State of Ohio Mathematics Standards

Geometry and Spatial Sense

Grade: K
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Identify and sort two- dimensional shapes and three- dimensional objects
a   Identify and describe two-dimensional figures and three-dimensional objects from the environment using the child’s own vocabulary
b   Sort shapes and objects into groups based on student-defined categories
c   Select all shapes or objects of one type from a group
d   Build two-dimensional figures using paper shapes or tangrams; build simple three-dimensional objects using blocks
   
Spatial Relationships
Standard Number: 2  
Name and demonstrate the relative position of objects as follows
a   place objects over, under, inside, outside, on, beside, between, above, below, on top of, upside-down, behind, in back of, in front of
b   describe placement of objects with terms such as on, inside, outside, above, below, over, under, beside, between, in front of, behind

Grade: Gr. 1
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Identify, compare, and sort two-dimensional shapes; i.e., square, circle, ellipse, triangle, rectangle, rhombus, trapezoid, parallelogram, pentagon, and hexagon
a   Recognize and identify triangles and rhombuses independent of position, shape or size
b   Describe two-dimensional shapes using attributes such as number of sides and number of vertices (corners, or angles)
   
Standard Number: 2  
Create new shapes by combining or cutting apart existing shapes

Standard Number: 3  
Identify the shapes of the faces of three-dimensional objects.

Spatial Relationships
Standard Number: 4  
Extend the use of location words to include distance (near, far, close to) and directional words (left, right).

Standard Number: 5  
Copy figures and draw simple two-dimensional shapes from memory.

Grade: Gr. 2
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Identify, describe, compare, and sort three-dimensional objects (i.e., cubes, spheres, prisms, cones, cylinders and pyramids) according to the shape of the faces or the numbers of faces, edges, or vertices

Standard Number: 2  
Predict what new shapes will be formed by combining or cutting apart existing shapes

Standard Number: 3  
Recognize two-dimensional shapes and three-dimensional objects from different positions

Spatial Relationships
Standard Number: 4  
Identify and determine whether two-dimensional shapes are congruent (same shape and size) or similar (same shape different size) by copying or using superposition (lay one thing on top of another).

Transformation and Symmetry
Standard Number: 5  
Create and identify two-dimensional figures with line symmetry; e.g., what letter shapes, logos, polygons are symmetrical?

Grade: Gr. 3
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Analyze and describe properties of two-dimensional shapes and three-dimensional objects using terms such as vertex, edge, angle, side and face

Standard Number: 2  
Identify and describe the relative size of angles with respect to right angles as follows:
a   Use physical models, like straws, to make different sized angles by opening and closing the sides, not by changing the side lengths
b   Identify, classify and draw right, acute, obtuse and straight angles.

Spatial Relationships
Standard Number: 3  
Find and name locations on a labeled grid or coordinate system; e.g., a map or graph

Transformation and Symmetry
Standard Number: 4  
Draw lines of symmetry to verify symmetrical two-dimensional shapes

Vizualization and Geometric Models
Standard Number: 5  
Build a three-dimensional model of an object composed of cubes; e.g., construct a model based on an illustration or actual object

Grade: Gr. 4
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Identify, describe and model intersecting, parallel and perpendicular lines and line segments; e.g., use straws or other material to model lines

Standard Number: 2
  
Describe, classify, compare and model two- and three-dimensional objects using their attributes

Standard Number: 3  
Identify similarities and differences of quadrilaterals; e.g., squares, rectangles, parallelograms and trapezoids

Standard Number: 4  
Identify and define triangles based on angle measures (equiangular, right, acute and obtuse triangles) and side lengths (isosceles, equilateral and scalene triangles)

Spatial Relationships
Standard Number: 5  
Describe points, lines and planes, and identify models in the environment

Standard Number: 6  
Specify locations and plot ordered pairs on a coordinate plane, using first quadrant points

Transformation and Symmetry
Standard Number: 7  
Identify, describe and use reflections (flips), rotations (turns), and translations (slides) in solving geometric problems; e.g., use transformations to determine if 2 shapes are congruent

Vizualization and Geometric Models
Standard Number: 8  
Use geometric models to solve problems in other areas of mathematics, such as number (multiplication/division) and measurement (area, perimeter, border)

Grade: Gr. 5
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Draw circles, and identify and determine relationships among the radius, diameter, center and circumference; e.g., radius is half the diameter, the ratio of the circumference of a circle to its diameter is an approximation of p

Standard Number: 2  
Use standard language to describe line, segment, ray, angle, skew, parallel and perpendicular

Standard Number: 3  
Label vertex, rays, interior and exterior for an angle

Standard Number: 4  
Describe and use properties of congruent figures to solve problems

Standard Number: 5  
Use physical models to determine the sum of the interior angles of triangles and quadrilaterals

Spatial Relationships
Standard Number: 6  
Extend understanding of coordinate system to include points whose x or y values may be negative numbers

Vizualization and Geometric Models
Standard Number: 7  
Understand that the measure of an angle is determined by the degree of rotation of an angle side rather than the length of either side

Standard Number: 8  
Predict what three-dimensional object will result from folding a two-dimensional net, then confirm the prediction by folding the net

Grade: Gr. 6
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Classify and describe two-dimensional and three-dimensional geometric figures and objects by using their properties; e.g., interior angle measures, perpendicular/parallel sides, congruent angles/sides

Standard Number: 2  
Use standard language to define geometric vocabulary: vertex, face, altitude, diagonal, isosceles, equilateral, acute, obtuse, etc

Standard Number: 3  
Use multiple classification criteria to classify triangles; e.g., right scalene triangle

Standard Number: 4  
Identify and define relationships between planes; i.e., parallel, perpendicular and intersecting

Spatial Relationships
Standard Number: 5  
Predict and describe sizes, positions and orientations of two-dimensional shapes after transformations such as reflections, rotations, translations and dilations

Transformation and Symmetry
Standard Number: 6  
Draw similar figures that model proportional relationships; e.g., model similar figures with a 1 to 2 relationship by sketching two of the same figure, one with corresponding sides twice the length of the other

Vizualization and Geometric Models
Standard Number: 7  
Build three-dimensional objects built with cubes and sketch the two-dimensional representations of each side.

Grade: Gr. 7
Geometry and Spatial Sense Standard
Characteristics and Properties
Standard Number: 1  
Use proportional reasoning to describe and express relationships between parts and attributes of similar and congruent figures

Standard Number: 2  
Determine sufficient (not necessarily minimal) properties that define a specific two-dimensional figure or three-dimensional object
a   Determine when one set of figures is a subset of another; e.g., all squares are rectangles
b   Develop a set of properties that eliminates all but the desired figure; e.g., only squares are quadrilaterals with all sides congruent and all angles congruent

Standard Number: 3  
Use and demonstrate understanding of the properties of triangles
a   Use Pythagorean Theorem to solve problems involving right triangles
b   Use triangle angle sum relationships to solve problems

Standard Number: 4  
Determine necessary conditions for congruence of triangles

Spatial Relationships
Standard Number: 5  
Apply properties of congruent or similar triangles to solve problems involving missing lengths and angle measures.

Transformation and Symmetry
Standard Number: 6  
Determine and use scale factors for similar figures to solve problems using proportional reasoning

Standard Number: 7  
Identify the line and rotation symmetries of two-dimensional figures to solve problems

Vizualization and Geometric Models
Standard Number: 8  
Perform translations, reflections, rotations and dilations of two-dimensional figures using a variety of methods (paper folding, tracing, graph paper)

Standard Number: 9  
Draw representations of three-dimensional geometric objects from different views.

Grade: Gr. 8
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects

Standard Number: 2  
Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal

Standard Number: 3  
Use proportions in several forms to solve problems involving similar figures (part-to-part, part-to-whole, corresponding sides between figures).

Spatial Relationships
Standard Number: 4  
Represent and analyze shapes using coordinate geometry; e.g., given three vertices and the type of quadrilateral, find the coordinates of the fourth vertex

Transformation and Symmetry
Standard Number: 5  
Draw the results of translations, reflections, rotations and dilations of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations

Vizualization and Geometric Models
Standard Number: 6  
Draw nets for a variety of prisms, pyramids, cylinders and cones

Grade: Gr. 9
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Define the basic trigonometric ratios in right triangles: sine, cosine and tangent

Standard Number: 2  
Apply proportions and right triangle trigonometric ratios to solve problems involving missing lengths and angle sizes in similar figures

Vizualization and Geometric Models
Standard Number: 3  
Analyze two-dimensional figures in a coordinate plane; e.g., use slope and distance formulas to show that a quadrilateral is a parallelogram

Grade: Gr. 10
Geometry and Spatial Sense Standard
Characteristics and Properties

Standard Number: 1  
Formally define and explain key aspects of geometric figures
a   interior and exterior angles of polygons
b   segments related to triangles (median, altitude, midsegment
c   points of concurrency related to triangles (centroid, incenter, orthocenter, and circumcenter);
d   circles (radius, diameter, chord, circumference, major arc, minor arc, sector, segment, inscribed angle).
Standard Number: 2  
Recognize and explain the necessity for certain terms to remain undefined, such as point, line and plane

Standard Number: 3  
Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof
a   prove the Pythagorean Theorem
b   prove theorems involving triangle similarity and congruence
c   prove theorems involving properties of lines, angles, triangles and quadrilaterals
d   test a conjecture using basic constructions made with a compass and straightedge or technology

Spatial Relationships
Standard Number: 4  
Construct right triangles, equilateral triangles, parallelograms, trapezoids, rectangles, rhombuses, squares and kites, using compass and straightedge or dynamic geometry software

Standard Number: 5  
Construct congruent or similar figures using tools, such as compass, straightedge, and protractor or dynamic geometry software

Transformation and Symmetry
Standard Number: 6  
Identify the reflection and rotation symmetries of two- and three-dimensional figures

Standard Number: 7  
Perform reflections and rotations using compass and straightedge constructions and dynamic geometry software

Standard Number: 8  
Derive coordinate rules for translations, reflections and rotations of geometric figures in the coordinate plane

Standard Number: 9  
Show and describe the results of combinations of translations, reflections and rotations (compositions); e.g., perform compositions and specify the result of a composition as the outcome of a single motion, when applicable

Vizualization and Geometric Models
Standard Number: 10  
Solve problems involving chords, radii, and arcs within the same circle

Grade: Gr. 11
Geometry and Spatial Sense Standard
Spatial Relationships

Standard Number: 1  
Use polar coordinates to specify locations on a plane

Transformation and Symmetry
Standard Number: 2  
Represent translations using vectors

Standard Number: 3  
Describe multiplication of a vector and a scalar graphically and Algebraically, and apply to problem situations

Standard Number: 4  
Use trigonometric relationships to determine lengths and angle measures; i.e., Law of Sines and Law of Cosines

Vizualization and Geometric Models
Standard Number: 5  
Identify, sketch and classify the cross sections of three-dimensional objects

Grade: Gr. 12
Geometry and Spatial Sense Standard
Transformation and Symmetry

Standard Number: 1  
Use matrices to represent translations, reflections, rotations, dilations and their compositions

Standard Number: 2  
Derive and apply the basic trigonometric identities; i.e., angle addition, angle subtraction, and double angle

Vizualization and Geometric Models
Standard Number: 3  
Relate graphical and Algebraic representations of lines, simple curves and conic sections

Standard Number: 4  
Recognize and compare specific shapes and properties in multiple geometries; e.g., plane, spherical and hyperbolic