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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

 A  = Assessment items available