### Standard 12: Geometry

In grades 5-8, the mathematics curriculum should include the study of geometry of one, two and three dimensions in a variety of situations so that students can:
• Identify, describe, compare, and classify geometric figures
• Benchmarks 9C (The Mathematical World: Shapes)
Grades K-2, page 223
Shapes such as circles, squares, and triangles can be used to describe many things that can be seen.

• Visualize and represent geometric figures with special attention to developing spatial sense
• Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades K-2, page 26
Patterns can be made by putting different shapes together or taking them apart.

• Explore transformations of geometric figures
• Benchmarks 11C (Common Themes: Constancy and Change)
Grades 3-5, page 273
Some features of things may stay the same even when other features change. Some patterns look the same when they are shifted over, or turned, or reflected or seen form different directions.

• Understand and apply geometric properties and relationships
• Benchmarks 9C (The Mathematical World: Shapes)
Grades 3-5, page 223
Many objects can be described in terms of simple plane figures and solids. Shapes can be compared in terms of concepts such as parallel and perpendicular, congruence, similarity, and symmetry. Symmetry can be found by reflection, turns, or slides.

Benchmarks 9C (The Mathematical World: Shapes)
Grades 3-5, page 223
Length can be thought of as unit lengths joined together, area as a collection of unit squares, and volume as a set of unit cubes.

Benchmarks 9C (The Mathematical World: Shapes)
Grades 3-5, page 223
Areas of irregular shapes can be found by dividing them into squares and triangles.

Benchmarks 9C (The Mathematical World: Shapes)
Grades 6-8, page 224
Lines can be parallel, perpendicular, or oblique.

Benchmarks 9C (The Mathematical World: Shapes)
Grades 6-8, page 224
Some shapes have special properties: Triangular shapes tend to make structures rigid, and found shapes give the least possible boundary for a given amount of interior area. Shapes can match exactly or have the same shape in different sizes.

• Develop an appreciation of geometry as a means of describing the physical world
• Benchmarks 2A (The Nature of Mathematics: Patterns and Relationships)
Grades K-2, page 26
Circles, squares, triangles, and other shapes can be found in things in nature and in things that people build.

Benchmarks 11D (Common Themes: Scale)
Grades 6-8, page 274
Symmetry (or the lack of it) may determine properties of many objects, from molecules and crystals to organisms and designed structures.

• Represent and solve problems using geometric models
• Benchmarks 9C (The Mathematical World: Shapes)
Grades 3-5, page 223
Scale drawings show shapes and compare locations of things very different in size.

Benchmarks 11B (Common Themes: Models)
Grades 3-5, page 268
Geometric figures, number sequences, graphs, diagrams, sketches, number lines, maps, and stories can be used to represent objects, events, and processes in the real world, although such representations can never be exact in every detail.

Benchmarks 9C (The Mathematical World: Shapes)
Grades 6-8, page 224
Shapes on a sphere like the earth cannot be depicted on a flat surface without distortion.