 Geometry

• Instructional programs from prekindergarten through grade 12 should enable each and every student to—

• Analyze characteristics and properties  of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
• Specify locations  and describe  spatial relationships using coordinate geometry and other representational systems
• Apply transformations  and use symmetry to analyze mathematical situations
• Use visualization , spatial reasoning, and geometric modeling to solve problems

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

Pre-K–2 Expectations: In pre-K through grade 2 each and every student should–

• recognize, name, build, draw, compare, and sort two- and three-dimensional shapes;
• describe attributes and parts of two- and three-dimensional shapes;
• investigate and predict the results of putting together and taking apart two- and three-dimensional shapes.

• identify, compare, and analyze attributes of two- and three-dimensional shapes and develop vocabulary to describe the attributes;
• classify two- and three-dimensional shapes according to their properties and develop definitions of classes of shapes such as triangles and pyramids;
• investigate, describe, and reason about the results of subdividing, combining, and transforming shapes;
• explore congruence and similarity;
• make and test conjectures about geometric properties and relationships and develop logical arguments to justify conclusions.

• precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
• understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects;
• create and critique inductive and deductive arguments concerning geometric ideas and relationships, such as congruence, similarity, and the Pythagorean relationship.

• 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

Pre-K–2 Expectations: In pre-K through grade 2 each and every student should–

• describe, name, and interpret relative positions in space and apply ideas about relative position;
• describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance;
• find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.

• describe location and movement using common language and geometric vocabulary;
• make and use coordinate systems to specify locations and to describe paths;
• find the distance between points along horizontal and vertical lines of a coordinate system.

• use coordinate geometry to represent and examine the properties of geometric shapes;
• use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.

• 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

Pre-K–2 Expectations: In pre-K through grade 2 each and every student should–

• recognize and apply slides, flips, and turns;
• recognize and create shapes that have symmetry.

• predict and describe the results of sliding, flipping, and turning two-dimensional shapes;
• describe a motion or a series of motions that will show that two shapes are congruent;
• identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.

• describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling;
• examine the congruence, similarity, and line or rotational symmetry of objects using transformations.

• 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

Pre-K–2 Expectations: In pre-K through grade 2 each and every student should–

• create mental images of geometric shapes using spatial memory and spatial visualization;
• recognize and represent shapes from different perspectives;
• relate ideas in geometry to ideas in number and measurement;
• recognize geometric shapes and structures in the environment and specify their location.

• build and draw geometric objects;
• create and describe mental images of objects, patterns, and paths;
• identify and build a three-dimensional object from two-dimensional representations of that object;
• identify and draw a two-dimensional representation of a three-dimensional object;
• use geometric models to solve problems in other areas of mathematics, such as number and measurement;
• recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

• draw geometric objects with specified properties, such as side lengths or angle measures;
• use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume;
• use visual tools such as networks to represent and solve problems;
• use geometric models to represent and explain numerical and algebraic relationships;
• recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.  