The Common Core in Maryland

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Skills available for Maryland high school math standards

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Maryland College and Career-Ready Frameworks: Algebra I Maryland College and Career-Ready Frameworks: Algebra I
Maryland College and Career-Ready Frameworks: Algebra II Maryland College and Career-Ready Frameworks: Algebra II
Maryland College and Career-Ready Frameworks: Geometry Maryland College and Career-Ready Frameworks: Geometry
Maryland College and Career-Ready Standards: Algebra 1 Maryland College and Career-Ready Standards: Algebra 1
Maryland College and Career-Ready Standards: Algebra 2 Maryland College and Career-Ready Standards: Algebra 2
Maryland College and Career-Ready Standards: All High School Pathways Maryland College and Career-Ready Standards: All High School Pathways
Maryland College and Career-Ready Standards: Geometry Maryland College and Career-Ready Standards: Geometry
Maryland College and Career-Ready Standards: Mathematical Practices Maryland College and Career-Ready Standards: Mathematical Practices
Maryland College and Career-Ready Standards: Mathematics I Maryland College and Career-Ready Standards: Mathematics I
Maryland College and Career-Ready Standards: Mathematics II Maryland College and Career-Ready Standards: Mathematics II
Maryland College and Career-Ready Standards: Mathematics III Maryland College and Career-Ready Standards: Mathematics III
Maryland College and Career-Ready Standards: Precalculus A Maryland College and Career-Ready Standards: Precalculus A
Maryland College and Career-Ready Standards: Precalculus B Maryland College and Career-Ready Standards: Precalculus B
Maryland College and Career-Ready Standards: Statistics Maryland College and Career-Ready Standards: Statistics

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Complex Numbers

N.CN The Complex Number System
- N.CN.A Perform arithmetic operations with complex numbers.
  - N.CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- N.CN.B Represent Complex Numbers and their operations on the complex plane.
  - N.CN.B.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
  - N.CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
  - N.CN.B.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
    - Midpoints in the complex plane (PC-W.7)
    - Distance in the complex plane (PC-W.8)

Vectors and Matrices

N.VM Vector and Matrix Quantities
- N.VM.A Represent and model with vector quantities.
  - N.VM.A .1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g. v, |v|, ||v||, v).
  - N.VM.A .2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
    - Find the component form of a vector (PC-Y.2)
    - Find the component form of a three-dimensional vector (PC-Z.2)
  - N.VM.A .3 Solve problems involving velocity and other quantities that can be represented by vectors.
- N.VM.B Perform operations on vectors.
  - N.VM.B.4 Add and subtract vectors.
    - N.VM.B.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
    - N.VM.B.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
      - Find the magnitude and direction of a vector sum (PC-Y.9)
    - N.VM.B.4.c Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.
      - Subtract vectors (PC-Y.8)
      - Add and subtract three-dimensional vectors (PC-Z.3)
  - N.VM.B.5 Multiply a vector by a scalar.
    - N.VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).
      - Multiply a vector by a scalar (PC-Y.10)
      - Scalar multiples of three-dimensional vectors (PC-Z.4)
    - N.VM.B.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c > 0).
      - Find the magnitude of a vector scalar multiple (PC-Y.11)
      - Determine the direction of a vector scalar multiple (PC-Y.12)
    - N.VM.B.5.c Determine the dot product of two vectors.
- N.VM.C Perform operations on matrices and use matrices in applications.
  - N.VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  - N.VM.C.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
    - Multiply a matrix by a scalar (PC-Q.4)
  - N.VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.
  - N.VM.C.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
    - Properties of matrices (PC-Q.8)
  - N.VM.C.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
  - N.VM.C.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
  - N.VM.C.12 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Sequences and Series

N.OA Operations and Algebraic Thinking
- N.OA.A Write and interpret numerical expressions.
  - N.OA.A.1 Use the notation for the factorial of a non-negative integer, n!, to evaluate expressions.
    - Factorials (PC)
A.SSE Seeing Structure in Expressions
- A.SSE.B Write expressions in equivalent forms to solve problems.
  - A.SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.
    - A.SSE.B.4.a Express the sums in a series using sigma notation.
      - Find the sum of a finite geometric series (PC-BB.4)
  - A.SSE.B.5 Determine the sum, if it exists, of an infinite geometric series.
    - Find the value of an infinite geometric series (PC-BB.10)
A.APR Arithmetic with Polynomials and Rational Expression
- A.APR.C Use polynomial identities to solve problems.
  - A.APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.
F.IF Interpreting Functions
- F.IF.A Understand the concept of a function and use function notation.
  - F.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.
- F.IF.C Analyze functions using different representations.
  - F.IF.C.10 Describe the behavior of a sequence.
F.BF Building Functions
- F.BF.C Build a function that models a relationship between two quantities.
  - F.BF.A.1 Write a function that describes a relationship between two quantities.
    - F.BF.A.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.
      - Find recursive and explicit formulas (PC-AA.5)
  - F.BF.A.2 Write sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
    - Convert a recursive formula to an explicit formula (PC-AA.6)
    - Convert an explicit formula to a recursive formula (PC-AA.7)

Limits

F.IF Interpreting Functions
- F.IF.A Understand the concept of a function and use function notation.
  - F.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
    - F.IF.A.2.a Understand the concept of limit of a function.
- F.IF.C Analyze functions using different representations.
  - F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
    - F.IF.C.8.c Interpret the behavior of the graph of a function using the concept of limits.
    - F.IF.C.8.d Estimate limits algebraically, numerically and graphically.

Conic Sections

A.SSE Seeing Structure in Expressions
- A.SSE.A Interpret the structure of expressions.
  - A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
    - A.SSE.A.2.a Analyze the structure of the general form of a second degree equation, Ax² + Bxy + Cy² + Dx + Ey + F = 0, to identify the conic section represented by the equation.
      - Using the discriminant (PC-D.10)
- A.SSE.B Write expressions in equivalent forms to solve problems.
  - A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
    - A.SSE.B.3.d Choose and produce an equivalent form of a second degree equation, Ax² + Cy² + Dx + Ey + F = 0, to reveal and explain properties of the conic section represented by the equation.
    - A.SSE.B.3.e Translate between the standard and general form, Ax² + Cy² + Dx + Ey + F = 0, of a second degree equation.
      - Convert equations of conic sections from general to standard form (PC-U.13)
G.GMD Geometric Measurement and Dimension
- G.GMD.B Visualize relationships between two-dimensional and three-dimensional objects.
  - G.GMD.B.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.B.4.a Identify the shapes of two-dimensional cross-sections of a right double cone.
G.GPE Expressing Geometric Properties with Equations
- G.GPE.A Translate between the geometric description and the equation for a conic section.
  - G.GPE.A.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.
  - G.GPE.A.2 Derive the equation of a parabola given a focus and directrix.
    - Write equations of parabolas in vertex form using properties (PC)
  - G.GPE.A.3 Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
    - Write equations of ellipses in standard form (PC-U.9)
    - Write equations of hyperbolas in standard form (PC-U.12)

Parametric Equations

P.IPE Interpreting Parametric Equations
- P.IPE.A Analyze parametric equations.
  - P.IPE.A.1 Sketch the curve defined by parametric equations.
  - P.IPE.A.2 Use parametric equations to model and solve motion problems.
P.CED Creating Equations
- P.CED.A Creating equations that describe numbers or relationships.
  - P.CED.A.1 Create a single equation, using rectangular coordinates, that is equivalent to a pair of parametric equations.
  - P.CED.A.2 Given a data set, create a parametric equation and a single equation using rectangular coordinates to fit the data.

Polar Coordinate System and Polar Equations

G.GPE Expressing Geometric Properties with Equations
- G.GPE.C Polar Coordinates.
  - G.GPE.C.8 Understand the relationship between polar coordinates and Cartesian coordinates.
    - Find the modulus and argument of a complex number (PC-X.1)
  - G.GPE.C.9 Convert between polar and rectangular coordinates.
  - G.GPE.C.10 Plot points on a polar coordinate grid.
- G.GPE.D Polar Equations.
  - G.GPE.D.11 Convert equations between polar and rectangular forms.
  - G.GPE.D.12 Graph polar equations by hand and using technology.
    - Match polar equations and graphs (PC-X.5)
  - G.GPE.D.13 Solve systems of polar equations.

The Common Core in Maryland

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Complex Numbers

N.CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.

N.CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

N.CN.B Represent Complex Numbers and their operations on the complex plane.

N.CN.B.4 Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

N.CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

N.CN.B.6 Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Vectors and Matrices

N.VM Vector and Matrix Quantities

N.VM.A Represent and model with vector quantities.

N.VM.A .1 Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g. v, |v|, ||v||, v).

N.VM.A .2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

N.VM.A .3 Solve problems involving velocity and other quantities that can be represented by vectors.

N.VM.B Perform operations on vectors.

N.VM.B.4 Add and subtract vectors.

N.VM.B.4.a Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

N.VM.B.4.b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

N.VM.B.5 Multiply a vector by a scalar.

N.VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

N.VM.B.5.b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c > 0).

N.VM.B.5.c Determine the dot product of two vectors.

N.VM.C Perform operations on matrices and use matrices in applications.

N.VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

N.VM.C.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

N.VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.

N.VM.C.9 Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

N.VM.C.10 Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

N.VM.C.11 Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

N.VM.C.12 Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Sequences and Series

N.OA Operations and Algebraic Thinking

N.OA.A Write and interpret numerical expressions.

N.OA.A.1 Use the notation for the factorial of a non-negative integer, n!, to evaluate expressions.

A.SSE Seeing Structure in Expressions

A.SSE.B Write expressions in equivalent forms to solve problems.

A.SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

A.SSE.B.4.a Express the sums in a series using sigma notation.

A.SSE.B.5 Determine the sum, if it exists, of an infinite geometric series.

A.APR Arithmetic with Polynomials and Rational Expression

A.APR.C Use polynomial identities to solve problems.

A.APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

F.IF Interpreting Functions

F.IF.A Understand the concept of a function and use function notation.

F.IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

F.IF.C Analyze functions using different representations.

F.IF.C.10 Describe the behavior of a sequence.

F.BF Building Functions

F.BF.C Build a function that models a relationship between two quantities.

F.BF.A.1 Write a function that describes a relationship between two quantities.

F.BF.A.1.a Determine an explicit expression, a recursive process, or steps for calculation from a context.

F.BF.A.2 Write sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Limits

F.IF Interpreting Functions

F.IF.A Understand the concept of a function and use function notation.

F.IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.A.2.a Understand the concept of limit of a function.

F.IF.C Analyze functions using different representations.

F.IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.C.8.c Interpret the behavior of the graph of a function using the concept of limits.

F.IF.C.8.d Estimate limits algebraically, numerically and graphically.

Conic Sections

A.SSE Seeing Structure in Expressions

A.SSE.A Interpret the structure of expressions.

A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.

A.SSE.A.2.a Analyze the structure of the general form of a second degree equation, Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, to identify the conic section represented by the equation.

A.SSE.B Write expressions in equivalent forms to solve problems.

A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

A.SSE.B.3.d Choose and produce an equivalent form of a second degree equation, Ax2 + Cy2 + Dx + Ey + F = 0, to reveal and explain properties of the conic section represented by the equation.

A.SSE.B.3.e Translate between the standard and general form, Ax2 + Cy2 + Dx + Ey + F = 0, of a second degree equation.

G.GMD Geometric Measurement and Dimension

G.GMD.B Visualize relationships between two-dimensional and three-dimensional objects.

G.GMD.B.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.B.4.a Identify the shapes of two-dimensional cross-sections of a right double cone.

G.GPE Expressing Geometric Properties with Equations

G.GPE.A Translate between the geometric description and the equation for a conic section.

G.GPE.A.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.

N.VM.B.5.a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(v_x, v_y) = (cv_x, cv_y).

A.APR.C.5 Know and apply the Binomial Theorem for the expansion of (x + y)ⁿ in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal's Triangle.

A.SSE.A.2.a Analyze the structure of the general form of a second degree equation, Ax² + Bxy + Cy² + Dx + Ey + F = 0, to identify the conic section represented by the equation.

A.SSE.B.3.d Choose and produce an equivalent form of a second degree equation, Ax² + Cy² + Dx + Ey + F = 0, to reveal and explain properties of the conic section represented by the equation.

A.SSE.B.3.e Translate between the standard and general form, Ax² + Cy² + Dx + Ey + F = 0, of a second degree equation.