Skill plan for the New York Regents Examinations - Geometry
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Conceptual Category
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Geometry
Congruence
Experiment with transformations in the plane (supporting cluster)
Experiment with transformations in the plane (supporting cluster)
G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Translations
Reflections
Rotations
Congruence transformations
Dilations
G-CO.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Translations
Reflections
Rotations
Congruence transformations
Dilations
G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-CO.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
G-CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Graph an image
Sequences of transformations
G-CO.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Graph an image
Sequences of transformations
Understand congruence in terms of rigid motions (major cluster)
Understand congruence in terms of rigid motions (major cluster)
G-CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.6: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G-CO.8: Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
Prove geometric theorems (major cluster)
Prove geometric theorems (major cluster)
G-CO.9: Prove theorems about lines and angles.
G-CO.9: Prove theorems about lines and angles.
G-CO.10: Prove theorems about triangles.
Congruent triangles
Isosceles triangles
Angles in triangles
Segments in triangles
G-CO.10: Prove theorems about triangles.
Congruent triangles
Isosceles triangles
Angles in triangles
Segments in triangles
G-CO.11: Prove theorems about parallelograms.
G-CO.11: Prove theorems about parallelograms.
Make geometric constructions (supporting cluster)
Make geometric constructions (supporting cluster)
G-CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Angles
Points, lines, and segments
Regular polygons
Triangles
Circles
G-CO.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Angles
Points, lines, and segments
Regular polygons
Triangles
Circles
G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G-CO.13: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Similarity, Right Triangles, & Trigonometry
Understand similarity in terms of similarity transformations (major cluster)
Understand similarity in terms of similarity transformations (major cluster)
G-SRT.1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G-SRT.1a: A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
G-SRT.1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.1b: The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-SRT.2: Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
G-SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
G-SRT.3: Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity (major cluster)
Prove theorems involving similarity (major cluster)
G-SRT.4: Prove theorems about triangles.
G-SRT.4: Prove theorems about triangles.
G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Congruent triangles
Proofs involving congruent triangles
Isosceles, equilateral, and right triangles
Similar triangles
Proofs involving similar triangles
G-SRT.5: Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Congruent triangles
Proofs involving congruent triangles
Isosceles, equilateral, and right triangles
Similar triangles
Proofs involving similar triangles
Define trigonometric ratios and solve problems involving right triangles (major cluster)
Define trigonometric ratios and solve problems involving right triangles (major cluster)
G-SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.6: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G-SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.7: Explain and use the relationship between the sine and cosine of complementary angles.
G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Pythagorean Theorem
Trigonometric ratios
Solve a right triangle
G-SRT.8: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Pythagorean Theorem
Trigonometric ratios
Solve a right triangle
Circles
Understand and apply theorems about circles (additional cluster)
Understand and apply theorems about circles (additional cluster)
G-C.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-C.1: Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G-C.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Translations
Reflections
Rotations
Congruence transformations
Dilations
G-C.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Translations
Reflections
Rotations
Congruence transformations
Dilations
G-C.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
G-C.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Find arc lengths and areas of sectors of circles (additional cluster)
Find arc lengths and areas of sectors of circles (additional cluster)
G-C.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Graph an image
Sequences of transformations
G-C.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Graph an image
Sequences of transformations
Expressing Geometric Properties with Equations
Translate between the geometric description and the equation for a conic section (additional cluster)
Translate between the geometric description and the equation for a conic section (additional cluster)
G-GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Complete the square
Write equations and graph circles
G-GPE.1: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
Complete the square
Write equations and graph circles
Use coordinates to prove simple geometric theorems algebraically (major cluster)
Use coordinates to prove simple geometric theorems algebraically (major cluster)
G-GPE.4: Use coordinates to prove simple geometric theorems algebraically.
G-GPE.4: Use coordinates to prove simple geometric theorems algebraically.
G-GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G-GPE.5: Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G-GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-GPE.6: Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
G-GPE.7: Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Geometric Measurement & Dimensions
Explain volume formulas and use them to solve problems (additional cluster)
Explain volume formulas and use them to solve problems (additional cluster)
G-GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
G-GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize relationships between two-dimensional and three-dimensional objects (additional cluster)
Visualize relationships between two-dimensional and three-dimensional objects (additional cluster)
G-GMD.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
G-GMD.4: Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Modeling with Geometry
Apply geometric concepts in modeling situations (major cluster)
Apply geometric concepts in modeling situations (major cluster)
G-MG.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
G-MG.1: Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
G-MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
G-MG.2: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
G-MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
G-MG.3: Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).
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