Topic: Shapes and figures 
Mathematics 
 Standard 5. Understands and applies basic and advanced properties of the concepts of geometry 
  Level PreK (Grade PreK) 
   Benchmark 1.  Knows basic geometric language for naming shapes (e.g., circle, triangle, square, rectangle) 
    Knowledge/skill statements 
     1.  Understands the term "circle" 
     2.  Understands the term "triangle" 
     3.  Understands the term "square" 
     4.  Understands the term "rectangle" 
     5.  Identifies a circle 
     6.  Identifies a triangle 
     7.  Identifies a square 
     8.  Identifies a rectangle 
   Benchmark 3.  Knows that geometric shapes can be put together or taken apart to form other shapes 
    Knowledge/skill statements 
     1.  Knows that geometric shapes can be put together to form other shapes 
     2.  Knows that geometric shapes can be taken apart to form other shapes 
   Benchmark 5.  Sorts and groups objects by attributes (e.g., shape, size, color) 
    Knowledge/skill statements 
     1.  Sorts objects by different attributes 
     2.  Groups objects by different attributes 
     3.  Sorts objects by shape 
     4.  Sorts objects by size 
     5.  Sorts objects by color 
     6.  Groups objects by shape 
     7.  Groups objects by size 
     8.  Groups objects by color 
     9.  Understands that objects can be sorted by their attributes 
     10.  Understands that objects can be grouped by their attributes 
  Level I (Grade K2) 
   Benchmark 1.  Understands basic properties of simple geometric shapes (e.g., number of sides, corners, square corners) and similarities and differences between simple geometric shapes 
    Knowledge/skill statements 
     1.  Understands the concept of sides 
     2.  Understands the concept of a triangle 
     3.  Understands the concept of a square 
     4.  Understands the differences between basic shapes 
     5.  Understands the concept of a circle 
     6.  Understands the concept of corners 
     7.  Understands the concept of square corners 
     8.  Understands the concept of a rectangle 
     9.  Understands the similarities among basic shapes 
   Benchmark 3.  Uses the names of simple geometric shapes (e.g., circles, squares, triangles) to represent and describe real world situations 
    Knowledge/skill statements 
     1.  Understands that geometric shapes are useful in representing the world 
     2.  Uses circles to represent real world situations 
     3.  Uses squares to represent real world situations 
     4.  Uses triangles to represent real world situations 
     5.  Uses circles to describe real world situations 
     6.  Uses squares to describe real world situations 
     7.  Uses triangles to describe real world situations 
   Benchmark 4.  Understands that patterns can be made by putting different shapes together or taking them apart 
    Knowledge/skill statements 
     1.  Understands that patterns can be made by putting shapes together 
     2.  Understands that patterns can be made by taking shapes apart 
  Level II (Grade 35) 
   Benchmark 1.  Knows basic geometric language for describing and naming shapes (e.g., trapezoid, parallelogram, cube, sphere) 
    Knowledge/skill statements 
     1.  Understands the concept of a trapezoid 
     2.  Understands the concept of a parallelogram 
     3.  Understands the concept of a sphere 
     4.  Understands the concept of a cube 
   Benchmark 2.  Understands basic properties of figures (e.g., two or threedimensionality, symmetry, number of faces, type of angle) 
    Knowledge/skill statements 
     1.  Understands the concept of twodimensionality 
     2.  Understands the concept of threedimensionality 
     3.  Understands the concept of symmetry 
     4.  Understand the defining number of faces for a threedimensional object 
     5.  Understands that objects have a defining number and types of angles 
   Benchmark 3.  Predicts and verifies the effects of combining, subdividing, and changing basic shapes 
    Knowledge/skill statements 
     1.  Predicts the effects of subdividing shapes 
     2.  Predicts the effects of changing shapes 
     3.  Predicts the effects of combining shapes 
     4.  Verifies the effects of subdividing shapes 
     5.  Verifies the effects of changing shapes 
     6.  Verifies the effects of combining shapes 
   Benchmark 4.  Understands that shapes can be congruent or similar 
    Knowledge/skill statements 
     1.  Understands the concept of similar shapes 
     2.  Understands the concept of congruent shapes 
  Level III (Grade 68) 
   Benchmark 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) 
    Knowledge/skill statements 
     1.  Uses a straight edge and compass to construct angle bisectors 
     2.  Uses a straight edge and compass to construct perpendicular bisectors 
     3.  Understands the concept of geometric constructions 
     4.  Understands the concept of an angle bisector 
     5.  Understands the concept of a perpendicular bisector 
   Benchmark 2.  Understands the defining properties of threedimensional figures (e.g., a cube has edges with equal lengths, faces with equal areas and congruent shapes, right angle corners) 
    Knowledge/skill statements 
     1.  Understands the concept of a threedimensional figure 
     2.  Understands the properties of threedimensional figures 
     3.  Understands that the properties of a cube include faces that are congruent, faces that have equal areas, edges with equal lengths, and right angle corners 
  Level IV (Grade 912) 
   Benchmark 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) 
    Knowledge/skill statements 
     1.  Uses synthetic geometric models to solve slope problems 
     2.  Uses synthetic geometric models to solve line symmetry problems 
     3.  Uses synthetic geometric models to solve point symmetry problems 
     4.  Uses synthetic geometric models to solve distance problems 
     5.  Uses synthetic geometric models to solve midpoint problems 
     6.  Understands the concept of a synthetic model (i.e., a pictorial representation) 
     7.  Uses analytic methods to solve slope problems 
     8.  Uses analytic methods to solve line symmetry problems 
     9.  Uses analytic methods to solve point symmetry problems 
     10.  Uses analytic methods to solve distance problems 
     11.  Uses analytic methods to solve midpoint problems 
   Benchmark 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 realworld problems 
    Knowledge/skill statements 
     1.  Constructs a line segment congruent to a line segment 
     2.  Understands the basic concept of a geometric construction 
     3.  Constructs a line through a given point not on the line 
     4.  Uses a line segment congruent to given line segment in proofs 
     5.  Uses geometric constructions to complete simple proofs 
     6.  Determines whether a problem can be modeled by geometric constructions 
     7.  Determines whether a problem can be solved by using geometric constructions 
   Benchmark 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) 
    Knowledge/skill statements 
     1.  Makes deductions about the relationships among geometric figures 
     2.  Makes inductions about the relationships among geometric figures 
     3.  Understands the concept of a transversal 
     4.  Makes deductions about the properties of figures 
     5.  Makes inductions about the properties of figures 
   Benchmark 11.  Uses properties of and relationships among figures to solve mathematical and realworld 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) 
    Knowledge/skill statements 
     1.  Determines whether a problem can be solved by making use of the relationships among angles 
     2.  Understands the concept of a chord 
     3.  Determines whether a problem can be solved by using chords 
     4.  Understands the concept of an arc 
     5.  Determines whether a problem can be solved by using arcs 
     6.  Understands the concept of a tangent to a circle 
     7.  Determines whether a problem can be solved by using tangents to a circle 
     8.  Determines whether a problem can be solved by using radii 
     9.  Determines whether a problem can be solved by making use of the properties of figures 
     10.  Determines whether a problem can be solved by making use of relationships among figures 
     11.  Understands the properties of circles 
Engineering Education 
 Standard 12. Understands the techniques, tools, and technologies related to the production of technical drawings 
  Level IV (Grade 912) 
   Benchmark 8.  Knows procedures for creating geometric constructions (e.g., arcs, angles, bisecting lines, perpendicular and parallel lines, geometric shapes)

    Knowledge/skill statements 
     1.  Knows procedures for creating geometric constructions such as arcs 
     5.  Knows procedures for creating geometric constructions such as geometric shapes 