Geometry and Measurement Concepts Elementary School The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student: a Gives and responds to directions about location, e.g., by using words such as “in front of,” “right,” and “above.” b Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes. c Uses simple two dimensional coordinate systems to find locations on a map and to represent points and simple figures. d Uses many types of figures (angles, triangles, squares, rectangles, rhombi, parallelograms, quadrilaterals, polygons, prisms, pyramids, cubes, circles, and spheres) and identifies the figures by their properties, e.g., symmetry, number of faces, two- or three-dimensionality, no right angles. e Solves problems by showing relationships between and among figures, e.g., using congruence and similarity, and using transformations including flips, slides, and rotations. f Extends and creates geometric patterns using concrete and pictorial models. g Uses basic ways of estimating and measuring the size of figures and objects in the real world, including length, width, perimeter, and area. h Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations. i Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time. j Carries out simple unit conversions, such as between cm and m, and between hours and minutes. k Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.   Middle School The student produces evidence that demonstrates understanding of geometry and measurement concepts in the following areas; that is, the student: a Is familiar with assorted two- and three-dimensional objects, including squares, triangles, other polygons, circles, cubes, rectangular prisms, pyramids, spheres, and cylinders. b Identifies similar and congruent shapes and uses transformations in the coordinate plane, i.e., translations, rotations, and reflections. c Identifies three dimensional shapes from two dimensional perspectives; draws two dimensional sketches of three dimensional objects that preserve significant features. d Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms. e Recognizes similarity and rotational and bilateral symmetry in two- and three-dimensional figures. f Analyzes and generalizes geometric patterns, such as tessellations and sequences of shapes. g Measures angles, weights, capacities, times, and temperatures using appropriate units. h Chooses appropriate units of measure and converts with ease between like units, e.g., inches and miles, within a customary or metric system. (Conversions between customary and metric are not required.) i Reasons proportionally in situations with similar figures. j Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings. k Models situations geometrically to formulate and solve problems.   High School The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student: a Models situations geometrically to formulate and solve problems. b Works with two- and three-dimensional figures and their properties, including polygons and circles, cubes and pyramids, and cylinders, cones, and spheres. c Uses congruence and similarity in describing relationships between figures. d Visualizes objects, paths, and regions in space, including intersections and cross sections of three dimensional figures, and describes these using geometric language. e Knows, uses, and derives formulas for perimeter, circumference, area, surface area, and volume of many types of figures. f Uses the Pythagorean Theorem in many types of situations, and works through more than one proof of this theorem. g Works with similar triangles, and extends the ideas to include simple uses of the three basic trigonometric functions. h Analyzes figures in terms of their symmetries using, for example, concepts of reflection, rotation, and translation. i Compares slope (rise over run) and angle of elevation as measures of steepness. j Investigates geometric patterns, including sequences of growing shapes. k Works with geometric measures of length, area, volume, and angle; and non-geometric measures such as weight and time. l Uses quotient measures, such as speed and density, that give “per unit” amounts; and uses product measures, such as person-hours. m Understands the structure of standard measurement systems, both SI and customary, including unit conversions and dimensional analysis. n Solves problems involving scale, such as in maps and diagrams. o Represents geometric curves and graphs of functions in standard coordinate systems. p Analyzes geometric figures and proves simple things about them using deductive methods. q Explores geometry using computer programs such as CAD software, Sketchpad programs, or LOGO.