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# National Standards for Grade 8 - Mathematics (page 4)

National Assessment Governing Board
Updated on Mar 14, 2011

#### Geometry

By grade 4, students are expected to be familiar with a library of simple figures and their attributes, both in the plane (lines, circles, triangles, rectangles, and squares) and in space (cubes, spheres, and cylinders). In middle school, understanding of these shapes deepens, with the study of cross-sections of solids and the beginnings of an analytical understanding of properties of plane figures, especially parallelism, perpendicularity, and angle relations in polygons. Right angles and the Pythagorean theorem are introduced, and geometry becomes more and more mixed with measurement. The basis for analytic geometry is laid by study of the number line.

Symmetry is an increasingly important component of geometry. Elementary students are expected to be familiar with the basic types of symmetry transformations of plane figures, including flips (reflection across lines), turns (rotations around points), and slides (translations). In middle school, this knowledge becomes more systematic and analytical, with each type of transformation being distinguished from other types by their qualitative effects. For example, a rigid motion of the plane that leaves at least two points fixed (but not all points) must be a reflection in a line.

 GRADE 8 1) Dimension and shape a) Draw or describe a path of shortest length between points to solve problems in context. b) Identify a geometric object given a written description of its properties. c) Identify, define, or describe geometric shapes in the plane and in three-dimensional space given a visual representation. d) Draw or sketch from a written description polygons, circles, or semicircles. e) Represent or describe a three-dimensional situation in a two-dimensional drawing from different views. f) Demonstrate an understanding about the two- and three-dimensional shapes in our world through identifying, drawing, modeling, building, or taking apart. 2) Transformation of shapes and preservation of properties a) Identify lines of symmetry in plane figures or recognize and classify types of symmetries of plane figures. b) Recognize or informally describe the effect of a transformation on two-dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, magnifications, and contractions). c) Predict results of combining, subdividing, and changing shapes of plane figures and solids (e.g., paper folding, tiling, and cutting up and rearranging pieces). d) Justify relationships of congruence and similarity, and apply these relationships using scaling and proportional reasoning. e) For similar figures, identify and use the relationships of conservation of angle and of proportionality of side length and perimeter. 3) Relationships between geometric figures a) Apply geometric properties and relationships in solving simple problems in two and three dimensions. b) Represent problem situations with simple geometric models to solve mathematical or real-world problems. c) Use the Pythagorean theorem to solve problems. d) Describe or analyze simple properties of, or relationships between, triangles, quadrilaterals, and other polygonal plane figures. e) Describe or analyze properties and relationships of parallel or intersecting lines. 4) Position and direction a) Describe relative positions of points and lines using the geometric ideas of midpoint, points on common line through a common point, parallelism, or perpendicularity. b) Describe the intersection of two or more geometric figures in the plane (e.g., intersection of a circle and a line). c) Visualize or describe the cross section of a solid. d) Represent geometric figures using rectangular coordinates on a plane. 5) Mathematical reasoning a) Make and test a geometric conjecture about regular polygons.