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Skills available for Massachusetts high school math standards

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Massachusetts Curriculum Frameworks: Advanced Quantitative Reasoning Massachusetts Curriculum Frameworks: Advanced Quantitative Reasoning
Massachusetts Curriculum Frameworks: Algebra I Massachusetts Curriculum Frameworks: Algebra I
Massachusetts Curriculum Frameworks: Algebra II Massachusetts Curriculum Frameworks: Algebra II
Massachusetts Curriculum Frameworks: All High School Pathways Massachusetts Curriculum Frameworks: All High School Pathways
Massachusetts Curriculum Frameworks: Geometry Massachusetts Curriculum Frameworks: Geometry
Massachusetts Curriculum Frameworks: Mathematics I Massachusetts Curriculum Frameworks: Mathematics I
Massachusetts Curriculum Frameworks: Mathematics II Massachusetts Curriculum Frameworks: Mathematics II
Massachusetts Curriculum Frameworks: Mathematics III Massachusetts Curriculum Frameworks: Mathematics III
Massachusetts Curriculum Frameworks: Precalculus Massachusetts Curriculum Frameworks: Precalculus

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N Number and Quantity

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-U.2)
    - Find the component form of a three-dimensional vector (PC-V.2)
  - N-VM.A.3 Solve problems involving velocity and other quantities that can be represented by 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-L.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-L.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.

A Algebra

A-APR Arithmetic with Polynomials and Rational Expressions
- 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.
A-REI Reasoning with Equations and Inequalities
- A-REI.C Solve systems of equations.
  - A-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.
    - Solve a system of equations using augmented matrices (PC-I.8)
    - Solve a system of equations using augmented matrices: word problems (PC-I.9)
  - A-REI.C.9 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

F Functions

F-TF Trigonometric Functions
- F-TF.A Extend the domain of trigonometric functions using the unit circle.
  - F-TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.
    - Find trigonometric ratios using right triangles (PC-M.5)
    - Find trigonometric ratios of special angles (PC-M.7)
  - F-TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
    - Symmetry and periodicity of trigonometric functions (PC-O.2)
- F-TF.B Model periodic phenomena with trigonometric functions.
  - F-TF.B.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
    - Solve trigonometric equations (PC-M.11)
- F-TF.C Prove and apply trigonometric identities.
  - F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

G Geometry

G-SRT Similarity, Right Triangles, and Trigonometry
- G-SRT.D Apply trigonometry to general triangles.
  - G-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
G-C Circles
- G-C.A Understand and apply theorems about circles.
  - G-C.A.4 Construct a tangent line from a point outside a given circle to the circle.
    - Tangent lines (PC)
    - Construct a tangent line to a circle (PC)
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.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-P.9)
    - Write equations of hyperbolas in standard form (PC-P.12)
    - G-GPE.A.3.a Use equations and graphs of conic sections to model real-world problems.
G-GMD Geometric Measurement and Dimension
- G-GMD.A Explain volume formulas and use them to solve problems.
  - G-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

S Statistics and Probability

S-CP Conditional Probability and the Rules of Probability
- S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
  - S-CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
  - S-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.
S-MD Using Probability to Make Decisions
- S-MD.A Calculate expected values and use them to solve problems.
  - S-MD.A.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
    - Write a discrete probability distribution (PC-Y.2)
    - Graph a discrete probability distribution (PC-Y.3)
  - S-MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
    - Expected values of random variables (PC-Y.4)
  - S-MD.A.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.
    - Write the probability distribution for a game of chance (PC-Y.7)
    - Expected values for a game of chance (PC-Y.8)
  - S-MD.A.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.
- S-MD.B Use probability to evaluate outcomes of decisions.
  - S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
    - S-MD.B.5.a Find the expected payoff for a game of chance.
      - Expected values for a game of chance (PC-Y.8)
    - S-MD.B.5.b Evaluate and compare strategies on the basis of expected values.
      - Choose the better bet (PC-Y.9)
  - S-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).
  - S-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, or pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).

Massachusetts

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N Number and Quantity

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.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.

A Algebra

A-APR Arithmetic with Polynomials and Rational Expressions

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.

A-REI Reasoning with Equations and Inequalities

A-REI.C Solve systems of equations.

A-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.

A-REI.C.9 Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

F Functions

F-TF Trigonometric Functions

F-TF.A Extend the domain of trigonometric functions using the unit circle.

F-TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number.

F-TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

F-TF.B Model periodic phenomena with trigonometric functions.

F-TF.B.7 Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

F-TF.C Prove and apply trigonometric identities.

F-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

G Geometry

G-SRT Similarity, Right Triangles, and Trigonometry

G-SRT.D Apply trigonometry to general triangles.

G-SRT.D.11 Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

G-C Circles

G-C.A Understand and apply theorems about circles.

G-C.A.4 Construct a tangent line from a point outside a given circle to the circle.

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.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.

G-GPE.A.3.a Use equations and graphs of conic sections to model real-world problems.

G-GMD Geometric Measurement and Dimension

G-GMD.A Explain volume formulas and use them to solve problems.

G-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

S Statistics and Probability

S-CP Conditional Probability and the Rules of Probability

S-CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.

S-CP.B.8 Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

S-CP.B.9 Use permutations and combinations to compute probabilities of compound events and solve problems.

S-MD Using Probability to Make Decisions

S-MD.A Calculate expected values and use them to solve problems.

S-MD.A.1 Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

S-MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

S-MD.A.3 Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

S-MD.A.4 Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

S-MD.B Use probability to evaluate outcomes of decisions.

S-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

S-MD.B.5.a Find the expected payoff for a game of chance.

S-MD.B.5.b Evaluate and compare strategies on the basis of expected values.

S-MD.B.6 Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).

S-MD.B.7 Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, or pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).

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.