Geometry
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Pre-K–2 Expectations
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Grades 3–5 Expectations
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Grades 6–8 Expectations
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Grades 9–12 Expectations
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Specify locations and describe spatial relationships using coordinate geometry and other representational systems
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describe, name, and interpret relative positions in space and apply ideas about relative position;
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describe, name, and interpret direction and distance in navigating space and apply ideas about direction and distance;
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find and name locations with simple relationships such as "near to" and in coordinate systems such as maps.
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describe location and movement using common language and geometric vocabulary;
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make and use coordinate systems to specify locations and to describe paths;
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find the distance between points along horizontal and vertical lines of a coordinate system.
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use coordinate geometry to represent and examine the properties of geometric shapes;
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use coordinate geometry to examine special geometric shapes, such as regular polygons or those with pairs of parallel or perpendicular sides.
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use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations;
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investigate conjectures and solve problems involving two- and three-dimensional objects represented with Cartesian coordinates.
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Apply transformations and use symmetry to analyze mathematical situations
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recognize and apply slides, flips, and turns;
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recognize and create shapes that have symmetry.
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predict and describe the results of sliding, flipping, and turning two-dimensional shapes;
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describe a motion or a series of motions that will show that two shapes are congruent;
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identify and describe line and rotational symmetry in two- and three-dimensional shapes and designs.
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describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling;
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examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
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understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices;
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use various representations to help understand the effects of simple transformations and their compositions.
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Use visualization, spatial reasoning, and geometric modeling to solve problems
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create mental images of geometric shapes using spatial memory and spatial visualization;
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recognize and represent shapes from different perspectives;
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relate ideas in geometry to ideas in number and measurement;
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recognize geometric shapes and structures in the environment and specify their location.
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build and draw geometric objects;
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create and describe mental images of objects, patterns, and paths;
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identify and build a three-dimensional object from two-dimensional representations of that object;
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identify and draw a two-dimensional representation of a three-dimensional object;
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use geometric models to solve problems in other areas of mathematics, such as number and measurement;
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recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.
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draw geometric objects with specified properties, such as side lengths or angle measures;
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use two-dimensional representations of three-dimensional objects to visualize and solve problems such as those involving surface area and volume;
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use visual tools such as networks to represent and solve problems;
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use geometric models to represent and explain numerical and algebraic relationships;
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recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.
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draw and construct representations of two- and three-dimensional geometric objects using a variety of tools;
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visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections;
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use vertex-edge graphs to model and solve problems;
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use geometric models to gain insights into, and answer questions in, other areas of mathematics;
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use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
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Measurement
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Pre-K–2 Expectations
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Grades 3–5 Expectations
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Grades 6–8 Expectations
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Grades 9–12 Expectations
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Understand measurable attributes of objects and the units, systems, and processes of measurement
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recognize the attributes of length, volume, weight, area, and time;
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compare and order objects according to these attributes;
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understand how to measure using nonstandard and standard units;
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select an appropriate unit and tool for the attribute being measured.
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understand such attributes as length, area, weight, volume, and size of angle and select the appropriate type of unit for measuring each attribute;
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understand the need for measuring with standard units and become familiar with standard units in the customary and metric systems;
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carry out simple unit conversions, such as from centimeters to meters, within a system of measurement;
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understand that measurements are approximations and how differences in units affect precision;
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explore what happens to measurements of a two-dimensional shape such as its perimeter and area when the shape is changed in some way.
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understand both metric and customary systems of measurement;
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understand relationships among units and convert from one unit to another within the same system;
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understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume.
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make decisions about units and scales that are appropriate for problem situations involving measurement.
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Apply appropriate techniques, tools, and formulas to determine measurements.
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measure with multiple copies of units of the same size, such as paper clips laid end to end;
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use repetition of a single unit to measure something larger than the unit, for instance, measuring the length of a room with a single meterstick;
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use tools to measure;
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develop common referents for measures to make comparisons and estimates.
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develop strategies for estimating the perimeters, areas, and volumes of irregular shapes;
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select and apply appropriate standard units and tools to measure length, area, volume, weight, time, temperature, and the size of angles;
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select and use benchmarks to estimate measurements;
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develop, understand, and use formulas to find the area of rectangles and related triangles and parallelograms;
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develop strategies to determine the surface areas and volumes of rectangular solids.
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use common benchmarks to select appropriate methods for estimating measurements;
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select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision;
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develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders;
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solve problems involving scale factors, using ratio and proportion;
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analyze precision, accuracy, and approximate error in measurement situations;
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understand and use formulas for the area, surface area, and volume of geometric figures, including cones, spheres, and cylinders;
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apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations;
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use unit analysis to check measurement computations.
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