e.g., circle, triangle, square, rectangle) |

| 2. | Understands the common language used to describe position and location (e.g., "up," "down," "below," "above," "beside," "inside," "outside") |

| 3. | Knows that geometric shapes can be put together or taken apart to form other shapes |

| 4. | Understands that a shape’s orientation in space can change |

| 5. | Sorts and groups objects by attributes (e.g., shape, size, color) |

| ** Level I (Grade K-2)** |

| 1. | Understands basic properties of simple geometric shapes (e.g., number of sides, corners, square corners) and similarities and differences between simple geometric shapes A |

| 2. | Understands the common language of spatial sense (e.g., "left," "right," "horizontal," "in front of") |

| 3. | Uses the names of simple geometric shapes (e.g., circles, squares, triangles) to represent and describe real world situations |

| 4. | Understands that patterns can be made by putting different shapes together or taking them apart |

| ** Level II (Grade 3-5)** |

| 1. | Knows basic geometric language for describing and naming shapes (e.g., trapezoid, parallelogram, cube, sphere) A |

| 2. | Understands basic properties of figures (e.g., two- or three-dimensionality, symmetry, number of faces, type of angle) A |

| 3. | Predicts and verifies the effects of combining, subdividing, and changing basic shapes A |

| 4. | Understands that shapes can be congruent or similar A |

| 5. | Uses motion geometry (e.g., turns, flips, slides) to understand geometric relationships A |

| 6. | Understands characteristics of lines (e.g., parallel, perpendicular, intersecting) and angles (e.g., right, acute) A |

| 7. | Understands how scale in maps and drawings shows relative size and distance A |

| ** Level III (Grade 6-8)** |

| 1. | Uses geometric methods (i.e., an unmarked straightedge and a compass using an algorithm) to complete basic geometric constructions (e.g., perpendicular bisector of a line segment, angle bisector) |

| 2. | Understands the defining properties of three-dimensional figures (e.g., a cube has edges with equal lengths, faces with equal areas and congruent shapes, right angle corners) A |

| 3. | Understands the defining properties of triangles (e.g., the sum of the measures of two sides of a triangle must be greater than the measure of the third side) A |

| 4. | Understands geometric transformations of figures (e.g., rotations, translations, dilations) A |

| 5. | Understands the relationships between two- and three-dimensional representations of a figure (e.g., scale drawings, blueprints, planar cross sections) A |

| 6. | Understands the mathematical concepts of similarity (e.g., scale, proportion, growth rates) and congruency A |

| 7. | Understands the basic concept of the Pythagorean theorem |

| ** Level IV (Grade 9-12)** |

| 1. | Understands that objects and relations in geometry correspond directly to objects and relations in algebra (e.g., a line in geometry corresponds to a set of ordered pairs satisfying an equation of the form ax + by = c) |

| 2. | Use the Pythagorean theorem and its converse and properties of special right triangles (e.g. 30° - 60° - 90° triangle) to solve mathematical and real-world problems |

| 3. | Uses synthetic (i.e., pictorial) representations and analytic (i.e., coordinate) methods to solve problems involving symmetry and transformations of figures (e.g., problems involving distance, midpoint, and slope; determination of symmetry with respect to a point or line) A |

| 4. | Understands the characteristics and uses of vectors (e.g., representations of velocity and force) |

| 5. | Uses geometric constructions (e.g., the parallel to a line through a given point not on the line, line segment congruent to a given line segment) to complete simple proofs, to model, and to solve mathematical and real-world problems |

| 6. | Uses the properties of vectors and basic operations on vectors (e.g., vector addition, scalar multiplication) to solve problems (e.g., involving velocity) A |

| 7. | Understands the basic concepts of right triangle trigonometry (e.g., basic trigonometric ratios such as sine, cosine, and tangent) |

| 8. | Uses trigonometric ratio methods to solve mathematical and real-world problems (e.g., determination of the angle of depression between two markers on a contour map with different elevations) |

| 9. | Understands the basic properties and uses of polar coordinates |

| 10. | Uses inductive and deductive reasoning to make observations about and to verify properties of and relationships among figures (e.g., the relationship among interior angles of parallel lines cut by a transversal) A |

| 11. | Uses properties of and relationships among figures to solve mathematical and real-world problems (e.g., uses the property that the sum of the angles in a quadrilateral is equal to 360 degrees to square up the frame for a building; uses understanding of arc, chord, tangents, and properties of circles to determine the radius given a circular edge of a circle without the center) A |

| ** Level V (Grade (College Readiness))** |

| 1. | Understands the effect that scale factors have on magnitude (e.g., area, volume) |

| 2. | Uses the distance formula |

| 3. | Understands the parallel postulate in relation to Euclidean geometry and understands that non-Euclidean geometries exist |

| 4. | Constructs geometric proofs (e.g., proves the Pythagorean theorem, proves there are 180 degrees in a triangle) |

| 5. | Uses algebraic methods to solve geometric problems |

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A = Assessment items available |