K-12 Mathematics Standards: Geometry
Skills available for Georgia Geometry standards
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MM
Mathematical Modeling
G.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.
G.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.
G.MM.1.1: Explain mathematically applicable problems using a mathematical model.
G.MM.1.1: Explain mathematically applicable problems using a mathematical model.
G.MM.1.2: Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.
G.MM.1.2: Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.
G.MM.1.3: Using abstract and quantitative reasoning, make decisions about information and data from a mathematically applicable situation.
G.MM.1.3: Using abstract and quantitative reasoning, make decisions about information and data from a mathematically applicable situation.
G.MM.1.4: Use various mathematical representations and structures with this information to represent and solve real-life problems.
G.MM.1.4: Use various mathematical representations and structures with this information to represent and solve real-life problems.
PAR
Patterning & Algebraic Reasoning
Polynomial expressions
G.PAR.2 Interpret the structure of and perform operations with polynomials within a geometric framework.
G.PAR.2 Interpret the structure of and perform operations with polynomials within a geometric framework.
G.PAR.2.1: Interpret polynomial expressions of varying degrees that represent a quantity in terms of its given geometric framework.
G.PAR.2.1: Interpret polynomial expressions of varying degrees that represent a quantity in terms of its given geometric framework.
G.PAR.2.2: Perform operations with polynomials and prove that polynomials form a system analogous to the integers in that they are closed under these operations.
G.PAR.2.2: Perform operations with polynomials and prove that polynomials form a system analogous to the integers in that they are closed under these operations.
G.PAR.2.3: Using algebraic reasoning, add, subtract, and multiply single variable polynomials.
G.PAR.2.3: Using algebraic reasoning, add, subtract, and multiply single variable polynomials.
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GSR
Geometric & Spatial Reasoning
Congruence
G.GSR.3 Experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena.
G.GSR.3 Experiment with transformations in the plane to develop precise definitions for translations, rotations, and reflections and use these to describe symmetries and congruence to model and explain real-life phenomena.
G.GSR.3.1: Use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations.
Symmetry
Rotations
Reflections
Translations
Congruence transformations
G.GSR.3.1: Use geometric reasoning and symmetries of regular polygons to develop definitions of rotations, reflections, and translations.
Symmetry
Rotations
Reflections
Translations
Congruence transformations
G.GSR.3.2: Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines.
G.GSR.3.2: Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines.
G.GSR.3.3: Use geometric descriptions of rigid motions to draw the transformed figures and to predict the effect on a given figure. Describe a sequence of transformations from one figure to another and use transformation properties to determine congruence.
Graph an image
Sequences of transformations
G.GSR.3.3: Use geometric descriptions of rigid motions to draw the transformed figures and to predict the effect on a given figure. Describe a sequence of transformations from one figure to another and use transformation properties to determine congruence.
Graph an image
Sequences of transformations
G.GSR.3.4: Explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. Use congruency criteria for triangles to solve problems and to prove relationships in geometric figures.
Congruent triangles
Proofs
Right triangles
G.GSR.3.4: Explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. Use congruency criteria for triangles to solve problems and to prove relationships in geometric figures.
Congruent triangles
Proofs
Right triangles
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Geometric foundations, constructions, and proof
G.GSR.4 Establish facts between angle relations and generate valid arguments to defend facts established. Prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena.
G.GSR.4 Establish facts between angle relations and generate valid arguments to defend facts established. Prove theorems and solve geometric problems involving lines and angles to model and explain real-life phenomena.
G.GSR.4.1: Use the undefined notions of point, line, line segment, plane, distance along a line segment, and distance around a circular arc to develop and use precise definitions and symbolic notations to prove theorems and solve geometric problems.
Lines and planes
Angles
Circles
G.GSR.4.1: Use the undefined notions of point, line, line segment, plane, distance along a line segment, and distance around a circular arc to develop and use precise definitions and symbolic notations to prove theorems and solve geometric problems.
Lines and planes
Angles
Circles
G.GSR.4.2: Classify quadrilaterals in the coordinate plane by proving simple geometric theorems algebraically.
G.GSR.4.2: Classify quadrilaterals in the coordinate plane by proving simple geometric theorems algebraically.
G.GSR.4.3: Make formal geometric constructions with a variety of tools and methods.
G.GSR.4.3: Make formal geometric constructions with a variety of tools and methods.
G.GSR.4.4: Prove and apply theorems about lines and angles to solve problems.
Angles
Parallel lines
Perpendicular bisectors
G.GSR.4.4: Prove and apply theorems about lines and angles to solve problems.
Angles
Parallel lines
Perpendicular bisectors
G.GSR.4.5: Use geometric reasoning to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
G.GSR.4.5: Use geometric reasoning to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.
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Similarity
G.GSR.5 Describe dilations in terms of center and scale factor and use these terms to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena.
G.GSR.5 Describe dilations in terms of center and scale factor and use these terms to describe properties of dilations; use the precise definition of a dilation to describe similarity and establish the criterion for triangles to be similar; use these terms, definitions, and criterion to prove similarity, model, and explain real-life phenomena.
G.GSR.5.1: Verify experimentally the properties of dilations.
G.GSR.5.1: Verify experimentally the properties of dilations.
G.GSR.5.2: Given two figures, use and apply the definition of similarity in terms of similarity transformations.
G.GSR.5.2: Given two figures, use and apply the definition of similarity in terms of similarity transformations.
G.GSR.5.3: Use the properties of similarity transformations to establish criterion for two triangles to be similar. Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.GSR.5.3: Use the properties of similarity transformations to establish criterion for two triangles to be similar. Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.GSR.5.4: Construct formal proofs to justify and apply theorems about triangles.
G.GSR.5.4: Construct formal proofs to justify and apply theorems about triangles.
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Right triangle trigonometry
G.GSR.6 Examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena.
G.GSR.6 Examine side ratios of similar triangles; use the relationship between right triangles to develop an understanding of sine, cosine, and tangent to solve mathematically applicable geometric problems and to model and explain real-life phenomena.
G.GSR.6.1: Explain 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.GSR.6.1: Explain 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.GSR.6.2: Explain and use the relationship between the sine and cosine of complementary angles.
G.GSR.6.2: Explain and use the relationship between the sine and cosine of complementary angles.
G.GSR.6.3: Use trigonometric ratios and the Pythagorean Theorem to solve for sides and angles of right triangles in applied problems.
Pythagorean Theorem
Trigonometric ratios
Solve a right triangle
G.GSR.6.3: Use trigonometric ratios and the Pythagorean Theorem to solve for sides and angles of right triangles in applied problems.
Pythagorean Theorem
Trigonometric ratios
Solve a right triangle
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Trigonometry and the unit circle
G.GSR.7 Explore the concept of a radian measure and special right triangles.
G.GSR.7 Explore the concept of a radian measure and special right triangles.
G.GSR.7.1: Explore and interpret a radian as the ratio of the arc length to the radius of a circle.
G.GSR.7.1: Explore and interpret a radian as the ratio of the arc length to the radius of a circle.
G.GSR.7.2: Explore and explain the relationship between radian measures and degree measures and convert fluently between degree and radian measures.
G.GSR.7.2: Explore and explain the relationship between radian measures and degree measures and convert fluently between degree and radian measures.
G.GSR.7.3: Use special right triangles on the unit circle to determine the values of sine, cosine, and tangent for 30° ( Π/6 ), 45° ( Π/4 ) and 60° ( Π/3 ) angle measures. Use reflections of triangles to determine reference angles and identify coordinate values in all four quadrants of the coordinate plane.
G.GSR.7.3: Use special right triangles on the unit circle to determine the values of sine, cosine, and tangent for 30° ( Π/6 ), 45° ( Π/4 ) and 60° ( Π/3 ) angle measures. Use reflections of triangles to determine reference angles and identify coordinate values in all four quadrants of the coordinate plane.
Circles
G.GSR.8 Examine and apply theorems involving circles; describe and derive arc length and area of a sector; and model and explain real-life frameworks involving circles.
G.GSR.8 Examine and apply theorems involving circles; describe and derive arc length and area of a sector; and model and explain real-life frameworks involving circles.
G.GSR.8.1: Identify and apply angle relationships formed by chords, tangents, secants and radii with circles.
Central and inscribed angles
Chords, secants, and tangents
G.GSR.8.1: Identify and apply angle relationships formed by chords, tangents, secants and radii with circles.
Central and inscribed angles
Chords, secants, and tangents
G.GSR.8.2: Using similarity, derive the fact that the length of the arc (arc length) intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. Solve mathematically applicable problems involving applications of arc length and area of sector.
G.GSR.8.2: Using similarity, derive the fact that the length of the arc (arc length) intercepted by an angle is proportional to the radius; derive the formula for the area of a sector. Solve mathematically applicable problems involving applications of arc length and area of sector.
G.GSR.8.3: Write and graph the equation of circles in standard form.
Equations
Graphs
G.GSR.8.3: Write and graph the equation of circles in standard form.
Equations
Graphs
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Equations and measurement
G.GSR.9 Develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve mathematically applicable problems involving volume; explore and visualize relationships between two dimensional and three-dimensional objects to model and explain real-life phenomena.
G.GSR.9 Develop informal arguments for geometric formulas using dissection arguments, limit arguments, and Cavalieri's principle; solve mathematically applicable problems involving volume; explore and visualize relationships between two dimensional and three-dimensional objects to model and explain real-life phenomena.
G.GSR.9.1: Use volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems including right and oblique solids.
G.GSR.9.1: Use volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems including right and oblique solids.
G.GSR.9.2: Use geometric shapes, their measures, and their properties to describe objects and approximate volumes.
G.GSR.9.2: Use geometric shapes, their measures, and their properties to describe objects and approximate volumes.
G.GSR.9.3: Apply concepts of density based on area and volume in modeling situations.
G.GSR.9.3: Apply concepts of density based on area and volume in modeling situations.
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PR
Probabilistic Reasoning
Compound events and expected values
G.PR.10 Solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions.
G.PR.10 Solve problems involving the probability of compound events to make informed decisions; interpret expected value and measures of variability to analyze probability distributions.
G.PR.10.1: Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events. Apply the Addition Rule conceptually, P(A or B)= P(A) + P(B) − P(A and B), and interpret the answers in context.
G.PR.10.1: Describe categories of events as subsets of a sample space using unions, intersections, or complements of other events. Apply the Addition Rule conceptually, P(A or B)= P(A) + P(B) − P(A and B), and interpret the answers in context.
G.PR.10.2: Apply and interpret the general Multiplication Rule conceptually to independent events of a sample space, P(A and B) = [P(A)] × [P(B|A)] = [P(B)] × [P(A|B)] using contingency tables or tree diagrams.
G.PR.10.2: Apply and interpret the general Multiplication Rule conceptually to independent events of a sample space, P(A and B) = [P(A)] × [P(B|A)] = [P(B)] × [P(A|B)] using contingency tables or tree diagrams.
G.PR.10.3: Use conditional probability to interpret risk in terms of decision-making and investigate questions such as those involving false positives or false negatives from screening tests.
G.PR.10.3: Use conditional probability to interpret risk in terms of decision-making and investigate questions such as those involving false positives or false negatives from screening tests.
G.PR.10.4: Define permutations and combinations and apply this understanding to compute probabilities of compound events and solve meaningful problems.
G.PR.10.4: Define permutations and combinations and apply this understanding to compute probabilities of compound events and solve meaningful problems.
G.PR.10.5: Interpret the probability distribution for a given random variable and interpret the expected value.
G.PR.10.5: Interpret the probability distribution for a given random variable and interpret the expected value.
G.PR.10.6: Develop a probability distribution for variables of interest using theoretical and empirical (observed) probabilities and calculate and interpret the expected value.
G.PR.10.6: Develop a probability distribution for variables of interest using theoretical and empirical (observed) probabilities and calculate and interpret the expected value.
G.PR.10.7: Calculate the expected value of a random variable and interpret it as the mean of a given probability distribution.
G.PR.10.7: Calculate the expected value of a random variable and interpret it as the mean of a given probability distribution.
G.PR.10.8: Compare the payoff values associated with the probability distribution for a random variable and make informed decisions based on expected value and measures of variability.
G.PR.10.8: Compare the payoff values associated with the probability distribution for a random variable and make informed decisions based on expected value and measures of variability.
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DSR
Data & Statistical Reasoning; Probabilistic Reasoning
Categorical data in two-way frequency tables, conditional probability
G.DSR.11 Examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena.
G.DSR.11 Examine real-life situations presented in a two-way frequency table to calculate probabilities, to model categorical data, and to explain real-life phenomena.
G.DSR.11.1: Construct and summarize categorical data for two categories in two-way frequency tables.
G.DSR.11.1: Construct and summarize categorical data for two categories in two-way frequency tables.
G.DSR.11.2: Use categorical data in two-way frequency tables to calculate and interpret probabilities based on the investigation.
G.DSR.11.2: Use categorical data in two-way frequency tables to calculate and interpret probabilities based on the investigation.
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