Secondary Education (9-12)

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NCTM Standards

In grades 9–12 all students should [1]:

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

  • analyze properties and determine attributes of two- and three-dimensional objects;
  • explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;
  • establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;
  • use trigonometric relationships to determine lengths and angle measures.

Specify locations and describe spatial relationships using coordinate geometry and other representational systems

  • use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;
  • investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.

Apply transformations and use symmetry to analyze mathematical situations

  • understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices;
  • use various representations to help understand the effects of simple transformations and their compositions.

Use visualization, spatial reasoning, and geometric modeling to solve problems

  • draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;
  • visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections;
  • use vertex-edge graphs to model and solve problems;
  • use geometric models to gain insights into, and answer questions in, other areas of mathematics;
  • use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.