
Recognize and describe shapes and structures in the physical environment.
 Identify, name, sort, and describe two and threedimensional shapes (including circles, triangles, rectangles, squares, cubes, and spheres), and realworld approximations of the shapes, regardless of size or orientation.


Compose and decompose geometric shapes, including plane and solid figures to develop a foundation for understanding area, volume, fractions, and proportions.
 Compose (combine) and decompose (take apart) two and threedimensional figures and analyze the results.
 Compose and decompose two and threedimensional shapes to develop a foundation of fractional relationships and proportions.
 Cover twodimensional objects with shapes to develop a foundation for area.
 Fill threedimensional objects to develop a foundation for volume.


Identify, name, sort, and describe two and threedimensional geometric figures regardless of size or orientation.
 Describe characteristics of two and threedimensional objects (number of corners, edges, and sides, length of sides, etc.).


Describe and specify space and location with simple relationships and with coordinate systems.
 Describe the location of one object relative to another object using words such as in, out, over, under, above, below, between, next to, behind, and in front of.
 Locate points on maps and simple coordinate grids with letters and numbers.
 Represent points and simple figures on maps using simple coordinate grids with letters and numbers.


Experience and recognize slides, flips, turns and symmetry to analyze mathematical situations.
 Identify shapes that have been rotated (turned), reflected (flipped), translated (slid), and enlarged. Describe the direction of the translation (left, right, up, down).


Use attributes of geometric figures to solve spatial problems.
 Describe and represent shapes from different perspective.
 Explore relationships of different attributes.
 Describe geometric shapes in the environment and specify their location.


Identify attributes that are measurable, such as length, weight, time and capacity, and use these attributes to order objects and make direct comparisons.
 Identify attributes that are measurable such as length, volume, weight, and area. Use these attributes and appropriate language to make direct comparisons. (Taller, shorter, longer, same length; heavier, lighter, same weight; holds more, holds less, holds the same amount).
 Recognize temporal concepts such as before, after, sooner, later, morning, afternoon, evening.
 Use a seriated set of objects to order and compare lengths.
 Recognize that objects used to measure an attribute (length, weight, capacity) must have that attribute and must be consistent in size.
 Determines the relationship between the size of the unit and the number of units needed to make a measurement.


Estimate, measure and compute measurable attributes while solving problems.
 Select appropriate measurement tools and units (standard and nonstandard) to solve problems.


Estimate and measure length using standard (customary and metric) and nonstandard units with comprehension.
 Understand the necessity for identical units (standard or nonstandard) for accurate measurements.
 Use a variety of nonstandard units to measure length without gaps or overlaps.
 Use nonstandard units to compare objects according to their capacities or weights.
 Associate the time of day with everyday events.
 Name standard units of time (day, week, month).
 Use both analog and digital clock to tell time to the hour and half hour.
 Estimate and measure length using metric and customary units.
 Select appropriate measurement tools and units (standard and nonstandard) to solve problems.
 Use both analog and digital clock to tell time to the nearest fiveminute interval.
 Describe the relationship among standard units of time: minutes, hours days, weeks, months and years.



Describe, analyze and classify twodimensional and threedimensional shapes.
 Describe, analyze, and compare twodimensional shapes by their sides and angles and connect these attributes to definitions of shapes.
 Relate twodimensional shapes to threedimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces.
 Classify two and threedimensional shapes according to their attributes and develop definitions of classes of shapes such as parallelograms and prisms.


Explore congruence and similarity.
 Understand attributes and properties of twodimensional space through building, drawing and analyzing twodimensional shapes and use the attributes and properties to solve problems, including applications involving congruence and symmetry.
 Apply congruence to other contexts such as threedimensional shapes and repeating the congruent shapes to build a similar shape.
 Explore similar shapes to determine that angle measure is the same and the related sides are proportional, that is, related by the same multiplicative or scale factor.


Predict and describe the results of sliding (translation), flipping (reflection), and turning (rotation) twodimensional shapes.
 Investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons.
 Investigate and describe line and rotational symmetry.
 Extend their understanding of twodimensional space by using transformations to design and analyze simple tilings and tessellations.


Use ordered pairs on a coordinate grid to describe points or paths (first quadrant).
 Learn how to use two numbers to name points on a coordinate grid and know this ordered pair corresponds to a particular point on the grid.
 Make and use coordinate systems to specify locations and to describe paths.
 Explore methods for measuring the distance between two locations on the grid along horizontal and vertical lines.


Use geometric models to solve problems, such as determining perimeter, area, volume, and surface area.
 Develop measurement concepts and skills through experiences in analyzing attributes and properties of two and threedimensional objects.
 Form an understanding of perimeter as a measurable attribute and quantify perimeter by finding the total distance or length around the shape.
 Recognize area as an attribute of twodimensional regions and that they can quantify area by finding the total number of samesized units of area that cover the shape without gaps or overlaps.
 Connect area measure to the area model that has been used to represent multiplication, and use this connection to justify the formula for the area of a rectangle.
 Develop, understand and use formulas to find the area of rectangles, related triangles and parallelograms and learn to measure the necessary attributes of shapes.
 Recognize volume as an attribute of threedimensional space and understand they can quantify volume by finding the total number of samesized units of volume that fill the space without gaps or overlaps.
 Decompose threedimensional shapes to develop strategies for determining surface area.
 Develop strategies to determine the volumes of prisms by layering.


Select and apply appropriate standard (customary and metric) units and tools to measure length, area, volume, weight, time, temperature, and the size of angles.
 Select appropriate units, strategies, and tools to solve problems that involve estimating and measuring perimeter, area and volume.
 Develop facility in measuring with fractional parts of linear units.
 Understand that a square that is 1 unit on a side is the standard unit for measuring area.
 Understand that a cube that is 1 unit on an edge is the standard unit for measuring volume.
 Select and apply appropriate units, strategies and tools to solve problems that involve estimating and measuring weight, time and temperature.
 Measure and classify angles.


Select and use benchmarks ( inch, 2 liters, 5 pounds, etc.) to estimate measurements.
 Develop strategies for estimating measurements using appropriate benchmarks, both standard units such as 1 foot and nonstandard units such as the length a book.
 Learn to use strategies involving multiplicative reasoning to estimate measurements (i.e. estimating their teacher's height to be one and a quarter times the student's own height).
 Estimate angle measure using a right angle as the benchmark.



Understand, determine, and apply area of polygons.
 Use physical models, such as geoboards, to develop and make sense of area formulas.
 Use knowledge of area of simpler shapes to help find area of more complex shapes.
 Understand and apply formulas to find area of triangles and quadrilaterals.
 Solve problems related to and using area, including in realworld settings.


Understand and apply similarity, with connections to proportion.
 Understand that two objects are similar if they have the same shape (i.e., corresponding angles are congruent) but not necessarily the same size.
 Understand similarity in terms of a scale factor between corresponding lengths in similar objects (i.e., similar objects are related by transformations of magnifying or shrinking).
 Understand that relationships of lengths within similar objects are preserved (i.e., ratios of corresponding sides in similar objects are equal).
 Understand that congruent figures are similar with a scale factor of 1.
 Use understanding of similarity to solve problems in a variety of contexts.


Understand, determine, and apply surface area and volume of prisms and cylinders and circumference and area of circles.
 Find the area of more complex twodimensional shapes, such as pentagons, hexagons, or irregular shaped regions, by decomposing the complex shapes into simpler shapes, such as triangles.
 Understand that the ratio of the circumference to the diameter of a circle is constant and equal to p, and use this fact to develop a formula for the circumference of a circle.
 Understand that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram.
 Develop and justify strategies for determining the surface area of prisms and cylinders by determining the areas of shapes that comprise the surface.
 By decomposing prisms and cylinders by slicing them, develop and understand formulas for their volumes (Volume = Area of base x Height).
 Select appropriate twoand threedimensional shapes to model realworld situations and solve a variety of problems (including multistep problems) involving surface area, area and circumference of circles, and volume of prisms and cylinders.


Analyze twodimensional space and figures by using distance, angle, coordinates, and transformations.
 Explore and explain the relationships among angles when a transversal cuts parallel lines.
 Use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and apply this fact about triangles to find unknown measures of angles.
 Understand and explain how particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines (e.g., "slope triangles").
 Use reasoning about similar triangles to solve a variety of problems, including those that involve determining heights and distances.
 Explain why the Pythagorean Theorem is valid by using a variety of methods — for example, by decomposing a square in different ways.
 Apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane and to measure lengths and analyze polygons.
 Understand and apply transformations — reflection, translation, rotation, and dilation, and understand similarity, congruence, and symmetry in terms of transformations.


Visualize, represent, and describe threedimensional shapes.
 Recognize and draw twodimensional representations of threedimensional figures, including nets, frontsidetop views, and perspective drawings.
 Identify and describe threedimensional shapes, including prisms, pyramids, cylinders, and spheres.
 Examine, build, compose, and decompose threedimensional objects, using a variety of tools, including paperandpencil, geometric models, and dynamic geometry software.
 Use visualization and threedimensional shapes to solve problems, especially in realworld settings.



Represent and solve geometric problems by specifying locations using coordinates.
 Rectangular coordinates are the focus of the study of coordinate geometry in the core curriculum. However, students should recognize that the location of a point can be described in other ways, such as by using angle and distance (as in polar coordinates or bearings) or using latitude and longitude. The study of coordinate geometry includes investigating conjectures, modeling, and solving problems. By using inductive and deductive reasoning with coordinates, properties of geometric objects can be conjectured and proven.
 Coordinates can be used to describe points, lines, and other two and threedimensional figures. Transformations of these objects can be described using coordinate rules. Analysis of the relationships of geometric objects includes the use of formulas for distance, midpoint, and slope, and the Pythagorean theorem. Students should find and analyze equations that represent lines, circles and parabolas. (Students should be introduced to the other conic sections—ellipses and hyperbolas). In three dimensions, students should be able to plot points using rectangular coordinates.


Understand and apply the basic principles of transformational geometry.
 Transformations, such as reflections and rotations, are mappings that move points. Students should be familiar with three classes of transformations: (1) transformations that preserve distance (called isometries or rigid motions, such as reflections, rotations, translations), (2) transformations that preserve shape (such as size transformations, dilations, or similarity transformations), and (3) transformations that change distance and shape (e.g., shears). Students should recognize similarity and congruence in terms of certain transformations.
 Students should be able to identify, create, describe, and justify transformations using multiple representations. They should be able to find and describe an image under a given transformation or composition of transformations. Students should also be able to identify the transformations that produce a given image. Transformations should be represented algebraically (using coordinate rules, matrices, vectors), and those representations should be used to analyze and reason about transformations.


Understand and apply properties and relationships of geometric figures.
 Students should be able to visualize, describe, reason about, prove, and apply properties and relationships of two and threedimensional objects. Specific geometric skills students should demonstrate include visualizing, drawing, geometric modeling, making and testing conjectures, and using inductive and deductive reasoning.
 Properties and relationships of geometric objects should be examined and justified, including similarity, congruence, and measurement. Objects should be represented with drawings, coordinates, and matrices; and transformations of the objects should be investigated.
 The primary focus should be on twodimensional objects, their properties and relationships. Particular emphasis should be given to properties of angles, lines, polygons, and circles. In three dimensions, students should be able to visualize, draw, and determine measurements of simple threedimensional shapes.
 Measurement skills and concepts should be included in the study of geometry, including finding perimeter, area, volume and surface area (much of which is studied in middle school). Estimation, appropriate units, dimensional analysis, and judgments about accuracy should be part of the study of measurement.


Use trigonometry based on triangles and circles to solve problems about length and angle measures.
 Students should study trigonometry with respect to right triangles, general triangles, circles, and periodic relationships. Included in the study of right triangle trigonometry are the trigonometric ratios, the Pythagorean theorem and its converse, and the two specialcase triangles, 30°—60°—90° and 45°—45°—90°. Trigonometry should be extended beyond right triangles to general triangles using the Law of Sines and Law of Cosines.
 Examining right triangles in relation to the unit circle extends analysis to general periodic relationships. Degree and radian measure should be studied. The analysis of trigonometric functions includes: domain and range, period, amplitude, and vertical and horizontal shifts. Students should be able to recognize and model relevant periodic phenomenon with trigonometric functions.
 Students should use trigonometry to solve problems. Students should reason about, reason with, and apply fundamental trigonometric relationships, including sin2 x + cos2x = 1, tanx = sinx/cosx, and cosx = sin (90 — x).


Uses diagrams consisting of vertices and edges (vertexedge graphs) to model and solve problems.
 Vertexedge graphs are diagrams consisting of vertices (points) and edges (line segments or arcs) connecting some of the vertices. The term "vertexedge graph" is used to distinguish this type of graph from other graphs, such as function graphs or data plots. Nevertheless, vertexedge graphs are often simply called graphs, especially in college mathematics courses. Vertexedge graphs are also sometimes called networks, discrete graphs, or finite graphs. Whatever term is used, a vertexedge graph shows relationships and connections among objects, such as in a road network, a telecommunications network, or a family tree.
 Within the context of school geometry, which is fundamentally the study of shape, vertexedge graphs represent, in a sense, the situation of no shape. That is, vertexedge graphs are geometric models consisting of vertices and edges in which shape is not essential, only the connections among vertices are essential. These graphs are widely used in business and industry to solve problems about networks, paths, and relationships among a finite number of objects (such as, analyzing a computer network; optimizing the route used for snowplowing, garbage collection, or visiting business clients; scheduling committee meetings to avoid conflicts; or planning a large construction project to finish on time).
 Students should understand, analyze, and apply vertexedge graphs to model and solve problems related to paths, circuits, networks, and relationships among a finite number of elements, in realworld and abstract settings. Important vertexedge graph topics for the high school curriculum include: Euler and Hamilton paths and circuits, the traveling salesman problem (TSP), minimum spanning trees, critical paths, shortest paths, and vertex coloring. These topics can be compared and contrasted in terms of algorithms, optimization, properties, and types of problems that can be solved. Students should represent and analyze vertexedge graphs using adjacency matrices. Some students may also analyze and interpret powers of an adjacency matrix. This important material on vertexedge graphs may be addressed as part of instruction in geometry or when teaching matrices or in separate miniunits.

