The Common Core in Maine

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Skills available for Maine high school math standards

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Common Core State Standards: Algebra 1 Common Core State Standards: Algebra 1
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9-12.HSN Number and Quantity

9-12.HSN-CN The Complex Number System
- 9-12.HSN-CN.A Perform arithmetic operations with complex numbers.
  - 9-12.HSN-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- 9-12.HSN-CN.B Represent complex numbers and their operations on the complex plane.
  - 9-12.HSN-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.
  - 9-12.HSN-CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.
  - 9-12.HSN-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-S.7)
    - Distance in the complex plane (PC-S.8)
9-12.HSN-VM Vector and Matrix Quantities
- 9-12.HSN-VM.A Represent and model with vector quantities.
  - 9-12.HSN-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).
  - 9-12.HSN-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)
  - 9-12.HSN-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.
- 9-12.HSN-VM.B Perform operations on vectors.
  - 9-12.HSN-VM.B.4 Add and subtract vectors.
    - 9-12.HSN-VM.B.4a 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.
    - 9-12.HSN-VM.B.4b 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-U.9)
    - 9-12.HSN-VM.B.4c 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.
  - 9-12.HSN-VM.B.5 Multiply a vector by a scalar.
    - 9-12.HSN-VM.B.5a 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-U.10)
      - Scalar multiples of three-dimensional vectors (PC-V.4)
    - 9-12.HSN-VM.B.5b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v is not equal to 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-U.11)
      - Determine the direction of a vector scalar multiple (PC-U.12)
- 9-12.HSN-VM.C Perform operations on matrices and use matrices in applications.
  - 9-12.HSN-VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
  - 9-12.HSN-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)
  - 9-12.HSN-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.
  - 9-12.HSN-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)
  - 9-12.HSN-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.
  - 9-12.HSN-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.
  - 9-12.HSN-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.

9-12.HSA Algebra

9-12.HSA-REI Reasoning with Equations and Inequalities
- 9-12.HSA-REI.C Solve systems of equations
  - 9-12.HSA-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 (A2-G.18)
    - Solve a system of equations using augmented matrices: word problems (A2-G.19)
  - 9-12.HSA-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).

9-12.HSF Functions

9-12.HSF-IF Interpreting Functions
- 9-12.HSF-IF.C Analyze functions using different representations
  - 9-12.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
    - 9-12.HSF-IF.C.7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
      - Rational functions: asymptotes and excluded values (PC-E.1)
9-12.HSF-BF Building Functions
- 9-12.HSF-BF.A Build a function that models a relationship between two quantities
  - 9-12.HSF-BF.A.1 Write a function that describes a relationship between two quantities.
    - 9-12.HSF-BF.A.1c Compose functions.
      - Composition of functions (PC-A.11)
- 9-12.HSF-BF.B Build new functions from existing functions
  - 9-12.HSF-BF.B.4 Find inverse functions.
    - 9-12.HSF-BF.B.4b Verify by composition that one function is the inverse of another.
      - Identify inverse functions (PC-A.12)
      - Check whether two rational functions are inverses (PC-E.3)
    - 9-12.HSF-BF.B.4c Read values of an inverse function from a graph or a table, given that the function has an inverse.
      - Find values of inverse functions from tables (PC-A.13)
      - Find values of inverse functions from graphs (PC-A.14)
    - 9-12.HSF-BF.B.4d Produce an invertible function from a non-invertible function by restricting the domain.
  - 9-12.HSF-BF.B.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
9-12.HSF-TF Trigonometric Functions
- 9-12.HSF-TF.A Extend the domain of trigonometric functions using the unit circle
  - 9-12.HSF-TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-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)
  - 9-12.HSF-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)
- 9-12.HSF-TF.B Model periodic phenomena with trigonometric functions
  - 9-12.HSF-TF.B.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.
    - Inverses of trigonometric functions (PC-M.9)
  - 9-12.HSF-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)
- 9-12.HSF-TF.C Prove and apply trigonometric identities
  - 9-12.HSF-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

9-12.HSG Geometry

9-12.HSG-GPE Expressing Geometric Properties with Equations
- 9-12.HSG-GPE.A Translate between the geometric description and the equation for a conic section
  - 9-12.HSG-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)
9-12.HSG-GMD Geometric Measurement and Dimension
- 9-12.HSG-GMD.A Explain volume formulas and use them to solve problems
  - 9-12.HSG-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

9-12.HSS Statistics and Probability

9-12.HSS-MD Using Probability to Make Decisions
- 9-12.HSS-MD.A Calculate expected values and use them to solve problems
  - 9-12.HSS-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)
  - 9-12.HSS-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)
  - 9-12.HSS-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)
  - 9-12.HSS-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.
- 9-12.HSS-MD.B Use probability to evaluate outcomes of decisions
  - 9-12.HSS-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
    - 9-12.HSS-MD.B.5a Find the expected payoff for a game of chance.
      - Expected values for a game of chance (PC-Y.8)
    - 9-12.HSS-MD.B.5b Evaluate and compare strategies on the basis of expected values.
      - Choose the better bet (PC-Y.9)

The Common Core in Maine

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9-12.HSN Number and Quantity

9-12.HSN-CN The Complex Number System

9-12.HSN-CN.A Perform arithmetic operations with complex numbers.

9-12.HSN-CN.A.3 Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

9-12.HSN-CN.B Represent complex numbers and their operations on the complex plane.

9-12.HSN-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.

9-12.HSN-CN.B.5 Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

9-12.HSN-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.

9-12.HSN-VM Vector and Matrix Quantities

9-12.HSN-VM.A Represent and model with vector quantities.

9-12.HSN-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).

9-12.HSN-VM.A.2 Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

9-12.HSN-VM.A.3 Solve problems involving velocity and other quantities that can be represented by vectors.

9-12.HSN-VM.B Perform operations on vectors.

9-12.HSN-VM.B.4 Add and subtract vectors.

9-12.HSN-VM.B.4a 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.

9-12.HSN-VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

9-12.HSN-VM.B.5 Multiply a vector by a scalar.

9-12.HSN-VM.B.5a 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).

9-12.HSN-VM.B.5b Compute the magnitude of a scalar multiple cv using ||cv|| = |c|·||v||. Compute the direction of cv knowing that when |c|v is not equal to 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

9-12.HSN-VM.C Perform operations on matrices and use matrices in applications.

9-12.HSN-VM.C.6 Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

9-12.HSN-VM.C.7 Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

9-12.HSN-VM.C.8 Add, subtract, and multiply matrices of appropriate dimensions.

9-12.HSN-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.

9-12.HSN-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.

9-12.HSN-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.

9-12.HSN-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.

9-12.HSA Algebra

9-12.HSA-REI Reasoning with Equations and Inequalities

9-12.HSA-REI.C Solve systems of equations

9-12.HSA-REI.C.8 Represent a system of linear equations as a single matrix equation in a vector variable.

9-12.HSA-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).

9-12.HSF Functions

9-12.HSF-IF Interpreting Functions

9-12.HSF-IF.C Analyze functions using different representations

9-12.HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

9-12.HSF-IF.C.7d Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

9-12.HSF-BF Building Functions

9-12.HSF-BF.A Build a function that models a relationship between two quantities

9-12.HSF-BF.A.1 Write a function that describes a relationship between two quantities.

9-12.HSF-BF.A.1c Compose functions.

9-12.HSF-BF.B Build new functions from existing functions

9-12.HSF-BF.B.4 Find inverse functions.

9-12.HSF-BF.B.4b Verify by composition that one function is the inverse of another.

9-12.HSF-BF.B.4c Read values of an inverse function from a graph or a table, given that the function has an inverse.

9-12.HSF-BF.B.4d Produce an invertible function from a non-invertible function by restricting the domain.

9-12.HSF-BF.B.5 Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

9-12.HSF-TF Trigonometric Functions

9-12.HSF-TF.A Extend the domain of trigonometric functions using the unit circle

9-12.HSF-TF.A.3 Use special triangles to determine geometrically the values of sine, cosine, tangent for pi/3, pi/4 and pi/6, and use the unit circle to express the values of sine, cosine, and tangent for pi-x, pi+x, and 2pi-x in terms of their values for x, where x is any real number.

9-12.HSF-TF.A.4 Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

9-12.HSF-TF.B Model periodic phenomena with trigonometric functions

9-12.HSF-TF.B.6 Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

9-12.HSF-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.

9-12.HSF-TF.C Prove and apply trigonometric identities

9-12.HSF-TF.C.9 Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

9-12.HSG Geometry

9-12.HSG-GPE Expressing Geometric Properties with Equations

9-12.HSG-GPE.A Translate between the geometric description and the equation for a conic section

9-12.HSG-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.

9-12.HSG-GMD Geometric Measurement and Dimension

9-12.HSG-GMD.A Explain volume formulas and use them to solve problems

9-12.HSG-GMD.A.2 Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

9-12.HSS Statistics and Probability

9-12.HSS-MD Using Probability to Make Decisions

9-12.HSS-MD.A Calculate expected values and use them to solve problems

9-12.HSS-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.

9-12.HSS-MD.A.2 Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

9-12.HSS-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.

9-12.HSS-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.

9-12.HSS-MD.B Use probability to evaluate outcomes of decisions

9-12.HSS-MD.B.5 Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

9-12.HSS-MD.B.5a Find the expected payoff for a game of chance.

9-12.HSS-MD.B.5b Evaluate and compare strategies on the basis of expected values.

9-12.HSN-VM.B.5a 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).