Virginia S.O.L Standards: Geometry
4.10 - The student will
a) identify and describe representations of points, lines, line segments, rays, and angles, including endpoints and vertices; and
b) identify representations of lines that illustrate intersection, parallelism, and perpendicularity.
Background Information:
- A point is a location in space. It has no length, width, or height. A point is usually named with a capital letter.
- A line is a collection of points going on and on infeinitely in both directions. It has no endpoints. When a line is drawn, at least two points on it can be marked and given capital letter names. Arrows must be drawn to show that the line goes on in both directions infinitely.
- A line segment is part of a line. It has two endpoints and includes all the points between those endpoints. To name a line segment, name the endpoints.
- Two rays that have the same endpoint form an angle. This endpoint is called the vertex. Angles are found wherever lines and line segments intersect. An angle can be named in three different ways by using
– one letter at the vertex; or
– a number written inside the rays of the angle.
- Intersecting lines have one point in common.
- Perpendicular lines are special intersecting lines that form right angles where they intersect.
- Parallel lines are lines that lie in the same place and do not intersect. Parallel lines are always the same distance apart and do not share any points.
- Students should explore intersection, parallelism, and perpendicularity in both two and three dimensions. For example, students should analyze the relationships between the edges of a cube. Which edges are parallel? Which are perpendicular? What plane contains the upper left edge and the lower right edge of the cube? Students can visualize this by using the classroom itself to notice the lines formed by the intersection of the ceiling and walls, of the floor and wall, and of two walls.
4.11 - The student will
a) investigate congruence of plane figures after geometric transformations, such as reflection, translation, and rotation, using mirrors, paper folding, and tracing; and
b) recognize the images of figures resulting from geometric transformations, such as translation, reflection, and rotation.
Background Information:
- The van Hiele theory of geometric understanding describes how students learn geometry and provides a framework for structuring student experiences that should lead to conceptual growth and understanding.
– Level 1: Visualization. Geometric figures are recognized as entities, without any awareness of parts of figures or relationships between components of a figure. Students should recognize and name figures and distinguish a given figure from others that look somewhat the same. (This is the expected level of student performance during grades K and 1.)
– Level 2: Analysis. Properties are perceived but are isolated and unrelated. Students should recognize and name properties of geometric figures. (Students are expected to transition to this level during grades 2 and 3.)
– Level 3: Abstraction. Definitions are meaningful, with relationships being perceived between properties and between figures. Logical implications and class inclusion are understood, but the role and significance of deduction is not understood. (Students should transition to this level during grades 5 and 6 and fully attain it before taking algebra.)
- Congruent figures are figures having exactly the same size and shape. Opportunities for exploring figures that are congruent and/or noncongruent can best be accomplished by using physical models.
- A translation is a transformation in which an image is formed by moving every point on a figure the same distance in the same direction.
- A reflection is a transformation in which a figure is flipped over a line called the line of reflection. All corresponding points in the image and preimage are equidistant from the line of reflection.
- A rotation is a transformation in which an image is formed by turning its preimage about a point.
- The resulting figure of a translation, reflection, or rotation is congruent to the original figure.
4.12 - The student will
a) define polygon; and
b) identify polygons with 10 or fewer sides.
Background Information:
- A polygon is a closed plane geometric figure composed of at least three line segments that do not cross. None of the sides are curved.
- A triangle is a polygon with three angles and three sides.
- A quadrilateral is a polygon with four sides.
- A rectangle is a quadrilateral with four right angles.
- A square is a rectangle with four sides of equal length.
- A trapezoid is a quadrilateral with exactly one pair of parallel sides.
- A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
- A rhombus is a quadrilateral with 4 congruent sides.
- A pentagon is a 5-sided polygon.
- A hexagon is a 6-sided polygon.
- A heptagon is a 7-sided polygon.
- An octagon is an 8-sided polygon.
- A nonagon is a 9-sided polygon.
- A decagon is a 10-sided polygon.