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Skills available for Georgia high school math standards

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K-12 Mathematics Standards: Advanced Algebra (Algebra 2) K-12 Mathematics Standards: Advanced Algebra (Algebra 2)
K-12 Mathematics Standards: Algebra 1 K-12 Mathematics Standards: Algebra 1
K-12 Mathematics Standards: Calculus K-12 Mathematics Standards: Calculus
K-12 Mathematics Standards: Geometry K-12 Mathematics Standards: Geometry
K-12 Mathematics Standards: Mathematical Practices K-12 Mathematics Standards: Mathematical Practices
K-12 Mathematics Standards: Precalculus K-12 Mathematics Standards: Precalculus

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MM Mathematical Modeling

PC.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.
- PC.MM.1.1 Explain contextual, mathematical problems using a mathematical model.
- PC.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.
- PC.MM.1.3 Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.
- PC.MM.1.4 Use various mathematical representations and structures with this information to represent and solve real-life problems.

FGR Functional & Graphical Reasoning

Functions and their Characteristics
- PC.FGR.2 Analyze the behaviors of rational and piecewise functions to model contextual mathematical problems.
  - Analyze piecewise-defined functions using different representations.
    - PC.FGR.2.1 Graph piecewise-defined functions, including step functions and absolute value functions.
      - Graph an absolute value function (PC)
      - Graph piecewise-defined functions (PC)
    - PC.FGR.2.2 Describe characteristics by interpreting the algebraic form and graph of a piecewise-defined function.
    - PC.FGR.2.3 Represent the limit of a function using both the informal definition and the graphical interpretation in the context of piecewise-defined functions; interpret limits expressed in analytic notation.
  - Analyze rational functions using different representations.
    - PC.FGR.2.4 Divide polynomials using various methods.
    - PC.FGR.2.5 Graph rational functions and identify key characteristics.
      - Rational functions: asymptotes and excluded values (PC-G.1)
    - PC.FGR.2.6 Represent the behavior of a rational function using limit notation for vertical and horizontal asymptotes and end behavior.
    - PC.FGR.2.7 Represent the limit of a function using both the informal definition and the graphical interpretation in the context of rational functions; interpret limits expressed in analytic notation.
      - Find limits at vertical asymptotes using graphs (PC-HH.4)
    - PC.FGR.2.8 Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.
      - Solve rational equations (PC-G.2)
    - PC.FGR.2.9 Perform partial fraction decomposition of rational functions using non-repeated linear factors.
Trigonometric Relationships and Functions
- PC.FGR.3 Utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions.
  - Define and analyze trigonometric relationships.
    - PC.FGR.3.1 Use the concept of a radian as the ratio of the arc length to the radius of a circle to establish the existence of 2? radians in one revolution.
      - Convert between radians and degrees (PC-R.1)
      - Radians and arc length (PC-R.2)
    - PC.FGR.3.2 Utilize right triangles on the unit circle to determine the values of the six trigonometric ratios for ^π⁄₆, ^π⁄₄, and ^π⁄₃. Use reflections of the triangles as reference angles to establish known values in all four quadrants of the coordinate plane.
      - Quadrants (PC-R.3)
      - Coterminal and reference angles (PC-R.4)
    - PC.FGR.3.3 Define the six trigonometric ratios in terms of x , y , and r using the unit circle centered at the origin of the coordinate plane. Interpret radian measures of angles as a rotation both counterclockwise and clockwise around the unit circle.
    - PC.FGR.3.4 Derive the fundamental trigonometric identities.
    - PC.FGR.3.5 Determine the value(s) of trigonometric functions for a set of given conditions.
      - Find trigonometric ratios using the unit circle (PC-R.6)
      - Find trigonometric ratios using reference angles (PC-R.8)
  - Analyze trigonometric functions and their inverses.
    - PC.FGR.3.6 Graph and write equations of trigonometric functions using period, phase shift, and amplitude in modeling contexts.
    - PC.FGR.3.7 Classify the six trigonometric functions as even or odd and describe the symmetry.
      - Symmetry and periodicity of trigonometric functions (PC-T.2)
    - PC.FGR.3.8 Restrict the domain of a trigonometric function to create an invertible function and graph the inverse function. Evaluate inverse trigonometric expressions.
      - Inverses of trigonometric functions (PC-R.9)
      - Inverses of trigonometric functions using a calculator (PC-R.10)

AGR Algebraic & Geometric Reasoning

Trigonometric Identities and Equations
- PC.AGR.4 Manipulate, prove, and apply trigonometric identities and equations to solve contextual mathematical problems.
  - Verify trigonometric identities and solve trigonometric equations.
    - PC.AGR.4.1 Apply the fundamental trigonometric identities to simplify expressions and verify other identities.
      - Find trigonometric ratios using a Pythagorean or reciprocal identity (PC-T.3)
      - Find trigonometric ratios using multiple identities (PC-T.4)
    - PC.AGR.4.2 Use sum, difference, double-angle, and half-angle formulas for sine, cosine, and tangent to establish other identities and apply them to solve problems.
      - Trigonometric sum and difference identities (PC-T.5)
      - Solve trigonometric equations using sum and difference identities (PC-T.6)
    - PC.AGR.4.3 Solve trigonometric equations arising in modeling contexts.
      - Solve trigonometric equations (PC-R.11)
  - Apply trigonometry to general triangles.
    - PC.AGR.4.4 Prove and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.
    - PC.AGR.4.5 Determine the area of an oblique triangle.
      - Area of a triangle: sine formula (PC-R.18)
      - Area of a triangle: Heron's formula (PC-R.19)

GSR Geometric & Spatial Reasoning

Conic Sections and Polar Equations
- PC.GSR.5 Analyze the behaviors of conic sections and polar equations to model contextual mathematical problems.
  - Analyze conic sections using different representations.
    - PC.GSR.5.1 Identify and graph different conic sections given the equations in standard form.
    - PC.GSR.5.2 Identify different conic sections in general form and complete the square to convert the equation of a conic section into standard form.
      - Convert equations of conic sections from general to standard form (PC-U.13)
  - Extend trigonometry to the polar plane.
    - PC.GSR.5.3 Define polar coordinates and relate polar coordinates to Cartesian coordinates.
    - PC.GSR.5.4 Classify special polar equations and apply to contextual situations.
    - PC.GSR.5.5 Graph equations in the polar coordinate plane with and without the use of technology.
      - Match polar equations and graphs (PC-X.5)

AGR Algebraic & Graphical Reasoning

Vectors and Parametric Equations
- PC.AGR.6 Represent and model vector quantities to solve problems in contextual situations.
  - Perform operations with vectors in context.
    - PC.AGR.6.1 Represent vector quantities as directed line segments; represent magnitude and direction of vectors in component form using appropriate mathematical notation.
    - PC.AGR.6.2 Add and subtract vectors and multiply vectors by a scalar to find the resultant vector.
    - PC.AGR.6.3 Add and subtract vectors on a coordinate plane using different methods.
      - Graph a resultant vector using the triangle method (PC-Y.5)
      - Graph a resultant vector using the parallelogram method (PC-Y.6)
    - PC.AGR.6.4 Solve contextual vector problems, such as those involving velocity, force, and other quantities.
  - Model situations with parametric equations.
    - PC.AGR.6.5 Sketch the graph of a curve represented parametrically, indicating the direction of motion.
    - PC.AGR.6.6 Apply parametric equations to contextual problems.

PAR Patterning & Algebraic Reasoning

Sequences and Series
- PC.PAR.7 Demonstrate how sequences and series apply to mathematical models in real-life situations.
  - Analyze sequences using multiple representations.
    - PC.PAR.7.1 Demonstrate that sequences are functions whose domain is the set of natural numbers.
    - PC.PAR.7.2 Represent sequences graphically, numerically, and symbolically.
    - PC.PAR.7.3 Determine the limit of a sequence if it exists.
  - Analyze series using multiple representations.
    - PC.PAR.7.4 Demonstrate that a series is the sum of the sequence and represent series graphically, numerically, and symbolically.
      - Identify arithmetic and geometric series (PC-BB.1)
      - Introduction to sigma notation (PC-BB.2)
    - PC.PAR.7.5 Describe the behavior of a series in terms of the limit of its partial sums.
    - PC.PAR.7.6 Derive and use the sum formula of a finite geometric series to solve contextual problems to model real-life situations.
      - Find the sum of a finite geometric series (PC-BB.4)
    - PC.PAR.7.7 Derive and use the sum formula of an infinite geometric series to solve contextual problems to model real-life situations.
      - Convergent and divergent geometric series (PC-BB.9)
      - Find the value of an infinite geometric series (PC-BB.10)

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MM Mathematical Modeling

PC.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

PC.MM.1.1 Explain contextual, mathematical problems using a mathematical model.

PC.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.

PC.MM.1.3 Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

PC.MM.1.4 Use various mathematical representations and structures with this information to represent and solve real-life problems.

FGR Functional & Graphical Reasoning

Functions and their Characteristics

PC.FGR.2 Analyze the behaviors of rational and piecewise functions to model contextual mathematical problems.

Analyze piecewise-defined functions using different representations.

PC.FGR.2.1 Graph piecewise-defined functions, including step functions and absolute value functions.

PC.FGR.2.2 Describe characteristics by interpreting the algebraic form and graph of a piecewise-defined function.

PC.FGR.2.3 Represent the limit of a function using both the informal definition and the graphical interpretation in the context of piecewise-defined functions; interpret limits expressed in analytic notation.

Analyze rational functions using different representations.

PC.FGR.2.4 Divide polynomials using various methods.

PC.FGR.2.5 Graph rational functions and identify key characteristics.

PC.FGR.2.6 Represent the behavior of a rational function using limit notation for vertical and horizontal asymptotes and end behavior.

PC.FGR.2.7 Represent the limit of a function using both the informal definition and the graphical interpretation in the context of rational functions; interpret limits expressed in analytic notation.

PC.FGR.2.8 Solve simple rational equations in one variable, and give examples showing how extraneous solutions may arise.

PC.FGR.2.9 Perform partial fraction decomposition of rational functions using non-repeated linear factors.

Trigonometric Relationships and Functions

PC.FGR.3 Utilize trigonometric expressions to solve problems and model periodic phenomena with trigonometric functions.

Define and analyze trigonometric relationships.

PC.FGR.3.1 Use the concept of a radian as the ratio of the arc length to the radius of a circle to establish the existence of 2? radians in one revolution.

PC.FGR.3.2 Utilize right triangles on the unit circle to determine the values of the six trigonometric ratios for π⁄6, π⁄4, and π⁄3. Use reflections of the triangles as reference angles to establish known values in all four quadrants of the coordinate plane.

PC.FGR.3.3 Define the six trigonometric ratios in terms of x , y , and r using the unit circle centered at the origin of the coordinate plane. Interpret radian measures of angles as a rotation both counterclockwise and clockwise around the unit circle.

PC.FGR.3.4 Derive the fundamental trigonometric identities.

PC.FGR.3.5 Determine the value(s) of trigonometric functions for a set of given conditions.

Analyze trigonometric functions and their inverses.

PC.FGR.3.6 Graph and write equations of trigonometric functions using period, phase shift, and amplitude in modeling contexts.

PC.FGR.3.7 Classify the six trigonometric functions as even or odd and describe the symmetry.

PC.FGR.3.8 Restrict the domain of a trigonometric function to create an invertible function and graph the inverse function. Evaluate inverse trigonometric expressions.

AGR Algebraic & Geometric Reasoning

Trigonometric Identities and Equations

PC.AGR.4 Manipulate, prove, and apply trigonometric identities and equations to solve contextual mathematical problems.

Verify trigonometric identities and solve trigonometric equations.

PC.AGR.4.1 Apply the fundamental trigonometric identities to simplify expressions and verify other identities.

PC.AGR.4.2 Use sum, difference, double-angle, and half-angle formulas for sine, cosine, and tangent to establish other identities and apply them to solve problems.

PC.AGR.4.3 Solve trigonometric equations arising in modeling contexts.

Apply trigonometry to general triangles.

PC.AGR.4.4 Prove and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles.

PC.AGR.4.5 Determine the area of an oblique triangle.

GSR Geometric & Spatial Reasoning

Conic Sections and Polar Equations

PC.GSR.5 Analyze the behaviors of conic sections and polar equations to model contextual mathematical problems.

Analyze conic sections using different representations.

PC.GSR.5.1 Identify and graph different conic sections given the equations in standard form.

PC.GSR.5.2 Identify different conic sections in general form and complete the square to convert the equation of a conic section into standard form.

Extend trigonometry to the polar plane.

PC.GSR.5.3 Define polar coordinates and relate polar coordinates to Cartesian coordinates.

PC.GSR.5.4 Classify special polar equations and apply to contextual situations.

PC.GSR.5.5 Graph equations in the polar coordinate plane with and without the use of technology.

AGR Algebraic & Graphical Reasoning

Vectors and Parametric Equations

PC.AGR.6 Represent and model vector quantities to solve problems in contextual situations.

Perform operations with vectors in context.

PC.AGR.6.1 Represent vector quantities as directed line segments; represent magnitude and direction of vectors in component form using appropriate mathematical notation.

PC.AGR.6.2 Add and subtract vectors and multiply vectors by a scalar to find the resultant vector.

PC.AGR.6.3 Add and subtract vectors on a coordinate plane using different methods.

PC.AGR.6.4 Solve contextual vector problems, such as those involving velocity, force, and other quantities.

Model situations with parametric equations.

PC.AGR.6.5 Sketch the graph of a curve represented parametrically, indicating the direction of motion.

PC.AGR.6.6 Apply parametric equations to contextual problems.

PAR Patterning & Algebraic Reasoning

Sequences and Series

PC.PAR.7 Demonstrate how sequences and series apply to mathematical models in real-life situations.

Analyze sequences using multiple representations.

PC.PAR.7.1 Demonstrate that sequences are functions whose domain is the set of natural numbers.

PC.PAR.7.2 Represent sequences graphically, numerically, and symbolically.

PC.PAR.7.3 Determine the limit of a sequence if it exists.

Analyze series using multiple representations.

PC.PAR.7.4 Demonstrate that a series is the sum of the sequence and represent series graphically, numerically, and symbolically.

PC.PAR.7.5 Describe the behavior of a series in terms of the limit of its partial sums.

PC.PAR.7.6 Derive and use the sum formula of a finite geometric series to solve contextual problems to model real-life situations.

PC.PAR.7.7 Derive and use the sum formula of an infinite geometric series to solve contextual problems to model real-life situations.

PC.FGR.3.2 Utilize right triangles on the unit circle to determine the values of the six trigonometric ratios for ^π⁄₆, ^π⁄₄, and ^π⁄₃. Use reflections of the triangles as reference angles to establish known values in all four quadrants of the coordinate plane.